Ref refsubwiki https://subwiki.org/wiki/Main_Page MediaWiki 1.41.2 first-letter Media Special Talk User User talk Ref Ref talk File File talk MediaWiki MediaWiki talk Template Template talk Help Help talk Category Category talk Subwiki Subwiki talk Resource Resource talk Phenomenon Phenomenon talk Property Property talk Concept Concept talk smw/schema smw/schema talk Rule Rule talk Module Module talk 0 1 1 2008-05-12T15:43:32Z MediaWiki default 0 wikitext text/x-wiki <big>'''MediaWiki has been successfully installed.'''</big> Consult the [http://meta.wikimedia.org/wiki/Help:Contents User's Guide] for information on using the wiki software. == Getting started == * [http://www.mediawiki.org/wiki/Manual:Configuration_settings Configuration settings list] * [http://www.mediawiki.org/wiki/Manual:FAQ MediaWiki FAQ] * [http://lists.wikimedia.org/mailman/listinfo/mediawiki-announce MediaWiki release mailing list] 08fc850f2898611c250d639e30f69532b5a016f8 2 1 2008-05-12T16:02:52Z 59.162.2.142 0 /* Getting started */ wikitext text/x-wiki <big>'''MediaWiki has been successfully installed.'''</big> Consult the [http://meta.wikimedia.org/wiki/Help:Contents User's Guide] for information on using the wiki software. == Getting started == * [http://www.mediawiki.org/wiki/Manual:Configuration_settings Configuration settings list] * [http://www.mediawiki.org/wiki/Manual:FAQ MediaWiki FAQ] * [http://lists.wikimedia.org/mailman/listinfo/mediawiki-announce MediaWiki release mailing list] == Testing Math Mode == <math>\pi=\frac{3}{4} \sqrt{3}+24 \int_0^{1/4}{\sqrt{x-x^2}dx}</math> e0ed9b1826243d67b512049c061b85a937a12406 4 2 2008-05-13T00:55:19Z Vipul 2 Replacing page with 'Welcome to the Subject Wikis Reference Guide!' wikitext text/x-wiki Welcome to the Subject Wikis Reference Guide! d0cf94233f937d1887e0a9e282d19abdc1aac661 2 2 3 2008-05-13T00:48:17Z WikiSysop 1 Creating user page with biography of new user. wikitext text/x-wiki I'm Vipul. I'm the main person managing this wiki, as of now. c243611f7f5e5cdad945951e22d6c2d6343d8347 0 3 5 2008-05-13T01:13:05Z Vipul 2 New page: The term ''normal'' in mathematics is used in the following broad senses: * To denote something upright or perpendicular * To denote something that is as it ''should be''. In this sense, ... wikitext text/x-wiki The term ''normal'' in mathematics is used in the following broad senses: * To denote something upright or perpendicular * To denote something that is as it ''should be''. In this sense, normal means ''good'' or ''desirable'' rather than ''typical'' ===In group theory=== [[Groupprops:Normal subgroup|Normal subgroup]]: A subgroup of a group that occurs as the kernel of a homomorphism, or equivalently, such that every left coset and right coset are equal. Related terms: Normality (the property of a subgroup being normal), [[Groupprops:Normal core|normal core]] (the largest normal subgroup contained in a given subgroup), [[Groupprops:Normal closure|normal closure]] (the smallest normal subgroup containing a given subgroup), [[Groupprops:Normalizer|normalizer]] (the largest subgroup containing a given subgroup, in which it is normal). Term variations: [[Groupprops:Subnormal subgroup|subnormal subgroup]], [[Groupprops:Abnormal subgroup|abnormal subgroup]], [[Groupprops:Quasinormal subgroup|quasinormal subgroup]], and others. See [[Groupprops:Category:Variations of normality]], [[Groupprops:Category:Opposites of normality]], and [[Groupprops:Category:Analogues of normality]]. ===In topology=== [[Topospaces:Normal space|Normal space]]: A topological space is termed normal if all points are closed sets (the <math>T_1</math> assumption), and any two disjoint closed subsets can be separated by disjoint open subsets. In some definitions, the <math>T_1</math> assumption is skipped. Related terms: normality (the property of a topological space being normal) Term variations: [[Topospaces:Category:Variations of normality|Variations of normality]] offers a list. ===In Galois theory=== Normal field extension: A field extension is termed normal if the fixed field under the automorphism group of the extension, is precisely the base field. ===In commutative algebra=== [[Commalg:Normal domain|Normal domain]]: An integral domain is termed normal if it is integrally closed in its field of fractions. [[Commalg:Normal ring|Normal ring]]: A commutative unital ring is termed normal if it is integrally closed in its total quotient ring ===In differential geometry=== [[Diffgeom:Normal bundle|Normal bundle]] to a submanifold, or immersed manifold: The quotient of the tangent bundle to the whole manifold (restricted to the submanifold), by the tangent bundle to the submanifold, or immersed manifold. When the manifolds are Riemannian, the normal bundle can be viewed as the orthogonal complement to the tangent bundle of the submanifold, in the tangent bundle to the whole manifold. Vectors in the normal bundle are termed normal vectors. ===In normed vector spaces=== In general normed vector spaces, as well as in tangent spaces to Riemannian manifolds, ''normalize'' means to scale by a factor to make the norm equal to 1. ===In probability/statistics=== Normal distribution: A particular kind of distribution parametrized by mean and variance, also called the Gaussian distribution. Normal random variable: A random variable whose distribution is a normal distribution. 50d1ba7b8e48bd8caadfc4e77666b424d145e672 6 5 2008-05-13T01:14:58Z Vipul 2 /* In differential geometry */ wikitext text/x-wiki The term ''normal'' in mathematics is used in the following broad senses: * To denote something upright or perpendicular * To denote something that is as it ''should be''. In this sense, normal means ''good'' or ''desirable'' rather than ''typical'' ===In group theory=== [[Groupprops:Normal subgroup|Normal subgroup]]: A subgroup of a group that occurs as the kernel of a homomorphism, or equivalently, such that every left coset and right coset are equal. Related terms: Normality (the property of a subgroup being normal), [[Groupprops:Normal core|normal core]] (the largest normal subgroup contained in a given subgroup), [[Groupprops:Normal closure|normal closure]] (the smallest normal subgroup containing a given subgroup), [[Groupprops:Normalizer|normalizer]] (the largest subgroup containing a given subgroup, in which it is normal). Term variations: [[Groupprops:Subnormal subgroup|subnormal subgroup]], [[Groupprops:Abnormal subgroup|abnormal subgroup]], [[Groupprops:Quasinormal subgroup|quasinormal subgroup]], and others. See [[Groupprops:Category:Variations of normality]], [[Groupprops:Category:Opposites of normality]], and [[Groupprops:Category:Analogues of normality]]. ===In topology=== [[Topospaces:Normal space|Normal space]]: A topological space is termed normal if all points are closed sets (the <math>T_1</math> assumption), and any two disjoint closed subsets can be separated by disjoint open subsets. In some definitions, the <math>T_1</math> assumption is skipped. Related terms: normality (the property of a topological space being normal) Term variations: [[Topospaces:Category:Variations of normality|Variations of normality]] offers a list. ===In Galois theory=== Normal field extension: A field extension is termed normal if the fixed field under the automorphism group of the extension, is precisely the base field. ===In commutative algebra=== [[Commalg:Normal domain|Normal domain]]: An integral domain is termed normal if it is integrally closed in its field of fractions. [[Commalg:Normal ring|Normal ring]]: A commutative unital ring is termed normal if it is integrally closed in its total quotient ring ===In differential geometry=== [[Diffgeom:Normal bundle|Normal bundle]] to a submanifold, or immersed manifold: The quotient of the tangent bundle to the whole manifold (restricted to the submanifold), by the tangent bundle to the submanifold, or immersed manifold. When the manifolds are Riemannian, the normal bundle can be viewed as the orthogonal complement to the tangent bundle of the submanifold, in the tangent bundle to the whole manifold. [[Diffgeom:Normal coordinate system|Normal coordinate system]] refers to a particular kind of local coordinate system for a Riemannian manifold. Vectors in the normal bundle are termed normal vectors. ===In normed vector spaces=== In general normed vector spaces, as well as in tangent spaces to Riemannian manifolds, ''normalize'' means to scale by a factor to make the norm equal to 1. ===In probability/statistics=== Normal distribution: A particular kind of distribution parametrized by mean and variance, also called the Gaussian distribution. Normal random variable: A random variable whose distribution is a normal distribution. 8ec91adddedc2839c19dff736c893a67311a7a56 8 6 2008-05-13T01:20:02Z Vipul 2 wikitext text/x-wiki The term ''normal'' in mathematics is used in the following broad senses: * To denote something upright or perpendicular * To denote something that is as it ''should be''. In this sense, normal means ''good'' or ''desirable'' rather than ''typical'' ===In group theory=== [[Groupprops:Normal subgroup|Normal subgroup]]: A subgroup of a group that occurs as the kernel of a homomorphism, or equivalently, such that every left coset and right coset are equal. Related terms: Normality (the property of a subgroup being normal), [[Groupprops:Normal core|normal core]] (the largest normal subgroup contained in a given subgroup), [[Groupprops:Normal closure|normal closure]] (the smallest normal subgroup containing a given subgroup), [[Groupprops:Normalizer|normalizer]] (the largest subgroup containing a given subgroup, in which it is normal). Term variations: [[Groupprops:Subnormal subgroup|subnormal subgroup]], [[Groupprops:Abnormal subgroup|abnormal subgroup]], [[Groupprops:Quasinormal subgroup|quasinormal subgroup]], and others. See [[Groupprops:Category:Variations of normality]], [[Groupprops:Category:Opposites of normality]], and [[Groupprops:Category:Analogues of normality]]. ===In topology=== [[Topospaces:Normal space|Normal space]]: A topological space is termed normal if all points are closed sets (the <math>T_1</math> assumption), and any two disjoint closed subsets can be separated by disjoint open subsets. In some definitions, the <math>T_1</math> assumption is skipped. Related terms: normality (the property of a topological space being normal) Term variations: [[Topospaces:Category:Variations of normality|Variations of normality]] offers a list. ===In linear algebra=== [[Find link::Normal form]], also called canonical form, is a standard form in which to put a matrix (typically, upto conjugation). Two normal forms commonly used are the Jordan normal form and the rational normal form. [[Find link::Normal matrix]]: A matrix over the complex numbers, with the property that it commutes with its conjugate-transpose. ===In Galois theory=== [[Find link::Normal field extension]]: A field extension is termed normal if the fixed field under the automorphism group of the extension, is precisely the base field. ===In commutative algebra=== [[Commalg:Normal domain|Normal domain]]: An integral domain is termed normal if it is integrally closed in its field of fractions. [[Commalg:Normal ring|Normal ring]]: A commutative unital ring is termed normal if it is integrally closed in its total quotient ring ===In differential geometry=== [[Diffgeom:Normal bundle|Normal bundle]] to a submanifold, or immersed manifold: The quotient of the tangent bundle to the whole manifold (restricted to the submanifold), by the tangent bundle to the submanifold, or immersed manifold. When the manifolds are Riemannian, the normal bundle can be viewed as the orthogonal complement to the tangent bundle of the submanifold, in the tangent bundle to the whole manifold. [[Diffgeom:Normal coordinate system|Normal coordinate system]] refers to a particular kind of local coordinate system for a Riemannian manifold. Vectors in the normal bundle are termed normal vectors. ===In normed vector spaces=== In general normed vector spaces, as well as in tangent spaces to Riemannian manifolds, ''normalize'' means to scale by a factor to make the norm equal to 1. ===In probability/statistics=== Normal distribution: A particular kind of distribution parametrized by mean and variance, also called the Gaussian distribution. Normal random variable: A random variable whose distribution is a normal distribution. 9738e323c548fdca064861d6a6005a6712f8291b 9 8 2008-05-13T01:20:52Z Vipul 2 /* In probability/statistics */ wikitext text/x-wiki The term ''normal'' in mathematics is used in the following broad senses: * To denote something upright or perpendicular * To denote something that is as it ''should be''. In this sense, normal means ''good'' or ''desirable'' rather than ''typical'' ===In group theory=== [[Groupprops:Normal subgroup|Normal subgroup]]: A subgroup of a group that occurs as the kernel of a homomorphism, or equivalently, such that every left coset and right coset are equal. Related terms: Normality (the property of a subgroup being normal), [[Groupprops:Normal core|normal core]] (the largest normal subgroup contained in a given subgroup), [[Groupprops:Normal closure|normal closure]] (the smallest normal subgroup containing a given subgroup), [[Groupprops:Normalizer|normalizer]] (the largest subgroup containing a given subgroup, in which it is normal). Term variations: [[Groupprops:Subnormal subgroup|subnormal subgroup]], [[Groupprops:Abnormal subgroup|abnormal subgroup]], [[Groupprops:Quasinormal subgroup|quasinormal subgroup]], and others. See [[Groupprops:Category:Variations of normality]], [[Groupprops:Category:Opposites of normality]], and [[Groupprops:Category:Analogues of normality]]. ===In topology=== [[Topospaces:Normal space|Normal space]]: A topological space is termed normal if all points are closed sets (the <math>T_1</math> assumption), and any two disjoint closed subsets can be separated by disjoint open subsets. In some definitions, the <math>T_1</math> assumption is skipped. Related terms: normality (the property of a topological space being normal) Term variations: [[Topospaces:Category:Variations of normality|Variations of normality]] offers a list. ===In linear algebra=== [[Find link::Normal form]], also called canonical form, is a standard form in which to put a matrix (typically, upto conjugation). Two normal forms commonly used are the Jordan normal form and the rational normal form. [[Find link::Normal matrix]]: A matrix over the complex numbers, with the property that it commutes with its conjugate-transpose. ===In Galois theory=== [[Find link::Normal field extension]]: A field extension is termed normal if the fixed field under the automorphism group of the extension, is precisely the base field. ===In commutative algebra=== [[Commalg:Normal domain|Normal domain]]: An integral domain is termed normal if it is integrally closed in its field of fractions. [[Commalg:Normal ring|Normal ring]]: A commutative unital ring is termed normal if it is integrally closed in its total quotient ring ===In differential geometry=== [[Diffgeom:Normal bundle|Normal bundle]] to a submanifold, or immersed manifold: The quotient of the tangent bundle to the whole manifold (restricted to the submanifold), by the tangent bundle to the submanifold, or immersed manifold. When the manifolds are Riemannian, the normal bundle can be viewed as the orthogonal complement to the tangent bundle of the submanifold, in the tangent bundle to the whole manifold. [[Diffgeom:Normal coordinate system|Normal coordinate system]] refers to a particular kind of local coordinate system for a Riemannian manifold. Vectors in the normal bundle are termed normal vectors. ===In normed vector spaces=== In general normed vector spaces, as well as in tangent spaces to Riemannian manifolds, ''normalize'' means to scale by a factor to make the norm equal to 1. ===In probability/statistics=== [[Find link::Normal distribution]]: A particular kind of distribution parametrized by mean and variance, also called the Gaussian distribution. [[Find link::Normal random variable]]: A random variable whose distribution is a normal distribution. 48388cda05a345469609b81c58b4b93ffdc2159e 15 9 2008-05-13T01:52:29Z Vipul 2 wikitext text/x-wiki The term ''normal'' in mathematics is used in the following broad senses: * To denote something upright or perpendicular * To denote something that is as it ''should be''. In this sense, normal means ''good'' or ''desirable'' rather than ''typical'' ===In group theory=== [[Groupprops:Normal subgroup|Normal subgroup]]: A subgroup of a group that occurs as the kernel of a homomorphism, or equivalently, such that every left coset and right coset are equal. Related terms: Normality (the property of a subgroup being normal), [[Groupprops:Normal core|normal core]] (the largest normal subgroup contained in a given subgroup), [[Groupprops:Normal closure|normal closure]] (the smallest normal subgroup containing a given subgroup), [[Groupprops:Normalizer|normalizer]] (the largest subgroup containing a given subgroup, in which it is normal). Term variations: [[Groupprops:Subnormal subgroup|subnormal subgroup]], [[Groupprops:Abnormal subgroup|abnormal subgroup]], [[Groupprops:Quasinormal subgroup|quasinormal subgroup]], and others. See [[Groupprops:Category:Variations of normality]], [[Groupprops:Category:Opposites of normality]], and [[Groupprops:Category:Analogues of normality]]. ===In topology=== [[Topospaces:Normal space|Normal space]]: A topological space is termed normal if all points are closed sets (the <math>T_1</math> assumption), and any two disjoint closed subsets can be separated by disjoint open subsets. In some definitions, the <math>T_1</math> assumption is skipped. Related terms: normality (the property of a topological space being normal) Term variations: [[Topospaces:Category:Variations of normality|Variations of normality]] offers a list. ===In linear algebra=== [[Find link::Normal form]], also called canonical form, is a standard form in which to put a matrix (typically, upto conjugation). Two normal forms commonly used are the Jordan normal form and the rational normal form. [[Find link::Normal matrix]]: A matrix over the complex numbers, with the property that it commutes with its conjugate-transpose. ===In Galois theory=== [[Find link::Normal field extension]]: A field extension is termed normal if the fixed field under the automorphism group of the extension, is precisely the base field. ===In commutative algebra=== [[Commalg:Normal domain|Normal domain]]: An integral domain is termed normal if it is integrally closed in its field of fractions. [[Commalg:Normal ring|Normal ring]]: A commutative unital ring is termed normal if it is integrally closed in its total quotient ring ===In differential geometry=== Normal to a curve in the plane: The line perpendicular to the tangent to the curve at a point. [[Diffgeom:Normal bundle|Normal bundle]] to a submanifold, or immersed manifold: The quotient of the tangent bundle to the whole manifold (restricted to the submanifold), by the tangent bundle to the submanifold, or immersed manifold. When the manifolds are Riemannian, the normal bundle can be viewed as the orthogonal complement to the tangent bundle of the submanifold, in the tangent bundle to the whole manifold. [[Diffgeom:Normal coordinate system|Normal coordinate system]] refers to a particular kind of local coordinate system for a Riemannian manifold. Vectors in the normal bundle are termed normal vectors. ===In normed vector spaces=== In general normed vector spaces, as well as in tangent spaces to Riemannian manifolds, ''normalize'' means to scale by a factor to make the norm equal to 1. ===In probability/statistics=== [[Find link::Normal distribution]]: A particular kind of distribution parametrized by mean and variance, also called the Gaussian distribution. [[Find link::Normal random variable]]: A random variable whose distribution is a normal distribution. ===Others=== [[mathworld:NormalNumber|Normal number]] to base <math>b</math>: A real number that, when written in base <math>b</math>, has all digits occurring with equal probability. [[mathworld:AbsolutelyNormal|Absolutely normal number]]: A real number that is positive to base <math>b</math> for every integer <math>b > 1</math>. d83d31e361f56885704f6cae304294dcbc15c311 20 15 2008-05-14T16:31:15Z Vipul 2 wikitext text/x-wiki The term ''normal'' in mathematics is used in the following broad senses: * To denote something upright or perpendicular * To denote something that is as it ''should be''. In this sense, normal means ''good'' or ''desirable'' rather than ''typical'' ===In group theory=== {{:Normal subgroup}} ===In topology=== {{:Normal space}} ===In linear algebra=== [[Find link::Normal form]], also called canonical form, is a standard form in which to put a matrix (typically, upto conjugation). Two normal forms commonly used are the Jordan normal form and the rational normal form. [[Find link::Normal matrix]]: A matrix over the complex numbers, with the property that it commutes with its conjugate-transpose. ===In Galois theory=== [[Find link::Normal field extension]]: A field extension is termed normal if the fixed field under the automorphism group of the extension, is precisely the base field. ===In commutative algebra=== [[Commalg:Normal domain|Normal domain]]: An integral domain is termed normal if it is integrally closed in its field of fractions. [[Commalg:Normal ring|Normal ring]]: A commutative unital ring is termed normal if it is integrally closed in its total quotient ring ===In differential geometry=== Normal to a curve in the plane: The line perpendicular to the tangent to the curve at a point. [[Diffgeom:Normal bundle|Normal bundle]] to a submanifold, or immersed manifold: The quotient of the tangent bundle to the whole manifold (restricted to the submanifold), by the tangent bundle to the submanifold, or immersed manifold. When the manifolds are Riemannian, the normal bundle can be viewed as the orthogonal complement to the tangent bundle of the submanifold, in the tangent bundle to the whole manifold. [[Diffgeom:Normal coordinate system|Normal coordinate system]] refers to a particular kind of local coordinate system for a Riemannian manifold. Vectors in the normal bundle are termed normal vectors. ===In normed vector spaces=== In general normed vector spaces, as well as in tangent spaces to Riemannian manifolds, ''normalize'' means to scale by a factor to make the norm equal to 1. ===In probability/statistics=== [[Find link::Normal distribution]]: A particular kind of distribution parametrized by mean and variance, also called the Gaussian distribution. [[Find link::Normal random variable]]: A random variable whose distribution is a normal distribution. ===Others=== [[mathworld:NormalNumber|Normal number]] to base <math>b</math>: A real number that, when written in base <math>b</math>, has all digits occurring with equal probability. [[mathworld:AbsolutelyNormal|Absolutely normal number]]: A real number that is positive to base <math>b</math> for every integer <math>b > 1</math>. 65722a4d2b010209b64d2dbdf59240561221be11 0 4 7 2008-05-13T01:17:31Z Vipul 2 New page: ===In classical mechanics=== * Normal force refers to the component of contact force between two bodies, that acts in the direction perpendicular to the plane of contact. It manifests a t... wikitext text/x-wiki ===In classical mechanics=== * Normal force refers to the component of contact force between two bodies, that acts in the direction perpendicular to the plane of contact. It manifests a tendency of the bodies to avoid moving into each other, and acts outwards on both bodies. 5c003436da7efd6dc0d860d0f3a63ddbfacc80ea 13 7 2008-05-13T01:35:46Z Vipul 2 wikitext text/x-wiki ===In classical mechanics=== * [[Find link::Normal force]]: The component of contact force between two bodies, that acts in the direction perpendicular to the plane of contact. It manifests a tendency of the bodies to avoid moving into each other, and acts outwards on both bodies. 19151acfd939cafe03bf9dd7fad8acfa07e0fd15 24 13 2008-05-16T13:40:49Z Vipul 2 wikitext text/x-wiki ===In classical mechanics=== {{:Normal force}} dd43b5674ef0e1031b030e550abe9129846980ba 0 5 10 2008-05-13T01:23:26Z Vipul 2 New page: ===In stoichiometry=== [[Find link::Normal solution]]: An aqueous solution that contains one gram-equivalent of the active reagent in 1 liter of solution. [[Find link::Normality]] of an ... wikitext text/x-wiki ===In stoichiometry=== [[Find link::Normal solution]]: An aqueous solution that contains one gram-equivalent of the active reagent in 1 liter of solution. [[Find link::Normality]] of an aqueous solution: Number of gram equivalents of the active reagent contained in 1 liter of solution. c54bb3d78b322b9d3e403ffdf91994636e90df4d 11 10 2008-05-13T01:30:49Z Vipul 2 wikitext text/x-wiki ===In organic chemistry nomenclature=== [[Find link::Normal alkane]]: A straight-chain alkane. For instance, normal pentane is the straight-chain pentane. Also written by prefix <math>n</math>, for instance, <math>n</math>-pentane. The term is used more generally for organic compounds, as a prefix denoting a straight-chain structure, with functional groups at one end. ===In stoichiometry=== [[Find link::Normal solution]]: An aqueous solution that contains one gram-equivalent of the active reagent in 1 liter of solution. [[Find link::Normality]] of an aqueous solution: Number of gram equivalents of the active reagent contained in 1 liter of solution. 821ae8e6b663c60be9ab3f424044461a3e5ed50a 30 11 2008-05-16T13:52:10Z Vipul 2 wikitext text/x-wiki ===In organic chemistry nomenclature=== {{:Normal alkane}} ===In stoichiometry=== {{:Normality of aqueous solution}} 173731fc34a03ebe281e175a0bb4120f5749bdf2 0 6 12 2008-05-13T01:35:01Z Vipul 2 New page: Main form: ''normal'', adjective Related forms: ''normality'' (how normal something is), ''normalcy'', ''normalize'' (make normal) Typical use: * Something typical, expected, or standar... wikitext text/x-wiki Main form: ''normal'', adjective Related forms: ''normality'' (how normal something is), ''normalcy'', ''normalize'' (make normal) Typical use: * Something typical, expected, or standard * Something good or desirable, or something one optimistically hopes for * Upright or perpendicular ==Mathematics== {{:Normal (mathematics)}} ==Physics== {{:Normal (physics)}} ==Chemistry== {{:Normal (chemistry)}} f40b846b8e51c791e1330d19d084700593a6504f 0 7 14 2008-05-13T01:45:06Z Vipul 2 New page: ===In group theory=== [[Groupprops:Perfect group|Perfect group]]: A group that equals its own commutator subgroup. ===In topology=== [[Topospaces:Perfect space|Perfect space]]: A topolo... wikitext text/x-wiki ===In group theory=== [[Groupprops:Perfect group|Perfect group]]: A group that equals its own commutator subgroup. ===In topology=== [[Topospaces:Perfect space|Perfect space]]: A topological space where every point is closed, and is an intersection of countably many open subsets containing it. [[Topospaces:Perfectly normal space|Perfectly normal space]]: A normal space where every closed subset is an intersection of countably many open subsets containing it. Perfect set: A set in a metric space that has no isolated points. ===In number theory=== [[Find link::Perfect power]], for instance, perfect square or perfect cube: A perfect <math>n^{th}</math> power is an integer that occurs as the <math>n^{th}</math> power of an integer. [[Find link::Perfect number]]: A natural number that equals the sum of all its proper divisors. ===In field theory=== [[Find link::Perfect field]]: A field that either has characteristic zero, or has <math>p</math> and <math>x \mapsto x^p</math> is a surjective map. ===In graph theory=== [[Find link::Perfect graph]]: A graph with the property that for every induced subgraph, the chromatic number equals the clique number. [[Find link::Perfect matching]]: A matching in a bipartite graph such that every element on one side gets matched to exactly one element on the other side. ce6a8d2750797d5adc261f5055bccc74594f6ca0 36 14 2008-05-16T14:15:26Z Vipul 2 wikitext text/x-wiki ===In group theory=== [[Groupprops:Perfect group|Perfect group]]: A group that equals its own commutator subgroup. ===In topology=== [[Topospaces:Perfect space|Perfect space]]: A topological space where every point is closed, and is an intersection of countably many open subsets containing it. [[Topospaces:Perfectly normal space|Perfectly normal space]]: A normal space where every closed subset is an intersection of countably many open subsets containing it. Perfect set: A set in a metric space that has no isolated points. ===In number theory=== [[Find link::Perfect power]], for instance, perfect square or perfect cube: A perfect <math>n^{th}</math> power is an integer that occurs as the <math>n^{th}</math> power of an integer. [[Find link::Perfect number]]: A natural number that equals the sum of all its proper divisors. ===In field theory=== [[Find link::Perfect field]]: A field that either has characteristic zero, or has <math>p</math> and <math>x \mapsto x^p</math> is a surjective map. ===In graph theory=== [[Find link::Perfect graph]]: A graph with the property that for every induced subgraph, the chromatic number equals the clique number. [[Find link::Perfect matching]]: A matching in a bipartite graph such that every element on one side gets matched to exactly one element on the other side. ===In measure theory=== {{:Perfect measure}} a517c1552d9e1e34ad7cb501ac7ad785c52967dd 40 36 2008-05-16T14:23:00Z Vipul 2 wikitext text/x-wiki ===In group theory=== {{:Perfect group}} ===In topology=== {{:Perfect space}} [[Topospaces:Perfectly normal space|Perfectly normal space]]: A normal space where every closed subset is an intersection of countably many open subsets containing it. Perfect set: A set in a metric space that has no isolated points. ===In number theory=== [[Find link::Perfect power]], for instance, perfect square or perfect cube: A perfect <math>n^{th}</math> power is an integer that occurs as the <math>n^{th}</math> power of an integer. [[Find link::Perfect number]]: A natural number that equals the sum of all its proper divisors. ===In field theory=== [[Find link::Perfect field]]: A field that either has characteristic zero, or has <math>p</math> and <math>x \mapsto x^p</math> is a surjective map. ===In graph theory=== [[Find link::Perfect graph]]: A graph with the property that for every induced subgraph, the chromatic number equals the clique number. [[Find link::Perfect matching]]: A matching in a bipartite graph such that every element on one side gets matched to exactly one element on the other side. ===In measure theory=== {{:Perfect measure}} dd4509cd1b2360c400a3cbc44b2b3442fa4c308f 0 8 16 2008-05-13T02:02:05Z Vipul 2 New page: The term ''characteristic'' in mathematics typically stands for: * Something specific, unique, intrinsic or invariant to the situation * Something that completely describes the given situ... wikitext text/x-wiki The term ''characteristic'' in mathematics typically stands for: * Something specific, unique, intrinsic or invariant to the situation * Something that completely describes the given situation Related terms are ''invariant''. ===In commutative algebra/field theory=== [[Commalg:Characteristic of a ring|Characteristic of a ring]]: The smallest positive integer <math>n</math> such that <math>n.1 = 0</math>. If no such positive integer exists, then the characteristic is said to be zero. A field either has characteristic zero or has characteristic equal to a prime number. ===In group theory=== [[Groupprops:Characteristic subgroup|Characteristic subgroup]]: A subgroup that is invariant (or, gets mapped to itself) under any automorphism of the group. ===In algebraic topology/differential geometry=== [[Diffgeom:Characteristic class|Characteristic class]]: A natural transformation from the vector bundle functor to the cohomology functor on a manifold; in other words, it assigns to every vector bundle over a manifold, a cohomology class, such that a certain naturality diagram commutes. ===In measure theory/analysis=== [[Find link::Characteristic function]] of a set, also called its indicator function, is a function that takes value 1 on the set and 0 outside it. ===In probability/statistics=== ===Others=== [[Find link::Characteristic series]] a571e75a27290038c8fcc61b2d80cea13af57739 0 9 17 2008-05-14T16:20:14Z Vipul 2 New page: <noinclude>[[Status::Basic definition|]][[Topic::Group theory]][[Primary wiki::Groupprops]]</noinclude> '''Normal subgroup''': A subgroup of a group that occurs as the kernel of a homomorp... wikitext text/x-wiki <noinclude>[[Status::Basic definition|:Basic definition]][[Topic::Group theory]][[Primary wiki::Groupprops]]</noinclude> '''Normal subgroup''': A subgroup of a group that occurs as the kernel of a homomorphism, or equivalently, such that every left coset and right coset are equal. Related terms: Normality (the property of a subgroup being normal), [[Groupprops:Normal core|normal core]] (the largest normal subgroup contained in a given subgroup), [[Groupprops:Normal closure|normal closure]] (the smallest normal subgroup containing a given subgroup), [[Groupprops:Normalizer|normalizer]] (the largest subgroup containing a given subgroup, in which it is normal). Term variations: [[Groupprops:Subnormal subgroup|subnormal subgroup]], [[Groupprops:Abnormal subgroup|abnormal subgroup]], [[Groupprops:Quasinormal subgroup|quasinormal subgroup]], and others. See [[Groupprops:Category:Variations of normality]], [[Groupprops:Category:Opposites of normality]], and [[Groupprops:Category:Analogues of normality]]. Primary subject wiki entry: [[Groupprops:Normal subgroup]] Other subject wiki entries: [[Diffgeom:Normal subgroup]] Also located at: [[Wikipedia:Normal subgroup]], [[Planetmath:NormalSubgroup]], [[Mathworld:NormalSubgroup]], [[Sor:N/n067690]], [[Citizendium::Normal subgroup]] da4ef51964d70130abea59c6ac439bc6e72753a5 18 17 2008-05-14T16:23:53Z Vipul 2 wikitext text/x-wiki <noinclude>[[Status::Basic definition| ]][[Topic::Group theory| ]][[Primary wiki::Groupprops| ]]</noinclude> '''Normal subgroup''': A subgroup of a group that occurs as the kernel of a homomorphism, or equivalently, such that every left coset and right coset are equal. Related terms: Normality (the property of a subgroup being normal), [[Groupprops:Normal core|normal core]] (the largest normal subgroup contained in a given subgroup), [[Groupprops:Normal closure|normal closure]] (the smallest normal subgroup containing a given subgroup), [[Groupprops:Normalizer|normalizer]] (the largest subgroup containing a given subgroup, in which it is normal). Term variations: [[Groupprops:Subnormal subgroup|subnormal subgroup]], [[Groupprops:Abnormal subgroup|abnormal subgroup]], [[Groupprops:Quasinormal subgroup|quasinormal subgroup]], and others. See [[Groupprops:Category:Variations of normality]], [[Groupprops:Category:Opposites of normality]], and [[Groupprops:Category:Analogues of normality]]. Primary subject wiki entry: [[Groupprops:Normal subgroup]] Other subject wiki entries: [[Diffgeom:Normal subgroup]] Also located at: [[Wikipedia:Normal subgroup]], [[Planetmath:NormalSubgroup]], [[Mathworld:NormalSubgroup]], [[Sor:N/n067690]], [[Citizendium:Normal subgroup]] 9be896e1ae11497dcc9c8f0140f91996ddb71d30 21 18 2008-05-16T13:32:47Z Vipul 2 wikitext text/x-wiki <noinclude>[[Status::Basic definition| ]][[Topic::Group theory| ]][[Primary wiki::Groupprops| ]]</noinclude> '''Normal subgroup''': A subgroup of a group that occurs as the kernel of a homomorphism, or equivalently, such that every left coset and right coset are equal. Related terms: Normality (the property of a subgroup being normal), [[Groupprops:Normal core|normal core]] (the largest normal subgroup contained in a given subgroup), [[Groupprops:Normal closure|normal closure]] (the smallest normal subgroup containing a given subgroup), [[Groupprops:Normalizer|normalizer]] (the largest subgroup containing a given subgroup, in which it is normal). Term variations: [[Groupprops:Subnormal subgroup|subnormal subgroup]], [[Groupprops:Abnormal subgroup|abnormal subgroup]], [[Groupprops:Quasinormal subgroup|quasinormal subgroup]], and others. See [[Groupprops:Category:Variations of normality]], [[Groupprops:Category:Opposites of normality]], and [[Groupprops:Category:Analogues of normality]]. Primary subject wiki entry: [[Groupprops:Normal subgroup]] Other subject wiki entries: [[Diffgeom:Normal subgroup]] Also located at: [[Wikipedia:Normal subgroup]], [[Planetmath:NormalSubgroup]], [[Mathworld:NormalSubgroup]], [[Sor:N/n067690|Springer Online Reference Works]], [[Citizendium:Normal subgroup]] 453e45d5888f5e8b00fba7944d94195aef032e5b 0 10 19 2008-05-14T16:28:56Z Vipul 2 New page: <noinclude>[[Status::Basic definition| ]][[Topic::Point-set topology| ]][[Primary wiki::Topospaces| ]]</noinclude> '''Normal space''': A topological space is termed normal if all points ar... wikitext text/x-wiki <noinclude>[[Status::Basic definition| ]][[Topic::Point-set topology| ]][[Primary wiki::Topospaces| ]]</noinclude> '''Normal space''': A topological space is termed normal if all points are closed sets (the <math>T_1</math> assumption), and any two disjoint closed subsets can be separated by disjoint open subsets. In some definitions, the <math>T_1</math> assumption is skipped. Related terms: normality (the property of a topological space being normal) Term variations: [[Topospaces:Category:Variations of normality|Variations of normality]] offers a list. Primary subject wiki entry: [[Topospaces:Normal space]] Also located at: [[Wikipedia:Normal space]], [[Planetmath:NormalSpace]], [[Mathworld:NormalSpace]] 398a278c5ab949e92fd7f7677bea87bcd4f8fce5 0 11 22 2008-05-16T13:37:17Z Vipul 2 New page: <noinclude>[[Status::Basic definition| ]][[Primary wiki::Groupprops| ]][[Topic::Group theory| ]]</noinclude> '''Characteristic subgroup''': A subgroup of a group that is invariant under a... wikitext text/x-wiki <noinclude>[[Status::Basic definition| ]][[Primary wiki::Groupprops| ]][[Topic::Group theory| ]]</noinclude> '''Characteristic subgroup''': A subgroup of a group that is invariant under all automorphisms of the group. Primary subject wiki entry: [[Groupprops:Characteristic subgroup]] Also located at: [[Wikipedia:Characteristic subgroup]], [[Planetmath:CharacteristicSubgroup]], [[Sor:C/c021740|Springer Online Reference Works:Characteristic subgroup]] f4d8ea2d096f19f0fc8c5780fa7875db5b1af7e4 0 12 23 2008-05-16T13:40:44Z Vipul 2 New page: <noinclude>[[Status::Basic definition| ]] [[Topic::Classical mechanics| ]]</noinclude> '''Normal force''': The component of contact force between two bodies, that acts in the direction pe... wikitext text/x-wiki <noinclude>[[Status::Basic definition| ]] [[Topic::Classical mechanics| ]]</noinclude> '''Normal force''': The component of contact force between two bodies, that acts in the direction perpendicular to the plane of contact. It manifests a tendency of the bodies to avoid moving into each other, and acts outwards on both bodies. No relevant subject wiki entry. Also located at: [[Wikipedia:Normal force]], [[Citizendium:Normal force]] 4d716ab82be5c7906460c94dc090b8bd5abf8f63 0 13 25 2008-05-16T13:47:23Z Vipul 2 New page: <noinclude>[[Topic::Organic chemistry]]</noinclude> '''Normal alkane'''A straight-chain alkane. For instance, normal pentane is the straight-chain pentane. Also written by prefix <math>n</... wikitext text/x-wiki <noinclude>[[Topic::Organic chemistry]]</noinclude> '''Normal alkane'''A straight-chain alkane. For instance, normal pentane is the straight-chain pentane. Also written by prefix <math>n</math>, for instance, <math>n</math>-pentane. The prefix ''normal'' is used more generally for organic compounds, as a prefix denoting a straight-chain structure, with functional groups at one end. 284beef90e4bbb5cd716426c6fbb4d2d4d729444 26 25 2008-05-16T13:48:41Z Vipul 2 wikitext text/x-wiki <noinclude>[[Topic::Organic chemistry| ]]</noinclude> '''Normal alkane'''A straight-chain alkane. For instance, normal pentane is the straight-chain pentane. Also written by prefix <math>n</math>, for instance, <math>n</math>-pentane. The prefix ''normal'' is used more generally for organic compounds, as a prefix denoting a straight-chain structure, with functional groups at one end. 4260dc1a68ac28b8bac86fc81d90052b693d4075 27 26 2008-05-16T13:49:26Z Vipul 2 wikitext text/x-wiki <noinclude>[[Topic::Organic chemistry| ]]</noinclude> '''Normal alkane''': A straight-chain alkane. For instance, normal pentane is the straight-chain pentane. Also written by prefix <math>n</math>, for instance, <math>n</math>-pentane. The prefix ''normal'' is used more generally for organic compounds, as a prefix denoting a straight-chain structure, with functional groups at one end. 6c9fb1e1d6926b3ca0ba7dece09a42baf427171c 0 14 28 2008-05-16T13:51:12Z Vipul 2 New page: <noinclude>[[Topic:Stoichiometry| ]]</noinclude> '''Normality of aqueous solution''': Number of gram equivalents of active reagent per liter (litre) of solution. For instance, an aqueous ... wikitext text/x-wiki <noinclude>[[Topic:Stoichiometry| ]]</noinclude> '''Normality of aqueous solution''': Number of gram equivalents of active reagent per liter (litre) of solution. For instance, an aqueous solution with 5 gram equivalents per solution is termed <math>5N</math>. 6b6f1d0ea0b1a2337bd6adf63ba4451bb3dc1d8a 29 28 2008-05-16T13:51:40Z Vipul 2 wikitext text/x-wiki <noinclude>[[Topic::Stoichiometry| ]]</noinclude> '''Normality of aqueous solution''': Number of gram equivalents of active reagent per liter (litre) of solution. For instance, an aqueous solution with 5 gram equivalents per solution is termed <math>5N</math>. 5dd17f55c10f43855b0874a53b91efc4ea6207a6 0 15 31 2008-05-16T13:57:07Z Vipul 2 New page: In economics, ''perfect'' is typically used in the sense of ''ideal'', or ''as good as it can get''. {{:Perfect information}} {{:Perfect competition}} wikitext text/x-wiki In economics, ''perfect'' is typically used in the sense of ''ideal'', or ''as good as it can get''. {{:Perfect information}} {{:Perfect competition}} 3a52325a2c4c9bcfbb32adf37a5ad7b60df3519b 0 16 32 2008-05-16T14:01:43Z Vipul 2 New page: <noinclude>[[Topic::Rational economics| ]][[Topic::Game theory| ]]</noinclude> '''Perfect information''' (also called '''complete information'''): Perfect information refers to a situatio... wikitext text/x-wiki <noinclude>[[Topic::Rational economics| ]][[Topic::Game theory| ]]</noinclude> '''Perfect information''' (also called '''complete information'''): Perfect information refers to a situation in a game where, at any given time, every player has complete information about the game. Equivalently, there is no existing piece of information that can be given to a player to make that player play better. The term is used in economics to describe a situation where all people in an economic transaction or market, have complete information. No related subject wiki entry. Also located at: [[Wikipedia:Perfect information]] 3516de9b05725f1daec44509106f174e7f2e29a2 0 17 33 2008-05-16T14:07:29Z Vipul 2 New page: <noinclude>[[Topic::Economics| ]]</noinclude> ''Perfect competition''' (also called '''pure competition'''): A market form where no buyer or seller can perceptibly influence the price of ... wikitext text/x-wiki <noinclude>[[Topic::Economics| ]]</noinclude> ''Perfect competition''' (also called '''pure competition'''): A market form where no buyer or seller can perceptibly influence the price of the good. It usually occurs when there is a large pool of buyers, and a large number of competing sellers, for the same good. Also located at: [[Wikipedia:Perfect competition]], [http://www.britannica.com/eb/article-34156/monopoly-and-competition Britannica:Perfect competition] 3dc72b68f4b35a82aaa348eadfcea65e55ae5c6a 34 33 2008-05-16T14:07:45Z Vipul 2 wikitext text/x-wiki <noinclude>[[Topic::Economics| ]]</noinclude> '''Perfect competition''' (also called '''pure competition'''): A market form where no buyer or seller can perceptibly influence the price of the good. It usually occurs when there is a large pool of buyers, and a large number of competing sellers, for the same good. Also located at: [[Wikipedia:Perfect competition]], [http://www.britannica.com/eb/article-34156/monopoly-and-competition Britannica:Perfect competition] 0a0c79950415c0a0d28fe8ca9e2bcb026876faac 0 18 35 2008-05-16T14:11:21Z Vipul 2 New page: <noinclude>[[Topic::Electomagnetism| ]]</noinclude> '''Perfect conductor''': An electrical conductor with zero resistivity. Certain materials become perfect conductors below a certain tem... wikitext text/x-wiki <noinclude>[[Topic::Electomagnetism| ]]</noinclude> '''Perfect conductor''': An electrical conductor with zero resistivity. Certain materials become perfect conductors below a certain temperature. A perfect conductor that also exhibits properties like the Meissner effect is called a [[superconductor]]. All known perfect conductors are superconductors. No subject wiki entry. Also located at: [[Wikipedia:Perfect conductor]] 7f3965a5ef0b00b6a76a23edf8d3f24dc96222da 38 35 2008-05-16T14:20:09Z Vipul 2 wikitext text/x-wiki <noinclude>[[Topic::Electromagnetism| ]]</noinclude> '''Perfect conductor''': An electrical conductor with zero resistivity. Certain materials become perfect conductors below a certain temperature. A perfect conductor that also exhibits properties like the Meissner effect is called a [[superconductor]]. All known perfect conductors are superconductors. No subject wiki entry. Also located at: [[Wikipedia:Perfect conductor]] cb917c82bde192a42212bf82638a939294f64617 0 19 37 2008-05-16T14:19:25Z Vipul 2 New page: <noinclude>[[Status::Semi-basic definition| ]][[Topic:Point-set topology| ]][[Primary wiki::Topospaces]]</noinclude> '''Perfect space''': A topological space where every point is closed, ... wikitext text/x-wiki <noinclude>[[Status::Semi-basic definition| ]][[Topic:Point-set topology| ]][[Primary wiki::Topospaces]]</noinclude> '''Perfect space''': A topological space where every point is closed, and is an intersection of countably many open subsets containing it. Main subject wiki entry: [[Topospaces:Perfect space]] 7502d0ab6e5f24c38a81e3395201315c17cb808a 0 20 39 2008-05-16T14:22:31Z Vipul 2 New page: <noinclude>[[Status::Semi-basic definition| ]][[Topic::Group theory| ]][[Primary wiki::Groupprops| ]]</noinclude> '''Perfect group''': A group that equals its own commutator subgroup (i.e... wikitext text/x-wiki <noinclude>[[Status::Semi-basic definition| ]][[Topic::Group theory| ]][[Primary wiki::Groupprops| ]]</noinclude> '''Perfect group''': A group that equals its own commutator subgroup (i.e. derived subgroup). Main subject wiki entry: [[Groupprops:Perfect group]] Also located at: [[Wikipedia:Perfect group]], [[Mathworld:PerfectGroup]], [[Planetmath:PerfectGroup]] c6cccccda274bca70bb15b6c5fbe16a7642e472b 8 21 41 2008-05-17T19:37:09Z Vipul 2 New page: [[Main Page|'''The Subject Wikis Reference Guide (in preparation)''']] wikitext text/x-wiki [[Main Page|'''The Subject Wikis Reference Guide (in preparation)''']] d0e86c4b003d12a343a72a11949c387f92e00c28 106 22 42 2008-05-17T19:46:03Z Vipul 2 New page: * [[Groupprops:Main Page|Groupprops]]: The Group Properties Wiki (currently, around 1200 pages). Main topic is group theory * [[Topospaces:Main Page|Topospaces]]: The Topology Wiki (curren... wikitext text/x-wiki * [[Groupprops:Main Page|Groupprops]]: The Group Properties Wiki (currently, around 1200 pages). Main topic is group theory * [[Topospaces:Main Page|Topospaces]]: The Topology Wiki (currently, around 400 pages). Main topics are point-set topology and algebraic topology * [[Commalg:Main Page|Commalg]]: The Commutative Algebra Wiki (currently, around 300 pages). Main topic is commutative algebra, with some algebraic geometry. * [[Diffgeom:Main Page|Diffgeom]]: The Differential Geometry Wiki (to be set up) * [[Measure:Main Page|Measure]]: The Measure Theory Wiki (to be set up) * [[Noncommutative:Main Page|Noncommutative]]: The Noncommutative Algebra Wiki (to be set up) * [[Companal:Main Page|Companal]]: The Complex Analysis Wiki (to be set up) * [[Cattheory:Main Page|Cattheory]]: The Category Theory Wiki (to be set up) d03496c3fa04b99da43d20af7210a71e88dc84a1 12 23 43 2008-05-17T19:48:23Z Vipul 2 New page: This is a help page, for term lookup. Are you looking for the meaning of a particular term? The Subject Wikis Reference Guide can help you to look up definitions of terms, as well as all ... wikitext text/x-wiki This is a help page, for term lookup. Are you looking for the meaning of a particular term? The Subject Wikis Reference Guide can help you to look up definitions of terms, as well as all kinds of information about the term you're looking for. ==A quick explanation of how we're organized== The subwiki.org domain contains a number of subject-specific wikis. Each subject-specific wiki has articles developed and organized from the viewpoint of a specific subject or discipline. As of now, all the subject-specific wikis are in mathematical subjects. Here's a list of all the subject-specific wikis. {{:Ref:Subwiki list}} The Subject Wikis Reference Guide is designed to let you reach, just from the name of a term, the appropriate subject wikis that contain that term. 39ee452b2f067422756aead50f65d7b82381e55d 44 43 2008-05-17T20:11:26Z Vipul 2 wikitext text/x-wiki This is a help page, for term lookup. Are you looking for the meaning of a particular term? The Subject Wikis Reference Guide can help you to look up definitions of terms, as well as all kinds of information about the term you're looking for. ==A quick explanation of how we're organized== The subwiki.org domain contains a number of subject-specific wikis. Each subject-specific wiki has articles developed and organized from the viewpoint of a specific subject or discipline. As of now, all the subject-specific wikis are in mathematical subjects. Here's a list of all the subject-specific wikis. {{:Ref:Subwiki list}} The Subject Wikis Reference Guide is designed to let you reach, just from the name of a term, the appropriate subject wikis that contain that term. ==Lookup when the term name is known== ===When the full term name is known=== Any term which occurs in one or more of the subject-specific wikis (and sometimes, a term that doesn't) should have a reference guide page. The reference guide page for a term has the same name as the term. Thus, for instance, to get the reference guide page about normal subgroup, you'll need to type the following URL: {{fullurl:Normal subgroup}} Alternatively, you can type normal subgroup (without quotes) in the search bar and press Enter (or the Go key). This will directly take you to the page titled "Normal subgroup". ===When part of the term name is known=== Suppose you don't know the full name of a term, but know that it is called "perfect something". Alternatively, suppose you're interested in all the possible uses of the word "perfect" in terminology in different subjects. Then, try searching for whatever part of the term you ''do'' know. For instance, in this case, you'll try the page: {{fullurl:Perfect}} or search for perfect in the search bar. If you're interested in the use of the term in a particular subject (for instance, economics), then type the prefix or partial term, followed by the subject in parentheses. For instance, to see all the uses of the term "perfect" in economics, try: {{fullurl:Perfect (economics)}} The generic page on Perfect includes the content from the pages on Perfect in each broad subject area, as sections. Further, the page on perfect (economics) includes content from the guide pages for each of the specific terms that have "perfect" in them. ==Proceeding from the guide page=== The Subject Wikis Reference guide page to a particular term gives the following: * A quick (and not necessarily comprehensive) definition, and common style of usage * A list of related terms and ideas. * A link to the primary subject wiki entry on the term (if such an entry exists). For instance, if you're looking for a term like "Hausdorff space", the Reference Guide page on it will tell you that the primary subject wiki entry is at [[Topospaces:Hausdorff space]]. * Links to entries on the term in other subject wikis. * Links to entries on the term in other encyclopedia-like web resources. Textbook and journal references are maintained in the subject wiki entries and are ''not'' to be found on the reference guide page. f9383a1682964a14cf5acbabdce8f6d441f683e0 0 24 45 2008-05-17T20:16:54Z Vipul 2 New page: The term ''group'' loosely refers to a collection of many similar items that behave in a cohesive manner. In particular subjects, the usage tends to emphasize the cohesiveness and group id... wikitext text/x-wiki The term ''group'' loosely refers to a collection of many similar items that behave in a cohesive manner. In particular subjects, the usage tends to emphasize the cohesiveness and group identity. ==In mathematics== {{:Group (mathematics)}} ==In chemistry== {{:Group (chemistry)}} ==In sociology== {{:Group (sociology)}} 213839c4d1509809b13da4c7763b30e7bf01c9f5 0 25 46 2008-05-17T20:32:55Z Vipul 2 New page: <noinclude>[[Status::Basic definition| ]][[Topic::Group theory| ]][[Primary wiki::Groupprops| ]]</noinclude> '''Group''': A set equipped with a binary operation (called multiplication) th... wikitext text/x-wiki <noinclude>[[Status::Basic definition| ]][[Topic::Group theory| ]][[Primary wiki::Groupprops| ]]</noinclude> '''Group''': A set equipped with a binary operation (called multiplication) that is associative, has an identity element (also known as neutral element or multiplicative unit) and has left and right inverses for every element. Groups occur throughout mathematics, and are the underlying structures for rings, fields, vector spaces, modules, function spaces, and other structures that arise naturally in algebra, analysis and topology. Primary subject wiki entry: [[Groupprops:Group]] Also located at: [[Wikipedia:Group (mathematics)]], [[Mathworld:Group]], [[Planetmath:Group]], [[Citizendium:Group]] f2e7fcf9035b8dc2371bf439c58e686557073c78 49 46 2008-05-18T00:01:38Z Vipul 2 wikitext text/x-wiki <noinclude>[[Status::Basic definition| ]][[Topic::Group theory| ]][[Primary wiki::Groupprops| ]]</noinclude> '''Group''': A set equipped with a binary operation (called multiplication) that is associative, has an identity element (also known as neutral element or multiplicative unit) and has left and right inverses for every element. Groups occur throughout mathematics, and are the underlying structures for rings, fields, vector spaces, modules, function spaces, and other structures that arise naturally in algebra, analysis and topology. Primary subject wiki entry: [[Groupprops:Group]] Related terms: [[Subgroup]], [[quotient group]], [[homomorphism of groups]], [[normal subgroup]] Survey articles related to this subject wiki entry: [[Groupprops:History of groups]], [[Groupprops:Understanding the definition of a group]], [[Groupprops:Verifying the group axioms]] Categories related to group: [[Groupprops:Category:Variations of group]] lists variations of the notion of group, [[Groupprops:Category:Particular groups]] lists some particular groups (upto isomorphism), [[Groupprops:Category:Group properties]] lists properties that can be evaluated for a group, [[Groupprops:Category:Views of the collection of groups]] gives different viewpoints for studying the collection of all groups Also located at: [[Wikipedia:Group (mathematics)]], [[Mathworld:Group]], [[Planetmath:Group]], [[Citizendium:Group]] 15f03c166fc5b748156f0176e3d3482af8ba37f1 50 49 2008-05-18T00:04:11Z Vipul 2 wikitext text/x-wiki <noinclude>[[Status::Basic definition| ]][[Topic::Group theory| ]][[Primary wiki::Groupprops| ]]</noinclude> '''Group''': A set equipped with a binary operation (called multiplication) that is associative, has an identity element (also known as neutral element or multiplicative unit) and has left and right inverses for every element. Groups occur throughout mathematics, and are the underlying structures for rings, fields, vector spaces, modules, function spaces, and other structures that arise naturally in algebra, analysis and topology. Related terms: [[subgroup]], [[quotient group]], [[homomorphism of groups]], [[normal subgroup]] Primary subject wiki entry: [[Groupprops:Group]] Survey articles related to this subject wiki entry: [[Groupprops:History of groups]], [[Groupprops:Understanding the definition of a group]], [[Groupprops:Verifying the group axioms]], [[Groupprops:Manipulating equations in groups]], [[Groupprops:Groups as symmetry]] Guided tours related to this subject wiki entry: [[Groupprops:Groupprops:Guided tour for beginners]] Categories related to group: [[Groupprops:Category:Variations of group]] lists variations of the notion of group, [[Groupprops:Category:Particular groups]] lists some particular groups (upto isomorphism), [[Groupprops:Category:Group properties]] lists properties that can be evaluated for a group, [[Groupprops:Category:Views of the collection of groups]] gives different viewpoints for studying the collection of all groups Also located at: [[Wikipedia:Group (mathematics)]], [[Mathworld:Group]], [[Planetmath:Group]], [[Citizendium:Group]] caac9877783a3a7594961267d286f547aabcc45f 0 26 47 2008-05-17T20:33:49Z Vipul 2 New page: {{:Periodic table group}} {{:Functional group}} wikitext text/x-wiki {{:Periodic table group}} {{:Functional group}} f23f379bcdbc9e555d856041f776303565e86b6a 0 27 48 2008-05-17T20:37:12Z Vipul 2 New page: <noinclude>[[Topic::Chemistry| ]]</noinclude> A ''group'' or ''family'' in the periodic table refers to the collection of elements in a column (vertical) in the periodic table. These elem... wikitext text/x-wiki <noinclude>[[Topic::Chemistry| ]]</noinclude> A ''group'' or ''family'' in the periodic table refers to the collection of elements in a column (vertical) in the periodic table. These elements usually share similar characteristics, because of similar electron configurations. For instance, the noble gases form Group Zero of the periodic table, and they share the attribute of being highly unreactive due to a stable electronic configuration. No subject wiki entry. Also located at: [[Wikipedia:Periodic table group| ]] 5405b8313f530f1fc42ba32e8917b6261df8d330 0 27 51 48 2008-05-18T00:05:09Z Vipul 2 wikitext text/x-wiki <noinclude>[[Topic::Chemistry| ]]</noinclude> '''Periodic table group''': A ''group'' or ''family'' in the periodic table refers to the collection of elements in a column (vertical) in the periodic table. These elements usually share similar characteristics, because of similar electron configurations. For instance, the noble gases form Group Zero of the periodic table, and they share the attribute of being highly unreactive due to a stable electronic configuration. No subject wiki entry. Also located at: [[Wikipedia:Periodic table group]] 130b88dd634e14c87040698bb39afc1a17d45bf5 0 28 52 2008-05-18T00:10:48Z Vipul 2 New page: <noinclude>[[Topic::Organic chemistry| ]]</noinclude> '''Functional group''': A collection of atoms in a molecule of an organic compound (with a specific pattern of bonds between them) th... wikitext text/x-wiki <noinclude>[[Topic::Organic chemistry| ]]</noinclude> '''Functional group''': A collection of atoms in a molecule of an organic compound (with a specific pattern of bonds between them) that is responsible for making the organic compound undergo certain characteristic reactions. It may also influence the reactivity of the molecule as a whole. No subject wiki entry Also located at: [[Wikipedia:Functional group]], [http://www.britannica.com/eb/article-9035655/functional-group Britannica:Functional group] c97c698625ec125847ea2163eff71da8ff5cd889 0 9 53 21 2008-05-22T18:16:35Z Vipul 2 wikitext text/x-wiki <noinclude>[[Status::Basic definition| ]][[Topic::Group theory| ]][[Primary wiki::Groupprops| ]]</noinclude> '''Normal subgroup''': A subgroup of a group that occurs as the kernel of a homomorphism, or equivalently, such that every left coset and right coset are equal. Related terms: Normality (the property of a subgroup being normal), [[Groupprops:Normal core|normal core]] (the largest normal subgroup contained in a given subgroup), [[Groupprops:Normal closure|normal closure]] (the smallest normal subgroup containing a given subgroup), [[Groupprops:Normalizer|normalizer]] (the largest subgroup containing a given subgroup, in which it is normal), [[Groupprops:Normal automorphism|normal automorphism]] (an automorphism that preserves every normal subgroup) Term variations: [[Groupprops:Subnormal subgroup|subnormal subgroup]], [[Groupprops:Abnormal subgroup|abnormal subgroup]], [[Groupprops:Quasinormal subgroup|quasinormal subgroup]], and others. See [[Groupprops:Category:Variations of normality]], [[Groupprops:Category:Opposites of normality]], and [[Groupprops:Category:Analogues of normality]]. Primary subject wiki entry: [[Groupprops:Normal subgroup]] Other subject wiki entries: [[Diffgeom:Normal subgroup]] Also located at: [[Wikipedia:Normal subgroup]], [[Planetmath:NormalSubgroup]], [[Mathworld:NormalSubgroup]], [[Sor:N/n067690|Springer Online Reference Works]], [[Citizendium:Normal subgroup]] b586f6857d365a3d98cbcbd9606fae0b42d236d2 0 1 54 4 2008-05-22T18:19:05Z Vipul 2 wikitext text/x-wiki Welcome to the Subject Wikis Reference Guide! Here is our list of subject wikis: {{:Ref:Subwiki list}} d13efd59934d38908d1b2052c815e2cbc1bc9926 108 29 55 2008-05-23T20:30:07Z Vipul 2 New page: Wikipedia is a free, online, open-content, collaborative, multilingual encyclopedia project. ==Use Wikipedia== Wikipedia is online, or Internet-based. It can be accessed using a web brow... wikitext text/x-wiki Wikipedia is a free, online, open-content, collaborative, multilingual encyclopedia project. ==Use Wikipedia== Wikipedia is online, or Internet-based. It can be accessed using a web browser and an Internet connection. Main website: http://www.wikipedia.org Language websites: * English: http://en.wikipedia.org * French: http://fr.wikipedia.org * German: http://de.wikipedia.org * Spanish: http://es.wikipedia.org f664fa6f5a9d80ff34d869a90e16802d2c168b40 56 55 2008-05-23T21:03:51Z Vipul 2 wikitext text/x-wiki Wikipedia is a free, online, open-content, collaborative, multilingual encyclopedia project. ==Use Wikipedia== ===Start Wikipedia=== Wikipedia is online, or Internet-based. It can be accessed using a web browser and an Internet connection. Main website: http://www.wikipedia.org Some major language websites: * English: http://en.wikipedia.org * French: http://fr.wikipedia.org * German: http://de.wikipedia.org * Polish: http://pl.wikipedia.org A complete list of language Wikipedias is available at: http://meta.wikimedia.org/wiki/List_of_Wikipedias ===Search Wikipedia=== * http://www.wikipedia.org allows search in any of the language Wikipedias * In any of the language Wikipedias, there is a search box in the left column that allows any search * http://en.wikipedia.org/Special:Search is Wikipedia's internal site search ==Wikipedia as a free resource== ===Free access=== All content on Wikipedia can be accessed for free (i.e., without any access or subscription fees). ===Copyright and reuse=== Any contributions made to Wikipedia are automatically considered to be released under the GNU Free Documentation License (GFDL), version 1.2. The GFDL is a ''copyleft'', ''open-content'' license, whose key feature is that it allows reuse without permission from the original author, as long as such reuse is also licensed under a similar license. Main reference pages: # http://en.wikipedia.org/wiki/Wikipedia:Copyrights explains Wikipedia's copyright policy # http://en.wikipedia.org/wiki/Wikipedia:Text_of_the_GNU_Free_Documentation_License is the full text of the license used by Wikipedia ==Management and organization== ===Parent organization=== Wikipedia is managed by the Wikimedia Foundation, a non-profit organization. The Wikimedia Foundation is responsible both for the servers that host Wikipedia content, and for maintaining and improving MediaWiki, the underlying software for Wikipedia. Main reference pages: # http://www.wikimedia.org is the Wikimedia Foundation's central page # http://meta.wikimedia.org is a wiki run by the Wikimedia Foundation to coordinate the Wikimedia Foundation's projects ===Software tools=== Wikipedia runs on MediaWiki, a software designed for web-base collaborative authoring of articles. MediaWiki is ''free software'', and is licensed under the GNU General Public License, Version 2.0. MediaWiki was originally written by Magnus Manske, specifically for Wikipedia. Its development is handled by software professionals working for the Wikimedia Foundation. However, MediaWiki is free software and can be both used and modified by anybody. A number of extensions to MediaWiki have been developed by people outside the Wikimedia Foundation. MediaWiki is also used as a software in a number of other collaborative software-based tools, such as the Subject Wikis Reference Guide. # http://www.mediawiki.org is the MediaWiki website. It has the MediaWiki manual, links for downloading the software, as well as a listing of extensions developed for MediaWiki # http://www.wikiindex.org gives a list of wikis (not comprehensive) 02bd7d28e5c9d79017f28dadd2d500dd3e05bf09 57 56 2008-05-23T21:15:21Z Vipul 2 /* Management and organization */ wikitext text/x-wiki Wikipedia is a free, online, open-content, collaborative, multilingual encyclopedia project. ==Use Wikipedia== ===Start Wikipedia=== Wikipedia is online, or Internet-based. It can be accessed using a web browser and an Internet connection. Main website: http://www.wikipedia.org Some major language websites: * English: http://en.wikipedia.org * French: http://fr.wikipedia.org * German: http://de.wikipedia.org * Polish: http://pl.wikipedia.org A complete list of language Wikipedias is available at: http://meta.wikimedia.org/wiki/List_of_Wikipedias ===Search Wikipedia=== * http://www.wikipedia.org allows search in any of the language Wikipedias * In any of the language Wikipedias, there is a search box in the left column that allows any search * http://en.wikipedia.org/Special:Search is Wikipedia's internal site search ==Wikipedia as a free resource== ===Free access=== All content on Wikipedia can be accessed for free (i.e., without any access or subscription fees). ===Copyright and reuse=== Any contributions made to Wikipedia are automatically considered to be released under the GNU Free Documentation License (GFDL), version 1.2. The GFDL is a ''copyleft'', ''open-content'' license, whose key feature is that it allows reuse without permission from the original author, as long as such reuse is also licensed under a similar license. Main reference pages: # http://en.wikipedia.org/wiki/Wikipedia:Copyrights explains Wikipedia's copyright policy # http://en.wikipedia.org/wiki/Wikipedia:Text_of_the_GNU_Free_Documentation_License is the full text of the license used by Wikipedia ==Management and organization== General information in this direction is available at: http://en.wikipedia.org/wiki/Wikipedia:About ===Parent organization=== Wikipedia is managed by the [[parent organization::Wikimedia Foundation]], a non-profit organization. The Wikimedia Foundation is responsible both for the servers that host Wikipedia content, and for maintaining and improving MediaWiki, the underlying software for Wikipedia. Main reference pages: # http://www.wikimedia.org is the Wikimedia Foundation's central landing page (listing Wikipedia and other sister projects) # http://www.wikimediafoundation.org is the Wikimedia Foundation's wiki # http://meta.wikimedia.org is a wiki run by the Wikimedia Foundation to coordinate the Wikimedia Foundation's projects ===Software tools=== Wikipedia runs on [[software::MediaWiki]], a software designed for web-base collaborative authoring of articles. MediaWiki is ''free software'', and is licensed under the GNU General Public License, Version 2.0. MediaWiki was originally written by Magnus Manske, specifically for Wikipedia. Its development is handled by software professionals working for the Wikimedia Foundation. However, MediaWiki is free software and can be both used and modified by anybody. A number of extensions to MediaWiki have been developed by people outside the Wikimedia Foundation. MediaWiki is also used as a software in a number of other collaborative software-based tools, such as the Subject Wikis Reference Guide. # http://www.mediawiki.org is the MediaWiki website. It has the MediaWiki manual, links for downloading the software, as well as a listing of extensions developed for MediaWiki # http://www.wikiindex.org gives a list of wikis (not comprehensive) ===Financial model=== The Wikimedia Foundation (Wikipedia's parent organization) is a non-profit organization (registered under Section 501 (c)(3)) founded in St. Petersburg, Florida, United States in 2002 and headquartered in San Francisco, California, United States. The financial model of the Wikimedia Foundation primarily rests on voluntary donation. # http://wikimediafoundation.org/wiki/Donate The donate page 84d38ba864fbc36379416336caa30ff98e0324be 58 57 2008-05-23T21:55:45Z Vipul 2 wikitext text/x-wiki Wikipedia is a free, online, open-content, collaborative, multilingual encyclopedia project. ==Use Wikipedia== ===Start Wikipedia=== Wikipedia is online, or Internet-based. It can be accessed using a web browser and an Internet connection. Main website: http://www.wikipedia.org Some major language websites: * English: http://en.wikipedia.org * French: http://fr.wikipedia.org * German: http://de.wikipedia.org * Polish: http://pl.wikipedia.org A complete list of language Wikipedias is available at: http://meta.wikimedia.org/wiki/List_of_Wikipedias ===Search Wikipedia=== # http://www.wikipedia.org allows search in any of the language Wikipedias # In any of the language Wikipedias, there is a search box in the left column that allows any search # http://en.wikipedia.org/Special:Search is the English Wikipedia's internal site search # If using Mozilla Firefox as the browser, you can add Wikipedia (English) to the list of Search Engines using https://addons.mozilla.org/ ===Article names and reaching a given article=== Wikipedia content is organized into articles. A Wikipedia article on a topic is named exactly by that topic. The URL for a given Wikipedia article in the English Wikipedia is given by: http://en.wikipedia.org/wiki/Name where Name denotes the name of the article. Thus, for instance, the English Wikipedia article on India is titled [[wikipedia:India|India]], and the full URL is given by: http://en.wikipedia.org/wiki/India To go to an article with a particular title, do any of the following: # Type the full URL: http://en.wikipedia.org/wiki/Name # Use the Wikipedia Search Engine for Firefox and type the name of the topic in the search bar # Open Wikipedia and type the name of the article in the search bar, and press Go (or enter) ==More about Wikipedia for users== ===Criteria for Wikipedia content=== Some important Wikipedia policies for content (nutshells as stated on the Wikipedia guideline pages): # [[Wikipedia:Wikipedia:Neutral point of view|Neutral Point of View policy]]: All Wikipedia articles and other encyclopedic content must be written from a neutral point of view, representing significant views fairly, proportionately and without bias. # [[Wikipedia:Wikipedia:No original research|No original research]]: #* Wikipedia does not publish original thought: all material in Wikipedia must be attributable to a reliable, published source. #* Articles may not contain any new analysis or synthesis of published material that serves to advance a position not clearly advanced by the sources. # [[Wikipedia:Wikipedia:Notability|Notability]]: If a topic has received significant coverage in reliable secondary sources that are independent of the subject, it is presumed to be notable. # [[Wikipedia:Wikipedia:What Wikipedia is not|What Wikipedia is not]]: Gives a list of things Wikipedia is not suitable for. # [[Wikipedia:Wikipedia:Verifiability|Verifiability policy]]: Material challenged or likely to be challenged, and all quotations, must be attributed to a reliable, published source. These criteria are taken as guidelines, and not every page on Wikipedia may conform to the criteria at any given time. ===Using Wikipedia=== Some pieces of information put up by Wikipedia for Wikipedia users: # [[Wikipedia:Wikipedia:Ten things you may not know about Wikipedia|Ten things about Wikipedia]]: Important information about Wikipedia # [[Wikipedia:Wikipedia:Researching with Wikipedia|Researching with Wikipedia]]: Information and suggestions about how to use Wikipedia for research ==Wikipedia as a free resource== ===Free access=== All content on Wikipedia can be accessed for free (i.e., without any access or subscription fees). ===Copyright and reuse=== Any contributions made to Wikipedia are automatically considered to be released under the GNU Free Documentation License ([[license::GFDL]]), version 1.2. The GFDL is a ''copyleft'', ''open-content'' license, whose key feature is that it allows reuse without permission from the original author, as long as such reuse is also licensed under a similar license. Main reference pages: # http://en.wikipedia.org/wiki/Wikipedia:Copyrights explains Wikipedia's copyright policy # http://en.wikipedia.org/wiki/Wikipedia:Text_of_the_GNU_Free_Documentation_License is the full text of the license used by Wikipedia ==Management and organization== General information in this direction is available at: http://en.wikipedia.org/wiki/Wikipedia:About ===Parent organization=== Wikipedia is managed by the [[parent organization::Wikimedia Foundation]], a non-profit organization. The Wikimedia Foundation is responsible both for the servers that host Wikipedia content, and for maintaining and improving MediaWiki, the underlying software for Wikipedia. Main reference pages: # http://www.wikimedia.org is the Wikimedia Foundation's central landing page (listing Wikipedia and other sister projects) # http://www.wikimediafoundation.org is the Wikimedia Foundation's wiki # http://meta.wikimedia.org is a wiki run by the Wikimedia Foundation to coordinate the Wikimedia Foundation's projects ===Software tools=== Wikipedia runs on [[software::MediaWiki]], a software designed for web-base collaborative authoring of articles. MediaWiki is ''free software'', and is licensed under the GNU General Public License, Version 2.0. MediaWiki was originally written by Magnus Manske, specifically for Wikipedia. Its development is handled by software professionals working for the Wikimedia Foundation. However, MediaWiki is free software and can be both used and modified by anybody. A number of extensions to MediaWiki have been developed by people outside the Wikimedia Foundation. MediaWiki is also used as a software in a number of other collaborative software-based tools, such as the Subject Wikis Reference Guide. # http://www.mediawiki.org is the MediaWiki website. It has the MediaWiki manual, links for downloading the software, as well as a listing of extensions developed for MediaWiki # http://www.wikiindex.org gives a list of wikis (not comprehensive) ===Financial model=== The Wikimedia Foundation (Wikipedia's parent organization) is a non-profit organization (registered under Section 501 (c)(3)) founded in St. Petersburg, Florida, United States in 2002 and headquartered in San Francisco, California, United States. The financial model of the Wikimedia Foundation primarily rests on voluntary donation. # http://wikimediafoundation.org/wiki/Donate The donate page ==Production model== ===Collaborative content creation=== Wikipedia uses a ''strongly collaborative'' model of content creation. c523467f22858fd00483a0df83507e12ce1b1b8f 59 58 2008-05-23T22:07:58Z Vipul 2 /* Production model */ wikitext text/x-wiki Wikipedia is a free, online, open-content, collaborative, multilingual encyclopedia project. ==Use Wikipedia== ===Start Wikipedia=== Wikipedia is online, or Internet-based. It can be accessed using a web browser and an Internet connection. Main website: http://www.wikipedia.org Some major language websites: * English: http://en.wikipedia.org * French: http://fr.wikipedia.org * German: http://de.wikipedia.org * Polish: http://pl.wikipedia.org A complete list of language Wikipedias is available at: http://meta.wikimedia.org/wiki/List_of_Wikipedias ===Search Wikipedia=== # http://www.wikipedia.org allows search in any of the language Wikipedias # In any of the language Wikipedias, there is a search box in the left column that allows any search # http://en.wikipedia.org/Special:Search is the English Wikipedia's internal site search # If using Mozilla Firefox as the browser, you can add Wikipedia (English) to the list of Search Engines using https://addons.mozilla.org/ ===Article names and reaching a given article=== Wikipedia content is organized into articles. A Wikipedia article on a topic is named exactly by that topic. The URL for a given Wikipedia article in the English Wikipedia is given by: http://en.wikipedia.org/wiki/Name where Name denotes the name of the article. Thus, for instance, the English Wikipedia article on India is titled [[wikipedia:India|India]], and the full URL is given by: http://en.wikipedia.org/wiki/India To go to an article with a particular title, do any of the following: # Type the full URL: http://en.wikipedia.org/wiki/Name # Use the Wikipedia Search Engine for Firefox and type the name of the topic in the search bar # Open Wikipedia and type the name of the article in the search bar, and press Go (or enter) ==More about Wikipedia for users== ===Criteria for Wikipedia content=== Some important Wikipedia policies for content (nutshells as stated on the Wikipedia guideline pages): # [[Wikipedia:Wikipedia:Neutral point of view|Neutral Point of View policy]]: All Wikipedia articles and other encyclopedic content must be written from a neutral point of view, representing significant views fairly, proportionately and without bias. # [[Wikipedia:Wikipedia:No original research|No original research]]: #* Wikipedia does not publish original thought: all material in Wikipedia must be attributable to a reliable, published source. #* Articles may not contain any new analysis or synthesis of published material that serves to advance a position not clearly advanced by the sources. # [[Wikipedia:Wikipedia:Notability|Notability]]: If a topic has received significant coverage in reliable secondary sources that are independent of the subject, it is presumed to be notable. # [[Wikipedia:Wikipedia:What Wikipedia is not|What Wikipedia is not]]: Gives a list of things Wikipedia is not suitable for. # [[Wikipedia:Wikipedia:Verifiability|Verifiability policy]]: Material challenged or likely to be challenged, and all quotations, must be attributed to a reliable, published source. These criteria are taken as guidelines, and not every page on Wikipedia may conform to the criteria at any given time. ===Using Wikipedia=== Some pieces of information put up by Wikipedia for Wikipedia users: # [[Wikipedia:Wikipedia:Ten things you may not know about Wikipedia|Ten things about Wikipedia]]: Important information about Wikipedia # [[Wikipedia:Wikipedia:Researching with Wikipedia|Researching with Wikipedia]]: Information and suggestions about how to use Wikipedia for research ==Wikipedia as a free resource== ===Free access=== All content on Wikipedia can be accessed for free (i.e., without any access or subscription fees). ===Copyright and reuse=== Any contributions made to Wikipedia are automatically considered to be released under the GNU Free Documentation License ([[license::GFDL]]), version 1.2. The GFDL is a ''copyleft'', ''open-content'' license, whose key feature is that it allows reuse without permission from the original author, as long as such reuse is also licensed under a similar license. Main reference pages: # http://en.wikipedia.org/wiki/Wikipedia:Copyrights explains Wikipedia's copyright policy # http://en.wikipedia.org/wiki/Wikipedia:Text_of_the_GNU_Free_Documentation_License is the full text of the license used by Wikipedia ==Management and organization== General information in this direction is available at: http://en.wikipedia.org/wiki/Wikipedia:About ===Parent organization=== Wikipedia is managed by the [[parent organization::Wikimedia Foundation]], a non-profit organization. The Wikimedia Foundation is responsible both for the servers that host Wikipedia content, and for maintaining and improving MediaWiki, the underlying software for Wikipedia. Main reference pages: # http://www.wikimedia.org is the Wikimedia Foundation's central landing page (listing Wikipedia and other sister projects) # http://www.wikimediafoundation.org is the Wikimedia Foundation's wiki # http://meta.wikimedia.org is a wiki run by the Wikimedia Foundation to coordinate the Wikimedia Foundation's projects ===Software tools=== Wikipedia runs on [[software::MediaWiki]], a software designed for web-base collaborative authoring of articles. MediaWiki is ''free software'', and is licensed under the GNU General Public License, Version 2.0. MediaWiki was originally written by Magnus Manske, specifically for Wikipedia. Its development is handled by software professionals working for the Wikimedia Foundation. However, MediaWiki is free software and can be both used and modified by anybody. A number of extensions to MediaWiki have been developed by people outside the Wikimedia Foundation. MediaWiki is also used as a software in a number of other collaborative software-based tools, such as the Subject Wikis Reference Guide. # http://www.mediawiki.org is the MediaWiki website. It has the MediaWiki manual, links for downloading the software, as well as a listing of extensions developed for MediaWiki # http://www.wikiindex.org gives a list of wikis (not comprehensive) ===Financial model=== The Wikimedia Foundation (Wikipedia's parent organization) is a non-profit organization (registered under Section 501 (c)(3)) founded in St. Petersburg, Florida, United States in 2002 and headquartered in San Francisco, California, United States. The financial model of the Wikimedia Foundation primarily rests on voluntary donation. # http://wikimediafoundation.org/wiki/Donate The donate page ==Production model== ===Collaborative content creation=== Wikipedia uses a ''strongly collaborative'' model of content creation, powered by a wiki software called MediaWiki. Some of the key features of the model used on Wikipedia: # Anybody can register an account on Wikipedia, without necessarily revealing his/her real name and identity # Anybody can start an article on a topic (as of now, only logged-in users can start new articles) if no article on the topic exists # Once an article is created, anybody (and not just the original author of the article) can edit the article # Every revision of the article is stored on the wiki, and different revisions can be viewed and compared using the "history" tab of the article # People seeking to collaborate on a given article can discuss material related to the article on its "discussion page" or "talk page", which can be accessed using the "discussion" tab of the article # A person can nominate an article for deletion on grounds that it violates Wikipedia policy. The issue is then discussed on a separate page, till a consensus is reached either way ===Compensation model=== Wikipedia does not, as a rule, compensate people for writing articles. In particular: # There is no monetary compensation or reimbursement for writing articles on Wikipedia # No credits for an article are given on the article page itself. The authorship of a given article may be seen using the "history" tab or the "credits" option. However, the information revealed by the history may b eincomplete because anonymous users may have made certain edits, and even the registered users may not have used their real names and identities ===Security model=== Wikipedia relies on soft security to combat problems such as vandalism and bias in articles. Dedicated volunteers patrol the wiki to locate acts of vandalism, or content that otherwise violates Wikipedia policy. User accounts or IP addresses associated with such editing may be banned from editing Wikipedia. ae4a0f90b1a35d9d09fb2dacc0c6270599e90305 60 59 2008-05-23T22:16:00Z Vipul 2 /* Using Wikipedia */ wikitext text/x-wiki Wikipedia is a free, online, open-content, collaborative, multilingual encyclopedia project. ==Use Wikipedia== ===Start Wikipedia=== Wikipedia is online, or Internet-based. It can be accessed using a web browser and an Internet connection. Main website: http://www.wikipedia.org Some major language websites: * English: http://en.wikipedia.org * French: http://fr.wikipedia.org * German: http://de.wikipedia.org * Polish: http://pl.wikipedia.org A complete list of language Wikipedias is available at: http://meta.wikimedia.org/wiki/List_of_Wikipedias ===Search Wikipedia=== # http://www.wikipedia.org allows search in any of the language Wikipedias # In any of the language Wikipedias, there is a search box in the left column that allows any search # http://en.wikipedia.org/Special:Search is the English Wikipedia's internal site search # If using Mozilla Firefox as the browser, you can add Wikipedia (English) to the list of Search Engines using https://addons.mozilla.org/ ===Article names and reaching a given article=== Wikipedia content is organized into articles. A Wikipedia article on a topic is named exactly by that topic. The URL for a given Wikipedia article in the English Wikipedia is given by: http://en.wikipedia.org/wiki/Name where Name denotes the name of the article. Thus, for instance, the English Wikipedia article on India is titled [[wikipedia:India|India]], and the full URL is given by: http://en.wikipedia.org/wiki/India To go to an article with a particular title, do any of the following: # Type the full URL: http://en.wikipedia.org/wiki/Name # Use the Wikipedia Search Engine for Firefox and type the name of the topic in the search bar # Open Wikipedia and type the name of the article in the search bar, and press Go (or enter) ==More about Wikipedia for users== ===Criteria for Wikipedia content=== Some important Wikipedia policies for content (nutshells as stated on the Wikipedia guideline pages): # [[Wikipedia:Wikipedia:Neutral point of view|Neutral Point of View policy]]: All Wikipedia articles and other encyclopedic content must be written from a neutral point of view, representing significant views fairly, proportionately and without bias. # [[Wikipedia:Wikipedia:No original research|No original research]]: #* Wikipedia does not publish original thought: all material in Wikipedia must be attributable to a reliable, published source. #* Articles may not contain any new analysis or synthesis of published material that serves to advance a position not clearly advanced by the sources. # [[Wikipedia:Wikipedia:Notability|Notability]]: If a topic has received significant coverage in reliable secondary sources that are independent of the subject, it is presumed to be notable. # [[Wikipedia:Wikipedia:What Wikipedia is not|What Wikipedia is not]]: Gives a list of things Wikipedia is not suitable for. # [[Wikipedia:Wikipedia:Verifiability|Verifiability policy]]: Material challenged or likely to be challenged, and all quotations, must be attributed to a reliable, published source. These criteria are taken as guidelines, and not every page on Wikipedia may conform to the criteria at any given time. ===Using Wikipedia=== Some pieces of information put up by Wikipedia for Wikipedia users: # [[Wikipedia:Wikipedia:Ten things you may not know about Wikipedia|Ten things about Wikipedia]]: Important information about Wikipedia # [[Wikipedia:Wikipedia:Researching with Wikipedia|Researching with Wikipedia]]: Information and suggestions about how to use Wikipedia for research # [[Wikipedia:Wikipedia:Basic navigation|Navigating and finding information on Wikipedia]] ==Wikipedia as a free resource== ===Free access=== All content on Wikipedia can be accessed for free (i.e., without any access or subscription fees). ===Copyright and reuse=== Any contributions made to Wikipedia are automatically considered to be released under the GNU Free Documentation License ([[license::GFDL]]), version 1.2. The GFDL is a ''copyleft'', ''open-content'' license, whose key feature is that it allows reuse without permission from the original author, as long as such reuse is also licensed under a similar license. Main reference pages: # http://en.wikipedia.org/wiki/Wikipedia:Copyrights explains Wikipedia's copyright policy # http://en.wikipedia.org/wiki/Wikipedia:Text_of_the_GNU_Free_Documentation_License is the full text of the license used by Wikipedia ==Management and organization== General information in this direction is available at: http://en.wikipedia.org/wiki/Wikipedia:About ===Parent organization=== Wikipedia is managed by the [[parent organization::Wikimedia Foundation]], a non-profit organization. The Wikimedia Foundation is responsible both for the servers that host Wikipedia content, and for maintaining and improving MediaWiki, the underlying software for Wikipedia. Main reference pages: # http://www.wikimedia.org is the Wikimedia Foundation's central landing page (listing Wikipedia and other sister projects) # http://www.wikimediafoundation.org is the Wikimedia Foundation's wiki # http://meta.wikimedia.org is a wiki run by the Wikimedia Foundation to coordinate the Wikimedia Foundation's projects ===Software tools=== Wikipedia runs on [[software::MediaWiki]], a software designed for web-base collaborative authoring of articles. MediaWiki is ''free software'', and is licensed under the GNU General Public License, Version 2.0. MediaWiki was originally written by Magnus Manske, specifically for Wikipedia. Its development is handled by software professionals working for the Wikimedia Foundation. However, MediaWiki is free software and can be both used and modified by anybody. A number of extensions to MediaWiki have been developed by people outside the Wikimedia Foundation. MediaWiki is also used as a software in a number of other collaborative software-based tools, such as the Subject Wikis Reference Guide. # http://www.mediawiki.org is the MediaWiki website. It has the MediaWiki manual, links for downloading the software, as well as a listing of extensions developed for MediaWiki # http://www.wikiindex.org gives a list of wikis (not comprehensive) ===Financial model=== The Wikimedia Foundation (Wikipedia's parent organization) is a non-profit organization (registered under Section 501 (c)(3)) founded in St. Petersburg, Florida, United States in 2002 and headquartered in San Francisco, California, United States. The financial model of the Wikimedia Foundation primarily rests on voluntary donation. # http://wikimediafoundation.org/wiki/Donate The donate page ==Production model== ===Collaborative content creation=== Wikipedia uses a ''strongly collaborative'' model of content creation, powered by a wiki software called MediaWiki. Some of the key features of the model used on Wikipedia: # Anybody can register an account on Wikipedia, without necessarily revealing his/her real name and identity # Anybody can start an article on a topic (as of now, only logged-in users can start new articles) if no article on the topic exists # Once an article is created, anybody (and not just the original author of the article) can edit the article # Every revision of the article is stored on the wiki, and different revisions can be viewed and compared using the "history" tab of the article # People seeking to collaborate on a given article can discuss material related to the article on its "discussion page" or "talk page", which can be accessed using the "discussion" tab of the article # A person can nominate an article for deletion on grounds that it violates Wikipedia policy. The issue is then discussed on a separate page, till a consensus is reached either way ===Compensation model=== Wikipedia does not, as a rule, compensate people for writing articles. In particular: # There is no monetary compensation or reimbursement for writing articles on Wikipedia # No credits for an article are given on the article page itself. The authorship of a given article may be seen using the "history" tab or the "credits" option. However, the information revealed by the history may b eincomplete because anonymous users may have made certain edits, and even the registered users may not have used their real names and identities ===Security model=== Wikipedia relies on soft security to combat problems such as vandalism and bias in articles. Dedicated volunteers patrol the wiki to locate acts of vandalism, or content that otherwise violates Wikipedia policy. User accounts or IP addresses associated with such editing may be banned from editing Wikipedia. c557d022845867cb0b74b4383aee994e8efb6474 61 60 2008-05-23T22:23:19Z Vipul 2 /* Production model */ wikitext text/x-wiki Wikipedia is a free, online, open-content, collaborative, multilingual encyclopedia project. ==Use Wikipedia== ===Start Wikipedia=== Wikipedia is online, or Internet-based. It can be accessed using a web browser and an Internet connection. Main website: http://www.wikipedia.org Some major language websites: * English: http://en.wikipedia.org * French: http://fr.wikipedia.org * German: http://de.wikipedia.org * Polish: http://pl.wikipedia.org A complete list of language Wikipedias is available at: http://meta.wikimedia.org/wiki/List_of_Wikipedias ===Search Wikipedia=== # http://www.wikipedia.org allows search in any of the language Wikipedias # In any of the language Wikipedias, there is a search box in the left column that allows any search # http://en.wikipedia.org/Special:Search is the English Wikipedia's internal site search # If using Mozilla Firefox as the browser, you can add Wikipedia (English) to the list of Search Engines using https://addons.mozilla.org/ ===Article names and reaching a given article=== Wikipedia content is organized into articles. A Wikipedia article on a topic is named exactly by that topic. The URL for a given Wikipedia article in the English Wikipedia is given by: http://en.wikipedia.org/wiki/Name where Name denotes the name of the article. Thus, for instance, the English Wikipedia article on India is titled [[wikipedia:India|India]], and the full URL is given by: http://en.wikipedia.org/wiki/India To go to an article with a particular title, do any of the following: # Type the full URL: http://en.wikipedia.org/wiki/Name # Use the Wikipedia Search Engine for Firefox and type the name of the topic in the search bar # Open Wikipedia and type the name of the article in the search bar, and press Go (or enter) ==More about Wikipedia for users== ===Criteria for Wikipedia content=== Some important Wikipedia policies for content (nutshells as stated on the Wikipedia guideline pages): # [[Wikipedia:Wikipedia:Neutral point of view|Neutral Point of View policy]]: All Wikipedia articles and other encyclopedic content must be written from a neutral point of view, representing significant views fairly, proportionately and without bias. # [[Wikipedia:Wikipedia:No original research|No original research]]: #* Wikipedia does not publish original thought: all material in Wikipedia must be attributable to a reliable, published source. #* Articles may not contain any new analysis or synthesis of published material that serves to advance a position not clearly advanced by the sources. # [[Wikipedia:Wikipedia:Notability|Notability]]: If a topic has received significant coverage in reliable secondary sources that are independent of the subject, it is presumed to be notable. # [[Wikipedia:Wikipedia:What Wikipedia is not|What Wikipedia is not]]: Gives a list of things Wikipedia is not suitable for. # [[Wikipedia:Wikipedia:Verifiability|Verifiability policy]]: Material challenged or likely to be challenged, and all quotations, must be attributed to a reliable, published source. These criteria are taken as guidelines, and not every page on Wikipedia may conform to the criteria at any given time. ===Using Wikipedia=== Some pieces of information put up by Wikipedia for Wikipedia users: # [[Wikipedia:Wikipedia:Ten things you may not know about Wikipedia|Ten things about Wikipedia]]: Important information about Wikipedia # [[Wikipedia:Wikipedia:Researching with Wikipedia|Researching with Wikipedia]]: Information and suggestions about how to use Wikipedia for research # [[Wikipedia:Wikipedia:Basic navigation|Navigating and finding information on Wikipedia]] ==Wikipedia as a free resource== ===Free access=== All content on Wikipedia can be accessed for free (i.e., without any access or subscription fees). ===Copyright and reuse=== Any contributions made to Wikipedia are automatically considered to be released under the GNU Free Documentation License ([[license::GFDL]]), version 1.2. The GFDL is a ''copyleft'', ''open-content'' license, whose key feature is that it allows reuse without permission from the original author, as long as such reuse is also licensed under a similar license. Main reference pages: # http://en.wikipedia.org/wiki/Wikipedia:Copyrights explains Wikipedia's copyright policy # http://en.wikipedia.org/wiki/Wikipedia:Text_of_the_GNU_Free_Documentation_License is the full text of the license used by Wikipedia ==Management and organization== General information in this direction is available at: http://en.wikipedia.org/wiki/Wikipedia:About ===Parent organization=== Wikipedia is managed by the [[parent organization::Wikimedia Foundation]], a non-profit organization. The Wikimedia Foundation is responsible both for the servers that host Wikipedia content, and for maintaining and improving MediaWiki, the underlying software for Wikipedia. Main reference pages: # http://www.wikimedia.org is the Wikimedia Foundation's central landing page (listing Wikipedia and other sister projects) # http://www.wikimediafoundation.org is the Wikimedia Foundation's wiki # http://meta.wikimedia.org is a wiki run by the Wikimedia Foundation to coordinate the Wikimedia Foundation's projects ===Software tools=== Wikipedia runs on [[software::MediaWiki]], a software designed for web-base collaborative authoring of articles. MediaWiki is ''free software'', and is licensed under the GNU General Public License, Version 2.0. MediaWiki was originally written by Magnus Manske, specifically for Wikipedia. Its development is handled by software professionals working for the Wikimedia Foundation. However, MediaWiki is free software and can be both used and modified by anybody. A number of extensions to MediaWiki have been developed by people outside the Wikimedia Foundation. MediaWiki is also used as a software in a number of other collaborative software-based tools, such as the Subject Wikis Reference Guide. # http://www.mediawiki.org is the MediaWiki website. It has the MediaWiki manual, links for downloading the software, as well as a listing of extensions developed for MediaWiki # http://www.wikiindex.org gives a list of wikis (not comprehensive) ===Financial model=== The Wikimedia Foundation (Wikipedia's parent organization) is a non-profit organization (registered under Section 501 (c)(3)) founded in St. Petersburg, Florida, United States in 2002 and headquartered in San Francisco, California, United States. The financial model of the Wikimedia Foundation primarily rests on voluntary donation. # http://wikimediafoundation.org/wiki/Donate The donate page ==Production model== ===Collaborative content creation=== Wikipedia uses a ''strongly collaborative'' model of content creation, powered by a wiki software called MediaWiki. Some of the key features of the model used on Wikipedia: # Anybody can register an account on Wikipedia, without necessarily revealing his/her real name and identity # Anybody can start an article on a topic (as of now, only logged-in users can start new articles) if no article on the topic exists # Once an article is created, anybody (and not just the original author of the article) can edit the article # Every revision of the article is stored on the wiki, and different revisions can be viewed and compared using the "history" tab of the article # People seeking to collaborate on a given article can discuss material related to the article on its "discussion page" or "talk page", which can be accessed using the "discussion" tab of the article # A person can nominate an article for deletion on grounds that it violates Wikipedia policy. The issue is then discussed on a separate page, till a consensus is reached either way Relevant links: # http://en.wikipedia.org/w/index.php?title=Special:UserLogin&type=signup is the page for account creation on English Wikipedia # [[Wikipedia:Help:Editing|Wikipedia Editing help]] ===Compensation model=== Wikipedia does not, as a rule, compensate people for writing articles. In particular: # There is no monetary compensation or reimbursement for writing articles on Wikipedia # No credits for an article are given on the article page itself. The authorship of a given article may be seen using the "history" tab or the "credits" option. However, the information revealed by the history may b eincomplete because anonymous users may have made certain edits, and even the registered users may not have used their real names and identities ===Security model=== Wikipedia relies on soft security to combat problems such as vandalism and bias in articles. Dedicated volunteers patrol the wiki to locate acts of vandalism, or content that otherwise violates Wikipedia policy. User accounts or IP addresses associated with such editing may be banned from editing Wikipedia. ==Related resources== ===Competitors== * [[Resource:Encyclopædia Britannica]]: Managed by Britannica Corporation, this has been one of the world's leading encyclopedia resources for many centuries. * [[Resource:Citizendium]] ===Sister projects=== * [[Resource:Wikibooks]] * [[Resource:Wiktionary]] * [[Resource:Wikiverse]] * [[Resource:Wikisource]] * [[Resource:Wikiversity]] * [[Resource:Wikinews]] e0d5178fcf005eadae5c9bdfba49e34888a0afb0 62 61 2008-05-23T22:24:08Z Vipul 2 /* =Competitors */ wikitext text/x-wiki Wikipedia is a free, online, open-content, collaborative, multilingual encyclopedia project. ==Use Wikipedia== ===Start Wikipedia=== Wikipedia is online, or Internet-based. It can be accessed using a web browser and an Internet connection. Main website: http://www.wikipedia.org Some major language websites: * English: http://en.wikipedia.org * French: http://fr.wikipedia.org * German: http://de.wikipedia.org * Polish: http://pl.wikipedia.org A complete list of language Wikipedias is available at: http://meta.wikimedia.org/wiki/List_of_Wikipedias ===Search Wikipedia=== # http://www.wikipedia.org allows search in any of the language Wikipedias # In any of the language Wikipedias, there is a search box in the left column that allows any search # http://en.wikipedia.org/Special:Search is the English Wikipedia's internal site search # If using Mozilla Firefox as the browser, you can add Wikipedia (English) to the list of Search Engines using https://addons.mozilla.org/ ===Article names and reaching a given article=== Wikipedia content is organized into articles. A Wikipedia article on a topic is named exactly by that topic. The URL for a given Wikipedia article in the English Wikipedia is given by: http://en.wikipedia.org/wiki/Name where Name denotes the name of the article. Thus, for instance, the English Wikipedia article on India is titled [[wikipedia:India|India]], and the full URL is given by: http://en.wikipedia.org/wiki/India To go to an article with a particular title, do any of the following: # Type the full URL: http://en.wikipedia.org/wiki/Name # Use the Wikipedia Search Engine for Firefox and type the name of the topic in the search bar # Open Wikipedia and type the name of the article in the search bar, and press Go (or enter) ==More about Wikipedia for users== ===Criteria for Wikipedia content=== Some important Wikipedia policies for content (nutshells as stated on the Wikipedia guideline pages): # [[Wikipedia:Wikipedia:Neutral point of view|Neutral Point of View policy]]: All Wikipedia articles and other encyclopedic content must be written from a neutral point of view, representing significant views fairly, proportionately and without bias. # [[Wikipedia:Wikipedia:No original research|No original research]]: #* Wikipedia does not publish original thought: all material in Wikipedia must be attributable to a reliable, published source. #* Articles may not contain any new analysis or synthesis of published material that serves to advance a position not clearly advanced by the sources. # [[Wikipedia:Wikipedia:Notability|Notability]]: If a topic has received significant coverage in reliable secondary sources that are independent of the subject, it is presumed to be notable. # [[Wikipedia:Wikipedia:What Wikipedia is not|What Wikipedia is not]]: Gives a list of things Wikipedia is not suitable for. # [[Wikipedia:Wikipedia:Verifiability|Verifiability policy]]: Material challenged or likely to be challenged, and all quotations, must be attributed to a reliable, published source. These criteria are taken as guidelines, and not every page on Wikipedia may conform to the criteria at any given time. ===Using Wikipedia=== Some pieces of information put up by Wikipedia for Wikipedia users: # [[Wikipedia:Wikipedia:Ten things you may not know about Wikipedia|Ten things about Wikipedia]]: Important information about Wikipedia # [[Wikipedia:Wikipedia:Researching with Wikipedia|Researching with Wikipedia]]: Information and suggestions about how to use Wikipedia for research # [[Wikipedia:Wikipedia:Basic navigation|Navigating and finding information on Wikipedia]] ==Wikipedia as a free resource== ===Free access=== All content on Wikipedia can be accessed for free (i.e., without any access or subscription fees). ===Copyright and reuse=== Any contributions made to Wikipedia are automatically considered to be released under the GNU Free Documentation License ([[license::GFDL]]), version 1.2. The GFDL is a ''copyleft'', ''open-content'' license, whose key feature is that it allows reuse without permission from the original author, as long as such reuse is also licensed under a similar license. Main reference pages: # http://en.wikipedia.org/wiki/Wikipedia:Copyrights explains Wikipedia's copyright policy # http://en.wikipedia.org/wiki/Wikipedia:Text_of_the_GNU_Free_Documentation_License is the full text of the license used by Wikipedia ==Management and organization== General information in this direction is available at: http://en.wikipedia.org/wiki/Wikipedia:About ===Parent organization=== Wikipedia is managed by the [[parent organization::Wikimedia Foundation]], a non-profit organization. The Wikimedia Foundation is responsible both for the servers that host Wikipedia content, and for maintaining and improving MediaWiki, the underlying software for Wikipedia. Main reference pages: # http://www.wikimedia.org is the Wikimedia Foundation's central landing page (listing Wikipedia and other sister projects) # http://www.wikimediafoundation.org is the Wikimedia Foundation's wiki # http://meta.wikimedia.org is a wiki run by the Wikimedia Foundation to coordinate the Wikimedia Foundation's projects ===Software tools=== Wikipedia runs on [[software::MediaWiki]], a software designed for web-base collaborative authoring of articles. MediaWiki is ''free software'', and is licensed under the GNU General Public License, Version 2.0. MediaWiki was originally written by Magnus Manske, specifically for Wikipedia. Its development is handled by software professionals working for the Wikimedia Foundation. However, MediaWiki is free software and can be both used and modified by anybody. A number of extensions to MediaWiki have been developed by people outside the Wikimedia Foundation. MediaWiki is also used as a software in a number of other collaborative software-based tools, such as the Subject Wikis Reference Guide. # http://www.mediawiki.org is the MediaWiki website. It has the MediaWiki manual, links for downloading the software, as well as a listing of extensions developed for MediaWiki # http://www.wikiindex.org gives a list of wikis (not comprehensive) ===Financial model=== The Wikimedia Foundation (Wikipedia's parent organization) is a non-profit organization (registered under Section 501 (c)(3)) founded in St. Petersburg, Florida, United States in 2002 and headquartered in San Francisco, California, United States. The financial model of the Wikimedia Foundation primarily rests on voluntary donation. # http://wikimediafoundation.org/wiki/Donate The donate page ==Production model== ===Collaborative content creation=== Wikipedia uses a ''strongly collaborative'' model of content creation, powered by a wiki software called MediaWiki. Some of the key features of the model used on Wikipedia: # Anybody can register an account on Wikipedia, without necessarily revealing his/her real name and identity # Anybody can start an article on a topic (as of now, only logged-in users can start new articles) if no article on the topic exists # Once an article is created, anybody (and not just the original author of the article) can edit the article # Every revision of the article is stored on the wiki, and different revisions can be viewed and compared using the "history" tab of the article # People seeking to collaborate on a given article can discuss material related to the article on its "discussion page" or "talk page", which can be accessed using the "discussion" tab of the article # A person can nominate an article for deletion on grounds that it violates Wikipedia policy. The issue is then discussed on a separate page, till a consensus is reached either way Relevant links: # http://en.wikipedia.org/w/index.php?title=Special:UserLogin&type=signup is the page for account creation on English Wikipedia # [[Wikipedia:Help:Editing|Wikipedia Editing help]] ===Compensation model=== Wikipedia does not, as a rule, compensate people for writing articles. In particular: # There is no monetary compensation or reimbursement for writing articles on Wikipedia # No credits for an article are given on the article page itself. The authorship of a given article may be seen using the "history" tab or the "credits" option. However, the information revealed by the history may b eincomplete because anonymous users may have made certain edits, and even the registered users may not have used their real names and identities ===Security model=== Wikipedia relies on soft security to combat problems such as vandalism and bias in articles. Dedicated volunteers patrol the wiki to locate acts of vandalism, or content that otherwise violates Wikipedia policy. User accounts or IP addresses associated with such editing may be banned from editing Wikipedia. ==Related resources== ===Competitors=== * [[Competitor::Resource:Encyclopædia Britannica]]: Managed by Britannica Corporation, this has been one of the world's leading encyclopedia resources for many centuries. * [[Competitor::Resource:Citizendium]] ===Sister projects=== * [[Resource:Wikibooks]] * [[Resource:Wiktionary]] * [[Resource:Wikiverse]] * [[Resource:Wikisource]] * [[Resource:Wikiversity]] * [[Resource:Wikinews]] acd17433e20aaa2747e79dce47b056f990a09202 63 62 2008-05-23T22:30:17Z Vipul 2 wikitext text/x-wiki Wikipedia is a free, online, open-content, collaborative, multilingual encyclopedia project. {{quotation|'''SLOGANS''': ''Wikipedia, the free encyclopedia''<br>''Wikipedia, the free encyclopedia that anyone can edit''<br>''Imagine a world in which every single human being can freely share in the sum of all knowledge. That's our commitment.''}} ==Use Wikipedia== ===Start Wikipedia=== Wikipedia is online, or Internet-based. It can be accessed using a web browser and an Internet connection. Main website: http://www.wikipedia.org Some major language websites: * English: http://en.wikipedia.org * French: http://fr.wikipedia.org * German: http://de.wikipedia.org * Polish: http://pl.wikipedia.org A complete list of language Wikipedias is available at: http://meta.wikimedia.org/wiki/List_of_Wikipedias ===Search Wikipedia=== # http://www.wikipedia.org allows search in any of the language Wikipedias # In any of the language Wikipedias, there is a search box in the left column that allows any search # http://en.wikipedia.org/Special:Search is the English Wikipedia's internal site search # If using Mozilla Firefox as the browser, you can add Wikipedia (English) to the list of Search Engines using https://addons.mozilla.org/ ===Article names and reaching a given article=== Wikipedia content is organized into articles. A Wikipedia article on a topic is named exactly by that topic. The URL for a given Wikipedia article in the English Wikipedia is given by: http://en.wikipedia.org/wiki/Name where Name denotes the name of the article. Thus, for instance, the English Wikipedia article on India is titled [[wikipedia:India|India]], and the full URL is given by: http://en.wikipedia.org/wiki/India To go to an article with a particular title, do any of the following: # Type the full URL: http://en.wikipedia.org/wiki/Name # Use the Wikipedia Search Engine for Firefox and type the name of the topic in the search bar # Open Wikipedia and type the name of the article in the search bar, and press Go (or enter) ==More about Wikipedia for users== See also: [[Wikipedia:Wikipedia:Introduction|The English Wikipedia's introduction page]] ===Criteria for Wikipedia content=== Some important Wikipedia policies for content (nutshells as stated on the Wikipedia guideline pages): # [[Wikipedia:Wikipedia:Neutral point of view|Neutral Point of View policy]]: All Wikipedia articles and other encyclopedic content must be written from a neutral point of view, representing significant views fairly, proportionately and without bias. # [[Wikipedia:Wikipedia:No original research|No original research]]: #* Wikipedia does not publish original thought: all material in Wikipedia must be attributable to a reliable, published source. #* Articles may not contain any new analysis or synthesis of published material that serves to advance a position not clearly advanced by the sources. # [[Wikipedia:Wikipedia:Notability|Notability]]: If a topic has received significant coverage in reliable secondary sources that are independent of the subject, it is presumed to be notable. # [[Wikipedia:Wikipedia:What Wikipedia is not|What Wikipedia is not]]: Gives a list of things Wikipedia is not suitable for. # [[Wikipedia:Wikipedia:Verifiability|Verifiability policy]]: Material challenged or likely to be challenged, and all quotations, must be attributed to a reliable, published source. These criteria are taken as guidelines, and not every page on Wikipedia may conform to the criteria at any given time. ===Using Wikipedia=== Some pieces of information put up by Wikipedia for Wikipedia users: # [[Wikipedia:Wikipedia:Ten things you may not know about Wikipedia|Ten things about Wikipedia]]: Important information about Wikipedia # [[Wikipedia:Wikipedia:Researching with Wikipedia|Researching with Wikipedia]]: Information and suggestions about how to use Wikipedia for research # [[Wikipedia:Wikipedia:Basic navigation|Navigating and finding information on Wikipedia]] ==Wikipedia as a free resource== ===Free access=== All content on Wikipedia can be accessed for free (i.e., without any access or subscription fees). ===Copyright and reuse=== Any contributions made to Wikipedia are automatically considered to be released under the GNU Free Documentation License ([[license::GFDL]]), version 1.2. The GFDL is a ''copyleft'', ''open-content'' license, whose key feature is that it allows reuse without permission from the original author, as long as such reuse is also licensed under a similar license. Main reference pages: # http://en.wikipedia.org/wiki/Wikipedia:Copyrights explains Wikipedia's copyright policy # http://en.wikipedia.org/wiki/Wikipedia:Text_of_the_GNU_Free_Documentation_License is the full text of the license used by Wikipedia ==Management and organization== General information in this direction is available at: http://en.wikipedia.org/wiki/Wikipedia:About ===Parent organization=== Wikipedia is managed by the [[parent organization::Wikimedia Foundation]], a non-profit organization. The Wikimedia Foundation is responsible both for the servers that host Wikipedia content, and for maintaining and improving MediaWiki, the underlying software for Wikipedia. Main reference pages: # http://www.wikimedia.org is the Wikimedia Foundation's central landing page (listing Wikipedia and other sister projects) # http://www.wikimediafoundation.org is the Wikimedia Foundation's wiki # http://meta.wikimedia.org is a wiki run by the Wikimedia Foundation to coordinate the Wikimedia Foundation's projects ===Software tools=== Wikipedia runs on [[software::MediaWiki]], a software designed for web-base collaborative authoring of articles. MediaWiki is ''free software'', and is licensed under the GNU General Public License, Version 2.0. MediaWiki was originally written by Magnus Manske, specifically for Wikipedia. Its development is handled by software professionals working for the Wikimedia Foundation. However, MediaWiki is free software and can be both used and modified by anybody. A number of extensions to MediaWiki have been developed by people outside the Wikimedia Foundation. MediaWiki is also used as a software in a number of other collaborative software-based tools, such as the Subject Wikis Reference Guide. # http://www.mediawiki.org is the MediaWiki website. It has the MediaWiki manual, links for downloading the software, as well as a listing of extensions developed for MediaWiki # http://www.wikiindex.org gives a list of wikis (not comprehensive) ===Financial model=== The Wikimedia Foundation (Wikipedia's parent organization) is a non-profit organization (registered under Section 501 (c)(3)) founded in St. Petersburg, Florida, United States in 2002 and headquartered in San Francisco, California, United States. The financial model of the Wikimedia Foundation primarily rests on voluntary donation. # http://wikimediafoundation.org/wiki/Donate The donate page ==Production model== ===Collaborative content creation=== Wikipedia uses a ''strongly collaborative'' model of content creation, powered by a wiki software called MediaWiki. Some of the key features of the model used on Wikipedia: # Anybody can register an account on Wikipedia, without necessarily revealing his/her real name and identity # Anybody can start an article on a topic (as of now, only logged-in users can start new articles) if no article on the topic exists # Once an article is created, anybody (and not just the original author of the article) can edit the article # Every revision of the article is stored on the wiki, and different revisions can be viewed and compared using the "history" tab of the article # People seeking to collaborate on a given article can discuss material related to the article on its "discussion page" or "talk page", which can be accessed using the "discussion" tab of the article # A person can nominate an article for deletion on grounds that it violates Wikipedia policy. The issue is then discussed on a separate page, till a consensus is reached either way Relevant links: # http://en.wikipedia.org/w/index.php?title=Special:UserLogin&type=signup is the page for account creation on English Wikipedia # [[Wikipedia:Help:Editing|Wikipedia Editing help]] ===Compensation model=== Wikipedia does not, as a rule, compensate people for writing articles. In particular: # There is no monetary compensation or reimbursement for writing articles on Wikipedia # No credits for an article are given on the article page itself. The authorship of a given article may be seen using the "history" tab or the "credits" option. However, the information revealed by the history may b eincomplete because anonymous users may have made certain edits, and even the registered users may not have used their real names and identities ===Security model=== Wikipedia relies on soft security to combat problems such as vandalism and bias in articles. Dedicated volunteers patrol the wiki to locate acts of vandalism, or content that otherwise violates Wikipedia policy. User accounts or IP addresses associated with such editing may be banned from editing Wikipedia. ==Related resources== ===Competitors=== * [[Competitor::Resource:Encyclopædia Britannica]]: Managed by Britannica Corporation, this has been one of the world's leading encyclopedia resources for many centuries. * [[Competitor::Resource:Citizendium]] ===Sister projects=== * [[Sister::Resource:Wikibooks]] * [[Sister::Resource:Wiktionary]] * [[Sister::Resource:Wikiverse]] * [[Sister::Resource:Wikisource]] * [[Sister::Resource:Wikiversity]] * [[Sister::Resource:Wikinews]] * [[Sister::Resource:Wikiquote]] * [[Sister::Resource:Wikispecies]] 76389ddad9c596242f58942fb9499d48915650fa 65 63 2008-05-23T22:39:24Z Vipul 2 /* Wikipedia as a free resource */ wikitext text/x-wiki Wikipedia is a free, online, open-content, collaborative, multilingual encyclopedia project. {{quotation|'''SLOGANS''': ''Wikipedia, the free encyclopedia''<br>''Wikipedia, the free encyclopedia that anyone can edit''<br>''Imagine a world in which every single human being can freely share in the sum of all knowledge. That's our commitment.''}} ==Use Wikipedia== ===Start Wikipedia=== Wikipedia is online, or Internet-based. It can be accessed using a web browser and an Internet connection. Main website: http://www.wikipedia.org Some major language websites: * English: http://en.wikipedia.org * French: http://fr.wikipedia.org * German: http://de.wikipedia.org * Polish: http://pl.wikipedia.org A complete list of language Wikipedias is available at: http://meta.wikimedia.org/wiki/List_of_Wikipedias ===Search Wikipedia=== # http://www.wikipedia.org allows search in any of the language Wikipedias # In any of the language Wikipedias, there is a search box in the left column that allows any search # http://en.wikipedia.org/Special:Search is the English Wikipedia's internal site search # If using Mozilla Firefox as the browser, you can add Wikipedia (English) to the list of Search Engines using https://addons.mozilla.org/ ===Article names and reaching a given article=== Wikipedia content is organized into articles. A Wikipedia article on a topic is named exactly by that topic. The URL for a given Wikipedia article in the English Wikipedia is given by: http://en.wikipedia.org/wiki/Name where Name denotes the name of the article. Thus, for instance, the English Wikipedia article on India is titled [[wikipedia:India|India]], and the full URL is given by: http://en.wikipedia.org/wiki/India To go to an article with a particular title, do any of the following: # Type the full URL: http://en.wikipedia.org/wiki/Name # Use the Wikipedia Search Engine for Firefox and type the name of the topic in the search bar # Open Wikipedia and type the name of the article in the search bar, and press Go (or enter) ==More about Wikipedia for users== See also: [[Wikipedia:Wikipedia:Introduction|The English Wikipedia's introduction page]] ===Criteria for Wikipedia content=== Some important Wikipedia policies for content (nutshells as stated on the Wikipedia guideline pages): # [[Wikipedia:Wikipedia:Neutral point of view|Neutral Point of View policy]]: All Wikipedia articles and other encyclopedic content must be written from a neutral point of view, representing significant views fairly, proportionately and without bias. # [[Wikipedia:Wikipedia:No original research|No original research]]: #* Wikipedia does not publish original thought: all material in Wikipedia must be attributable to a reliable, published source. #* Articles may not contain any new analysis or synthesis of published material that serves to advance a position not clearly advanced by the sources. # [[Wikipedia:Wikipedia:Notability|Notability]]: If a topic has received significant coverage in reliable secondary sources that are independent of the subject, it is presumed to be notable. # [[Wikipedia:Wikipedia:What Wikipedia is not|What Wikipedia is not]]: Gives a list of things Wikipedia is not suitable for. # [[Wikipedia:Wikipedia:Verifiability|Verifiability policy]]: Material challenged or likely to be challenged, and all quotations, must be attributed to a reliable, published source. These criteria are taken as guidelines, and not every page on Wikipedia may conform to the criteria at any given time. ===Using Wikipedia=== Some pieces of information put up by Wikipedia for Wikipedia users: # [[Wikipedia:Wikipedia:Ten things you may not know about Wikipedia|Ten things about Wikipedia]]: Important information about Wikipedia # [[Wikipedia:Wikipedia:Researching with Wikipedia|Researching with Wikipedia]]: Information and suggestions about how to use Wikipedia for research # [[Wikipedia:Wikipedia:Basic navigation|Navigating and finding information on Wikipedia]] ==Wikipedia as a free resource== ===Free access=== All content on Wikipedia can be accessed for free (i.e., without any access or subscription fees). ===Copyright and reuse=== Any contributions made to Wikipedia are automatically considered to be released under the GNU Free Documentation License ([[license::GFDL]]), version 1.2. The GFDL is a ''copyleft'', ''open-content'' license, whose key feature is that it allows reuse without permission from the original author, as long as such reuse is also licensed under a similar license. Main reference pages: # [[Wikipedia:Wikipedia:Copyrights]] explains Wikipedia's copyright policy # [[Wikipedia:Wikipedia:Text of the GNU Free Documentation License]] is the full text of the license used by Wikipedia ==Management and organization== General information in this direction is available at: http://en.wikipedia.org/wiki/Wikipedia:About ===Parent organization=== Wikipedia is managed by the [[parent organization::Wikimedia Foundation]], a non-profit organization. The Wikimedia Foundation is responsible both for the servers that host Wikipedia content, and for maintaining and improving MediaWiki, the underlying software for Wikipedia. Main reference pages: # http://www.wikimedia.org is the Wikimedia Foundation's central landing page (listing Wikipedia and other sister projects) # http://www.wikimediafoundation.org is the Wikimedia Foundation's wiki # http://meta.wikimedia.org is a wiki run by the Wikimedia Foundation to coordinate the Wikimedia Foundation's projects ===Software tools=== Wikipedia runs on [[software::MediaWiki]], a software designed for web-base collaborative authoring of articles. MediaWiki is ''free software'', and is licensed under the GNU General Public License, Version 2.0. MediaWiki was originally written by Magnus Manske, specifically for Wikipedia. Its development is handled by software professionals working for the Wikimedia Foundation. However, MediaWiki is free software and can be both used and modified by anybody. A number of extensions to MediaWiki have been developed by people outside the Wikimedia Foundation. MediaWiki is also used as a software in a number of other collaborative software-based tools, such as the Subject Wikis Reference Guide. # http://www.mediawiki.org is the MediaWiki website. It has the MediaWiki manual, links for downloading the software, as well as a listing of extensions developed for MediaWiki # http://www.wikiindex.org gives a list of wikis (not comprehensive) ===Financial model=== The Wikimedia Foundation (Wikipedia's parent organization) is a non-profit organization (registered under Section 501 (c)(3)) founded in St. Petersburg, Florida, United States in 2002 and headquartered in San Francisco, California, United States. The financial model of the Wikimedia Foundation primarily rests on voluntary donation. # http://wikimediafoundation.org/wiki/Donate The donate page ==Production model== ===Collaborative content creation=== Wikipedia uses a ''strongly collaborative'' model of content creation, powered by a wiki software called MediaWiki. Some of the key features of the model used on Wikipedia: # Anybody can register an account on Wikipedia, without necessarily revealing his/her real name and identity # Anybody can start an article on a topic (as of now, only logged-in users can start new articles) if no article on the topic exists # Once an article is created, anybody (and not just the original author of the article) can edit the article # Every revision of the article is stored on the wiki, and different revisions can be viewed and compared using the "history" tab of the article # People seeking to collaborate on a given article can discuss material related to the article on its "discussion page" or "talk page", which can be accessed using the "discussion" tab of the article # A person can nominate an article for deletion on grounds that it violates Wikipedia policy. The issue is then discussed on a separate page, till a consensus is reached either way Relevant links: # http://en.wikipedia.org/w/index.php?title=Special:UserLogin&type=signup is the page for account creation on English Wikipedia # [[Wikipedia:Help:Editing|Wikipedia Editing help]] ===Compensation model=== Wikipedia does not, as a rule, compensate people for writing articles. In particular: # There is no monetary compensation or reimbursement for writing articles on Wikipedia # No credits for an article are given on the article page itself. The authorship of a given article may be seen using the "history" tab or the "credits" option. However, the information revealed by the history may b eincomplete because anonymous users may have made certain edits, and even the registered users may not have used their real names and identities ===Security model=== Wikipedia relies on soft security to combat problems such as vandalism and bias in articles. Dedicated volunteers patrol the wiki to locate acts of vandalism, or content that otherwise violates Wikipedia policy. User accounts or IP addresses associated with such editing may be banned from editing Wikipedia. ==Related resources== ===Competitors=== * [[Competitor::Resource:Encyclopædia Britannica]]: Managed by Britannica Corporation, this has been one of the world's leading encyclopedia resources for many centuries. * [[Competitor::Resource:Citizendium]] ===Sister projects=== * [[Sister::Resource:Wikibooks]] * [[Sister::Resource:Wiktionary]] * [[Sister::Resource:Wikiverse]] * [[Sister::Resource:Wikisource]] * [[Sister::Resource:Wikiversity]] * [[Sister::Resource:Wikinews]] * [[Sister::Resource:Wikiquote]] * [[Sister::Resource:Wikispecies]] c9916569c6bb143014a18bbd341916a432c30eac 10 30 64 2008-05-23T22:35:53Z Vipul 2 New page: <blockquote class="toccolours" style="float:none; padding: 10px 15px 10px 15px; display:table;"> {{{1<noinclude>|&nbsp;&nbsp;{{Lorem}}</noinclude>}}}</blockquote><noinclude>''This template... wikitext text/x-wiki <blockquote class="toccolours" style="float:none; padding: 10px 15px 10px 15px; display:table;"> {{{1<noinclude>|&nbsp;&nbsp;{{Lorem}}</noinclude>}}}</blockquote><noinclude>''This template is copied from Wikipedia (source [[wp:Template:Quotation|here]]) and is released under the GFDL 1.2''</noinclude> ccf615a7461b17e4b9a39d37b8603c286b3b6878 0 31 66 2008-06-08T11:59:36Z Vipul 2 New page: Main forms: ''perfect'', adjective Related forms: ''perfectness'' (extent to which something is perfect), ''perfection'' (being perfect) Typical use: * Something special, good in a cert... wikitext text/x-wiki Main forms: ''perfect'', adjective Related forms: ''perfectness'' (extent to which something is perfect), ''perfection'' (being perfect) Typical use: * Something special, good in a certain peculiar way ==Economics== {{:Perfect (economics)}} ==Mathematics== {{:Perfect (mathematics)}} f1d0de58740b66a79d7a22818fe1a132978c9b46 0 32 67 2008-06-08T12:04:11Z Vipul 2 New page: ''Dimension'' in mathematics is a generalization of the idea that a point is zero-dimensional, a line is one-dimensional, a plane is two-dimensional, and space is three-dimensional. Loosel... wikitext text/x-wiki ''Dimension'' in mathematics is a generalization of the idea that a point is zero-dimensional, a line is one-dimensional, a plane is two-dimensional, and space is three-dimensional. Loosely, ''dimension'' describes the number of independent freely varying parameters. ===In linear algebra=== {{:Dimension of a vector space}} ===In commutative and noncommutative algebra=== {{:Krull dimension}} {{:Homological dimension of a module}} {{:Cohomological dimension of a module}} {{:Global dimension of a ring}} ===In point-set topology and algebraic topology=== {{fillin}} 028dd3bc12c6ef7e5929763887f49a9bb4d20421 71 67 2008-06-08T12:13:59Z Vipul 2 /* In point-set topology and algebraic topology */ wikitext text/x-wiki ''Dimension'' in mathematics is a generalization of the idea that a point is zero-dimensional, a line is one-dimensional, a plane is two-dimensional, and space is three-dimensional. Loosely, ''dimension'' describes the number of independent freely varying parameters. ===In linear algebra=== {{:Dimension of a vector space}} ===In commutative and noncommutative algebra=== {{:Krull dimension}} {{:Homological dimension of a module}} {{:Cohomological dimension of a module}} {{:Global dimension of a ring}} ===In point-set topology=== {{:Topological dimension}} {{:Large inductive dimension}} {{:Small inductive dimension}} ===In differential geometry=== {{:Dimension of a manifold}} 1431ed493a1d2d601ba75fc7efacebaf652a7d6c 0 33 68 2008-06-08T12:06:49Z Vipul 2 New page: <noinclude>[[Status::Basic definition| ]][[Topic::Linear algebra| ]]</noinclude> '''Dimension of a vector space''': The dimension of a vector space over a field equals the cardinality of a... wikitext text/x-wiki <noinclude>[[Status::Basic definition| ]][[Topic::Linear algebra| ]]</noinclude> '''Dimension of a vector space''': The dimension of a vector space over a field equals the cardinality of a basis for that vector space (we can always find a basis for any vector space, using the axiom of choice, and any two bases have equal cardinality). A field has dimension one as a vector space over itself. Dimension of a direct sum of vector spaces is the sum of their dimensions, and dimension of a tensor product of vector spaces is the product of their dimensions. No subject wiki entry 80a8a6cb22a2bf85b7cb25db32933063312e12bd 0 34 69 2008-06-08T12:10:44Z Vipul 2 New page: <noinclude>[[Status:Semi-basic definition| ]][[Topic::Commutative algebra| ]][[Primary wiki::Commalg| ]]</noinclude> '''Krull dimension''': The Krull dimension of a commutative unital rin... wikitext text/x-wiki <noinclude>[[Status:Semi-basic definition| ]][[Topic::Commutative algebra| ]][[Primary wiki::Commalg| ]]</noinclude> '''Krull dimension''': The Krull dimension of a commutative unital ring is the maximum possible length of an ascending chain of prime ideals in the ring. Primary subject wiki entry: [[Commalg:Krull dimension]] Also located at: [[Mathworld:KrullDimension]], [[Planetmath:KrullDimension]], [[Wikipedia:Krull dimension]] ae07d76106073e3742b60dcba4263f2fad811ae1 70 69 2008-06-08T12:11:03Z Vipul 2 wikitext text/x-wiki <noinclude>[[Status::Semi-basic definition| ]][[Topic::Commutative algebra| ]][[Primary wiki::Commalg| ]]</noinclude> '''Krull dimension''': The Krull dimension of a commutative unital ring is the maximum possible length of an ascending chain of prime ideals in the ring. Primary subject wiki entry: [[Commalg:Krull dimension]] Also located at: [[Mathworld:KrullDimension]], [[Planetmath:KrullDimension]], [[Wikipedia:Krull dimension]] d61524f151836ef83d897af6d6f594bad7c7855a 0 35 72 2008-06-08T12:20:24Z Vipul 2 New page: <noinclude>[[Status::Standard non-basic definition| ]][[Topic::Point-set topology| ]][[Primary wiki::Topospaces| ]]</noinclude> ''Topological dimension''': The '''topological dimension'''... wikitext text/x-wiki <noinclude>[[Status::Standard non-basic definition| ]][[Topic::Point-set topology| ]][[Primary wiki::Topospaces| ]]</noinclude> ''Topological dimension''': The '''topological dimension''' or '''covering dimension''' or '''Lebesgue covering dimension''' of a [[topological space]] is defined as the smallest integer <math>m</math> such that any [[open cover]] of the topological space has an open [[refinement]] that has [[order of a collection of subsets|order]] at most <math>m + 1</math>. Primary subject wiki entry: [[Topospaces:Topological dimension]] Also located at: [[Wikipedia:Lebesgue covering dimension]] ce122302b1a77c72ed4a51e1f8031e62099c9dce 73 72 2008-06-08T12:20:38Z Vipul 2 wikitext text/x-wiki <noinclude>[[Status::Standard non-basic definition| ]][[Topic::Point-set topology| ]][[Primary wiki::Topospaces| ]]</noinclude> '''Topological dimension''': The '''topological dimension''' or '''covering dimension''' or '''Lebesgue covering dimension''' of a [[topological space]] is defined as the smallest integer <math>m</math> such that any [[open cover]] of the topological space has an open [[refinement]] that has [[order of a collection of subsets|order]] at most <math>m + 1</math>. Primary subject wiki entry: [[Topospaces:Topological dimension]] Also located at: [[Wikipedia:Lebesgue covering dimension]] f80abb3517d170d721d7e16a4ff866bd55dd15a5 0 36 74 2008-06-08T12:24:22Z Vipul 2 New page: ===In commutative algebra=== {{:Degree of a polynomial}} {{:Degree of a rational function}} ===In algebraic topology=== Degree of a map between compact, connected, oriented manifolds ... wikitext text/x-wiki ===In commutative algebra=== {{:Degree of a polynomial}} {{:Degree of a rational function}} ===In algebraic topology=== Degree of a map between compact, connected, oriented manifolds Degree of a self-map on a compact, connected, orientable manifold ===In differential geometry=== Degree of a map between compact, oriented differential manifolds ===In complex analysis=== Degree of an analytic map between compact complex manifolds 0fe69d2a03f9b9ff4a41e321b77c2de3cb56b5e9 0 37 75 2008-06-08T12:34:05Z Vipul 2 New page: ===Typical uses in elementary mathematics=== {{:Product of numbers}} {{:Product of functions}} ===In a semigroup, monoid or group=== {{:Product in a semigroup}} {{:Product in a ring}}... wikitext text/x-wiki ===Typical uses in elementary mathematics=== {{:Product of numbers}} {{:Product of functions}} ===In a semigroup, monoid or group=== {{:Product in a semigroup}} {{:Product in a ring}} ===In a category=== {{:Direct product of groups}} {{:Direct product in a variety of algebras}} {{:Categorical product}} 057c5d4ceb63ac736646459c40cc0cd7934cef12 76 75 2008-06-08T12:35:42Z Vipul 2 wikitext text/x-wiki The term ''product'' is typically used for the result obtained on performing a ''multiplication''. Here, the multiplication is a binary operation on a set or structure. The multiplication is usually taken to be associative (so that parenthesization does not matter for the product) and often taken to be commutative (so that the order of terms doesn't matter). ===Typical uses in elementary mathematics=== {{:Product of numbers}} {{:Product of functions}} ===In a semigroup, monoid or group=== {{:Product in a semigroup}} {{:Product in a ring}} ===In a category=== {{:Direct product of groups}} {{:Direct product in a variety of algebras}} {{:Categorical product}} 7db2e29a5ae04ebffec968082ed1463513f776e5 0 38 77 2008-06-08T12:41:33Z Vipul 2 New page: The term ''prime'' is typically used to denote ''something that cannot be decomposed into smaller pieces''. Alternative words used in mathematics to describe such things are [[indecomposab... wikitext text/x-wiki The term ''prime'' is typically used to denote ''something that cannot be decomposed into smaller pieces''. Alternative words used in mathematics to describe such things are [[indecomposable (mathematics)|indecomposable]], [[simple (mathematics)|simple]], [[irreducible (mathematics)|irreducible]]. It is not necessary that every object is built from prime objects, though in various specific cases, that does happen; for instance, the fundamental theorem of arithmetic states that every natural number is a product of primes in a unique way (upto ordering). ===In elementary mathematics/number theory=== {{:Prime number}} ===In knot theory=== {{:Prime knot}} ===In field theory=== {{:Prime field}} ===In commutative algebra=== {{:Prime ideal}} {{:Prime element}} ===In noncommutative algebra=== {{:Prime ring}} ===In universal algebra=== {{:Prime variety}} 9d39d9180c7573d68ee24161698d48da17b42695 0 39 78 2008-06-08T12:44:02Z Vipul 2 New page: Main form: ''prime'', adjective Related forms: ''primality'' (whether or not something is prime) Typical use: * Something first, optimal, or best * Something smallest, indecomposable, o... wikitext text/x-wiki Main form: ''prime'', adjective Related forms: ''primality'' (whether or not something is prime) Typical use: * Something first, optimal, or best * Something smallest, indecomposable, or atomic ==Mathematics== {{:Prime (mathematics)}} d0bbdcc6c041d8b335aa534dda1e9d9707e0a3ef 0 40 79 2008-06-08T12:45:58Z Vipul 2 New page: Main form: ''simple'', adjective Related forms: ''simplicity'' (whether or not something is simple, extent to which something is simple) ==Mathematics== {{:Simple (mathematics)}} wikitext text/x-wiki Main form: ''simple'', adjective Related forms: ''simplicity'' (whether or not something is simple, extent to which something is simple) ==Mathematics== {{:Simple (mathematics)}} 33b531842005fb2fb34cb707f8efb5eb33f25f30 0 41 80 2008-06-08T12:48:52Z Vipul 2 New page: ===In commutative algebra=== {{:Principal ideal}} {{:Principal ideal ring}} ===In complex analysis=== {{:Principal part of a meromorphic function}} {{:Principal divisor}} wikitext text/x-wiki ===In commutative algebra=== {{:Principal ideal}} {{:Principal ideal ring}} ===In complex analysis=== {{:Principal part of a meromorphic function}} {{:Principal divisor}} 28a2b664e395e32b0492123b8cd178f4549144dc 0 42 81 2008-06-09T13:07:06Z Vipul 2 Redirecting to [[Normal]] wikitext text/x-wiki #redirect [[Normal]] c15e0668b0c1a1d255e5d8e61d80060a4098ea86 0 6 82 12 2008-06-09T13:14:10Z Vipul 2 wikitext text/x-wiki Main form: ''normal'', adjective Related forms: ''normality'' (how normal something is), ''normalcy'', ''normalize'' (make normal) Same roots: [[Root related::norm]] Typical use: * Something typical, expected, or standard. Similar words: [[Similar::ordinary]], [[Similar::typical]], [[Similar::standard]], [[Similar::average]] * Something good or desirable, or something one optimistically hopes for. Similar words: [[Similar::conventional]], [[Similar::standard]] * Upright or perpendicular. [[Similar::orthogonal]], [[Similar::perpendicular]] ==Mathematics== {{:Normal (mathematics)}} ==Physics== {{:Normal (physics)}} ==Chemistry== {{:Normal (chemistry)}} 8640e4bdb208033536f87d5e108ef1694cc98595 83 82 2008-06-09T13:24:49Z Vipul 2 wikitext text/x-wiki Main form: ''normal'', [[verb type::adjective]]. Used both descriptively (adding detail) and restrictively (restricting the subject). In sciences, typically used in a binary sense (something is either normal or is not normal), whereas in the social sciences and daily parlance, typically used to describe position on a wider scale. Related forms: ''normality'' (how normal something is), ''normalcy'' (the condition of being normal), ''normalize'' (make normal) Same roots: [[Root related::norm]] Typical use: * Something typical, expected, or standard. Similar words: [[Similar::ordinary]], [[Similar::typical]], [[Similar::standard]], [[Similar::average]] * Something good or desirable, or something one optimistically hopes for. Similar words: [[Similar::conventional]], [[Similar::standard]] * Upright or perpendicular. [[Similar::orthogonal]], [[Similar::perpendicular]] Opposite words: [[Opposite::Abnormal]], [[Opposite::Paranormal]] ==Mathematics== {{:Normal (mathematics)}} ==Physics== {{:Normal (physics)}} ==Chemistry== {{:Normal (chemistry)}} 283b35cde5aa48589dc1ea6f4a8a2a781c9a3c20 84 83 2008-06-09T13:26:52Z Vipul 2 wikitext text/x-wiki Main form: ''normal'', [[verb type::adjective]]. Used both descriptively (adding detail) and restrictively (restricting the subject). In sciences, typically used in a binary sense (something is either normal or is not normal), whereas in the social sciences and daily parlance, typically used to describe position on a wider scale. Related forms: ''normality'' (how normal something is), ''normalcy'' (the condition of being normal), ''normalize''/''normalise'' (make normal) Same roots: [[Root related::norm]] Typical use: * Something typical, expected, or standard. Similar words: [[Similar::ordinary]], [[Similar::typical]], [[Similar::standard]], [[Similar::average]] * Something good or desirable, or something one optimistically hopes for. Similar words: [[Similar::conventional]], [[Similar::standard]] * Upright or perpendicular. [[Similar::orthogonal]], [[Similar::perpendicular]] Opposite words: [[Opposite::Abnormal]], [[Opposite::Paranormal]] ==Mathematics== {{:Normal (mathematics)}} ==Physics== {{:Normal (physics)}} ==Chemistry== {{:Normal (chemistry)}} 3d01c342eb09c98adada9b3f2c9f023f04a90e00 85 84 2008-06-09T13:28:00Z Vipul 2 wikitext text/x-wiki Main form: ''normal'', [[verb type::adjective]]. Used both descriptively (adding detail) and restrictively (restricting the subject). In sciences, typically used in a binary sense (something is either normal or is not normal), whereas in the social sciences and daily parlance, typically used to describe position on a wider scale. Related forms: ''normality'' (how normal something is), ''normalcy'' (the condition of being normal), ''normalize''/''normalise'' (make normal) Same roots: [[Root related::norm]] Typical use: * Something typical, expected, or standard. Similar words: [[Similar::ordinary]], [[Similar::typical]], [[Similar::standard]], [[Similar::average]] * Something good or desirable, or something one optimistically hopes for. Similar words: [[Similar::conventional]], [[Similar::standard]] * Upright or perpendicular. [[Similar::orthogonal]], [[Similar::perpendicular]] Opposite words: [[Opposite::abnormal]], [[Opposite::paranormal]] ==Mathematics== {{:Normal (mathematics)}} ==Physics== {{:Normal (physics)}} ==Chemistry== {{:Normal (chemistry)}} e440cc8099a9e84bff4c29756b88b9e0669eece2 0 43 86 2008-06-09T13:46:58Z Vipul 2 New page: <noinclude>[[Status::Basic definition| ]][[Topic::Linear algebra| ]]</noinclude> '''Normal form''': also called canonical form, is a standard form in which to put a matrix (typically, upt... wikitext text/x-wiki <noinclude>[[Status::Basic definition| ]][[Topic::Linear algebra| ]]</noinclude> '''Normal form''': also called canonical form, is a standard form in which to put a matrix (typically, upto conjugation). Two normal forms commonly used are the Jordan normal form and the rational normal form. fb6e4fd45bf2d2e75a939eb81ef80a4c5a0aa826 0 3 87 20 2008-06-09T13:48:30Z Vipul 2 wikitext text/x-wiki The term ''normal'' in mathematics is used in the following broad senses: * To denote something upright or perpendicular * To denote something that is as it ''should be''. In this sense, normal means ''good'' or ''desirable'' rather than ''typical'' It is typically used as a limiting adjective, and in a binary sense. In other words, an object in a certain class is ''normal'' if it satisfies certain conditions, and every object in that class is either normal or not normal. The noun form is ''normality''. Thus, normality can be viewed as a property in various contexts. ===In group theory=== {{:Normal subgroup}} ===In topology=== {{:Normal space}} ===In linear algebra=== {{:Normal form (linear algebra)}} {{:Normal matrix}} {{:Normal operator}} ===In Galois theory=== [[Find link::Normal field extension]]: A field extension is termed normal if the fixed field under the automorphism group of the extension, is precisely the base field. ===In commutative algebra=== [[Commalg:Normal domain|Normal domain]]: An integral domain is termed normal if it is integrally closed in its field of fractions. [[Commalg:Normal ring|Normal ring]]: A commutative unital ring is termed normal if it is integrally closed in its total quotient ring ===In differential geometry=== Normal to a curve in the plane: The line perpendicular to the tangent to the curve at a point. [[Diffgeom:Normal bundle|Normal bundle]] to a submanifold, or immersed manifold: The quotient of the tangent bundle to the whole manifold (restricted to the submanifold), by the tangent bundle to the submanifold, or immersed manifold. When the manifolds are Riemannian, the normal bundle can be viewed as the orthogonal complement to the tangent bundle of the submanifold, in the tangent bundle to the whole manifold. [[Diffgeom:Normal coordinate system|Normal coordinate system]] refers to a particular kind of local coordinate system for a Riemannian manifold. Vectors in the normal bundle are termed normal vectors. ===In normed vector spaces=== In general normed vector spaces, as well as in tangent spaces to Riemannian manifolds, ''normalize'' means to scale by a factor to make the norm equal to 1. ===In probability/statistics=== [[Find link::Normal distribution]]: A particular kind of distribution parametrized by mean and variance, also called the Gaussian distribution. [[Find link::Normal random variable]]: A random variable whose distribution is a normal distribution. ===Others=== [[mathworld:NormalNumber|Normal number]] to base <math>b</math>: A real number that, when written in base <math>b</math>, has all digits occurring with equal probability. [[mathworld:AbsolutelyNormal|Absolutely normal number]]: A real number that is positive to base <math>b</math> for every integer <math>b > 1</math>. 35e6ae43cf68ffa74fca67dd6062b071056ee2b9 95 87 2008-06-09T13:59:04Z Vipul 2 wikitext text/x-wiki The term ''normal'' in mathematics is used in the following broad senses: * To denote something upright or perpendicular * To denote something that is as it ''should be''. In this sense, normal means ''good'' or ''desirable'' rather than ''typical'' It is typically used as a limiting adjective, and in a binary sense. In other words, an object in a certain class is ''normal'' if it satisfies certain conditions, and every object in that class is either normal or not normal. The noun form is ''normality''. Thus, normality can be viewed as a property in various contexts. ===In group theory=== {{:Normal subgroup}} ===In topology=== {{:Normal space}} ===In linear algebra=== {{:Normal form (linear algebra)}} {{:Normal matrix}} {{:Normal operator}} ===In Galois theory=== {{:Normal field extension}} ===In commutative algebra=== {{:Normal domain}} {{:Normal ring}} ===In differential geometry=== Normal to a curve in the plane: The line perpendicular to the tangent to the curve at a point. [[Diffgeom:Normal bundle|Normal bundle]] to a submanifold, or immersed manifold: The quotient of the tangent bundle to the whole manifold (restricted to the submanifold), by the tangent bundle to the submanifold, or immersed manifold. When the manifolds are Riemannian, the normal bundle can be viewed as the orthogonal complement to the tangent bundle of the submanifold, in the tangent bundle to the whole manifold. [[Diffgeom:Normal coordinate system|Normal coordinate system]] refers to a particular kind of local coordinate system for a Riemannian manifold. Vectors in the normal bundle are termed normal vectors. ===In normed vector spaces=== In general normed vector spaces, as well as in tangent spaces to Riemannian manifolds, ''normalize'' means to scale by a factor to make the norm equal to 1. ===In probability/statistics=== [[Find link::Normal distribution]]: A particular kind of distribution parametrized by mean and variance, also called the Gaussian distribution. [[Find link::Normal random variable]]: A random variable whose distribution is a normal distribution. ===Others=== [[mathworld:NormalNumber|Normal number]] to base <math>b</math>: A real number that, when written in base <math>b</math>, has all digits occurring with equal probability. [[mathworld:AbsolutelyNormal|Absolutely normal number]]: A real number that is positive to base <math>b</math> for every integer <math>b > 1</math>. 6d476d4eb85137dc73884de7e39aa48645352f86 98 95 2008-06-09T14:09:42Z Vipul 2 /* In differential geometry */ wikitext text/x-wiki The term ''normal'' in mathematics is used in the following broad senses: * To denote something upright or perpendicular * To denote something that is as it ''should be''. In this sense, normal means ''good'' or ''desirable'' rather than ''typical'' It is typically used as a limiting adjective, and in a binary sense. In other words, an object in a certain class is ''normal'' if it satisfies certain conditions, and every object in that class is either normal or not normal. The noun form is ''normality''. Thus, normality can be viewed as a property in various contexts. ===In group theory=== {{:Normal subgroup}} ===In topology=== {{:Normal space}} ===In linear algebra=== {{:Normal form (linear algebra)}} {{:Normal matrix}} {{:Normal operator}} ===In Galois theory=== {{:Normal field extension}} ===In commutative algebra=== {{:Normal domain}} {{:Normal ring}} ===In differential geometry=== Normal to a curve in the plane: The line perpendicular to the tangent to the curve at a point. {{:Normal bundle}} {{:Normal coordinate system}} ===In normed vector spaces=== In general normed vector spaces, as well as in tangent spaces to Riemannian manifolds, ''normalize'' means to scale by a factor to make the norm equal to 1. ===In probability/statistics=== [[Find link::Normal distribution]]: A particular kind of distribution parametrized by mean and variance, also called the Gaussian distribution. [[Find link::Normal random variable]]: A random variable whose distribution is a normal distribution. ===Others=== [[mathworld:NormalNumber|Normal number]] to base <math>b</math>: A real number that, when written in base <math>b</math>, has all digits occurring with equal probability. [[mathworld:AbsolutelyNormal|Absolutely normal number]]: A real number that is positive to base <math>b</math> for every integer <math>b > 1</math>. 22bf0002aa79a414243c97a9f57d5af1c7bc7c08 99 98 2008-06-09T14:11:36Z Vipul 2 wikitext text/x-wiki The term ''normal'' in mathematics is used in the following broad senses: * To denote something upright or perpendicular * To denote something that is as it ''should be''. In this sense, normal means ''good'' or ''desirable'' rather than ''typical'' It is typically used as a limiting adjective, and in a binary sense. In other words, an object in a certain class is ''normal'' if it satisfies certain conditions, and every object in that class is either normal or not normal. The noun form is ''normality''. Thus, normality can be viewed as a property in various contexts. ===In group theory=== {{:Normal subgroup}} ===In topology=== {{:Normal space}} ===In linear algebra=== {{:Normal form (linear algebra)}} {{:Normal matrix}} {{:Normal operator}} ===In Galois theory=== {{:Normal field extension}} ===In commutative algebra=== {{:Normal domain}} {{:Normal ring}} ===In differential geometry=== Normal to a curve in the plane: The line perpendicular to the tangent to the curve at a point. {{:Normal bundle}} {{:Normal coordinate system}} ===In category theory=== {{:Normal monomorphism}} {{:Normal epimorphism}} ===In probability/statistics=== {{:Normal distribution}} ===Others=== {{:Normal number}} 4e4d4135f47a09baf2fa57b7efbf54a691204097 100 99 2008-06-09T14:12:47Z Vipul 2 wikitext text/x-wiki The term ''normal'' in mathematics is used in the following broad senses: * To denote something upright or perpendicular * To denote something that is as it ''should be''. In this sense, normal means ''good'' or ''desirable'' rather than ''typical'' It is typically used as a limiting adjective, and in a binary sense. In other words, an object in a certain class is ''normal'' if it satisfies certain conditions, and every object in that class is either normal or not normal. The noun form is ''normality''. Thus, normality can be viewed as a property in various contexts. ===In group theory=== {{:Normal subgroup}} ===In topology=== {{:Normal space}} ===In linear algebra=== {{:Normal form (linear algebra)}} {{:Normal matrix}} {{:Normal operator}} ===In field theory=== {{:Normal field extension}} ===In commutative algebra=== {{:Normal domain}} {{:Normal ring}} ===In differential geometry=== Normal to a curve in the plane: The line perpendicular to the tangent to the curve at a point. {{:Normal bundle}} {{:Normal coordinate system}} ===In category theory=== {{:Normal monomorphism}} {{:Normal epimorphism}} ===In probability/statistics=== {{:Normal distribution}} ===Others=== {{:Normal number}} 999acca2d1a1b61cc26646fcce0a21cf1de871fb 0 44 88 2008-06-09T13:51:11Z Vipul 2 New page: <noinclude>[[Status::Semi-basic definition| ]][[Topic::Linear algebra| ]]</noinclude> '''Normal matrix''': A matrix over the complex numbers, that commutes with its conjugate-transpose. ... wikitext text/x-wiki <noinclude>[[Status::Semi-basic definition| ]][[Topic::Linear algebra| ]]</noinclude> '''Normal matrix''': A matrix over the complex numbers, that commutes with its conjugate-transpose. No primary subject wiki entry. Also located at: [[Wikipedia:Normal matrix]], [[Mathworld:NormalMatrix]], [[Planetmath:NormalMatrix]] 0dd374c88317e13b35334a8dfd4c368d9f6e5fd9 0 45 89 2008-06-09T13:55:09Z Vipul 2 New page: <noinclude>[[Status::Basic definition| ]][[Topic::Field theory| ]]</noinclude> '''Normal field extension''': A field extension is termed normal if the fixed field under the automorphism g... wikitext text/x-wiki <noinclude>[[Status::Basic definition| ]][[Topic::Field theory| ]]</noinclude> '''Normal field extension''': A field extension is termed normal if the fixed field under the automorphism group of the extension, is precisely the base field. No primary subject wiki entry. Also located at: [[Planetmath:NormalExtension]], [[Wikipedia::Normal extension]], [[Mathworld:NormalExtension]] 0ace5b4bef412c0b5f63391b451f325a7dc18347 90 89 2008-06-09T13:55:31Z Vipul 2 wikitext text/x-wiki <noinclude>[[Status::Basic definition| ]][[Topic::Field theory| ]]</noinclude> '''Normal field extension''': A field extension is termed normal if the fixed field under the automorphism group of the extension, is precisely the base field. No primary subject wiki entry. Also located at: [[Planetmath:NormalExtension]], [[Wikipedia:Normal extension]], [[Mathworld:NormalExtension]] 1168380a3a566a83eabd24556ef60310a36ac9ca 0 46 91 2008-06-09T13:55:50Z Vipul 2 Redirecting to [[Normal field extension]] wikitext text/x-wiki #redirect [[Normal field extension]] 94178a0d677f9748c1db241c7aa2d63c76d06cf5 0 47 92 2008-06-09T13:56:57Z Vipul 2 New page: <noinclude>[[Status::Semi-basic definition| ]][[Topic::Commutative algebra| ]][[Primary wiki::Commalg| ]]</noinclude> '''Normal domain''': An integral domain is termed normal if it is int... wikitext text/x-wiki <noinclude>[[Status::Semi-basic definition| ]][[Topic::Commutative algebra| ]][[Primary wiki::Commalg| ]]</noinclude> '''Normal domain''': An integral domain is termed normal if it is integrally closed in its field of fractions. 057d4f7a07a2ce51bf0a3e7b8c845d0be6367e36 93 92 2008-06-09T13:58:46Z Vipul 2 wikitext text/x-wiki <noinclude>[[Status::Semi-basic definition| ]][[Topic::Commutative algebra| ]][[Primary wiki::Commalg| ]]</noinclude> '''Normal domain''': An integral domain is termed normal if it is integrally closed in its field of fractions. Primary subject wiki entry: [[Commalg:Normal domain]] f9b1845f76b6ff4de0ca306b3e29b5d782879a22 0 48 94 2008-06-09T13:58:47Z Vipul 2 New page: <noinclude>[[Status::Semi-basic definition| ]][[Topic::Commutative algebra| ]][[Primary wiki::Commalg| ]]</noinclude> '''Normal ring''': A commutative unital ring is termed normal if it i... wikitext text/x-wiki <noinclude>[[Status::Semi-basic definition| ]][[Topic::Commutative algebra| ]][[Primary wiki::Commalg| ]]</noinclude> '''Normal ring''': A commutative unital ring is termed normal if it is integrally closed in its total quotient ring. Primary subject wiki entry: [[Commalg:Normal ring]] d356c8d482992586d16343f1334ed5a6b18819cf 0 49 96 2008-06-09T14:08:34Z Vipul 2 New page: <noinclude>[[Status::Semi-basic definition| ]][[Topic::Differential geometry| ]][[Primary wiki::Diffgeom| ]]</noinclude> '''Normal bundle''' to a submanifold, or immersed manifold: The qu... wikitext text/x-wiki <noinclude>[[Status::Semi-basic definition| ]][[Topic::Differential geometry| ]][[Primary wiki::Diffgeom| ]]</noinclude> '''Normal bundle''' to a submanifold, or immersed manifold: The quotient of the tangent bundle to the whole manifold (restricted to the submanifold), by the tangent bundle to the submanifold, or immersed manifold. When the manifolds are Riemannian, the normal bundle can be viewed as the orthogonal complement to the tangent bundle of the submanifold, in the tangent bundle to the whole manifold. Main subject wiki entry: [[Diffgeom:Normal bundle]] Also located at: [[Wikipedia:Normal bundle]], [[Mathworld:NormalBundle]], [[Planetmath:NormalBundle]] eb1b30aefaa317c9a123ba54c90780ef9c63871c 97 96 2008-06-09T14:09:29Z Vipul 2 wikitext text/x-wiki <noinclude>[[Status::Semi-basic definition| ]][[Topic::Differential geometry| ]][[Primary wiki::Diffgeom| ]]</noinclude> '''Normal bundle''' to a submanifold, or immersed manifold: The quotient of the tangent bundle to the whole manifold (restricted to the submanifold), by the tangent bundle to the submanifold, or immersed manifold. When the manifolds are Riemannian, the normal bundle can be viewed as the orthogonal complement to the tangent bundle of the submanifold, in the tangent bundle to the whole manifold. Vectors in the normal bundle are termed normal vectors. Main subject wiki entry: [[Diffgeom:Normal bundle]] Also located at: [[Wikipedia:Normal bundle]], [[Mathworld:NormalBundle]], [[Planetmath:NormalBundle]] 94f78ef155f82f49a7400ccd7057be757b6ba96f 0 6 101 85 2008-06-09T14:14:12Z Vipul 2 wikitext text/x-wiki Main form: ''normal'', [[verb type::adjective]]. Used both descriptively (adding detail) and restrictively (restricting the subject). In sciences, typically used in a binary sense (something is either normal or is not normal), whereas in the social sciences and daily parlance, typically used to describe position on a wider scale. Related forms: ''normality'' (how normal something is), ''normalcy'' (the condition of being normal), ''normalize''/''normalise'' (make normal) Same roots: [[Root related::norm]] Typical use: * Something typical, expected, or standard. Similar words: [[Similar::ordinary]], [[Similar::typical]], [[Similar::standard]], [[Similar::average]] * Something good or desirable, or something one optimistically hopes for. Similar words: [[Similar::conventional]], [[Similar::standard]], [[Similar::canonical]] * Upright or perpendicular. [[Similar::orthogonal]], [[Similar::perpendicular]] Opposite words: [[Opposite::abnormal]], [[Opposite::paranormal]] ==Mathematics== {{:Normal (mathematics)}} ==Physics== {{:Normal (physics)}} ==Chemistry== {{:Normal (chemistry)}} 66bf2eaf5eb9ec75a6cde516ba5f1c52b19c636a 102 101 2008-06-09T14:16:31Z Vipul 2 wikitext text/x-wiki Main form: ''normal'', [[verb type::adjective]]. Used both descriptively (adding detail) and restrictively (restricting the subject). In sciences, typically used in a binary sense (something is either normal or is not normal), whereas in the social sciences and daily parlance, typically used to describe position on a wider scale. Related forms: ''normality'' (how normal something is), ''normalcy'' (the condition of being normal), ''normalize''/''normalise'' (make normal) Same roots: [[Root related::norm]] Typical use: * Something typical, expected, or standard. Similar words: [[Similar::ordinary]], [[Similar::typical]], [[Similar::standard]], [[Similar::average]] * Something good or desirable, or something one optimistically hopes for. Similar words: [[Similar::conventional]], [[Similar::standard]], [[Similar::canonical]] * Upright or perpendicular. [[Similar::orthogonal]], [[Similar::perpendicular]] Opposite words: [[Opposite::abnormal]], [[Opposite::paranormal]] Derived words: [[Derived::conormal]], [[Derived::abnormal]], [[Derived::subnormal]], [[Derived::supernormal]], [[Derived::paranormal]], [[Derived::contranormal]], [[Derived::malnormal]], [[Derived::Pseudonormal]], [[Derived::orthonormal]], [[Derived::seminormal]], [[Derived::quasinormal]] ==Mathematics== {{:Normal (mathematics)}} ==Physics== {{:Normal (physics)}} ==Chemistry== {{:Normal (chemistry)}} 978962b80c83bc287e47fac7c1bfe77cc571bee4 113 102 2008-06-09T19:55:10Z Vipul 2 wikitext text/x-wiki Main form: ''normal'', [[word type::adjective]]. Used both descriptively (adding detail) and restrictively (restricting the subject). In sciences, typically used in a binary sense (something is either normal or is not normal), whereas in the social sciences and daily parlance, typically used to describe position on a wider scale. Related forms: ''normality'' (how normal something is), ''normalcy'' (the condition of being normal), ''normalize''/''normalise'' (make normal) Same roots: [[Root related::norm]] Typical use: * Something typical, expected, or standard. Similar words: [[Similar::ordinary]], [[Similar::typical]], [[Similar::standard]], [[Similar::average]] * Something good or desirable, or something one optimistically hopes for. Similar words: [[Similar::conventional]], [[Similar::standard]], [[Similar::canonical]] * Upright or perpendicular. [[Similar::orthogonal]], [[Similar::perpendicular]] Opposite words: [[Opposite::abnormal]], [[Opposite::paranormal]] Derived words: [[Derived::conormal]], [[Derived::abnormal]], [[Derived::subnormal]], [[Derived::supernormal]], [[Derived::paranormal]], [[Derived::contranormal]], [[Derived::malnormal]], [[Derived::Pseudonormal]], [[Derived::orthonormal]], [[Derived::seminormal]], [[Derived::quasinormal]] ==Mathematics== {{:Normal (mathematics)}} ==Physics== {{:Normal (physics)}} ==Chemistry== {{:Normal (chemistry)}} 562be39cd67d10fca6c4a9bd6afc4a70ba54127b 118 113 2008-06-09T20:27:31Z Vipul 2 wikitext text/x-wiki Main form: ''normal'', [[word type::adjective]]. Used both descriptively (adding detail) and restrictively (restricting the subject). In sciences, typically used in a binary sense (something is either normal or is not normal), whereas in the social sciences and daily parlance, typically used to describe position on a wider scale. Related forms: ''normality'' (how normal something is), ''normalcy'' (the condition of being normal), ''normalize''/''normalise'' (make normal) Same roots: [[Root related::norm]] Typical use: * Something typical, expected, or standard. Similar words: [[Similar::ordinary]], [[Similar::typical]], [[Similar::standard]], [[Similar::average]] * Something good or desirable, or something one optimistically hopes for. Similar words: [[Similar::conventional]], [[Similar::standard]], [[Similar::canonical]] * Upright or perpendicular. [[Similar::orthogonal]], [[Similar::perpendicular]] Opposite words: [[Opposite::abnormal]], [[Opposite::paranormal]] Derived words: [[Derived::conormal]], [[Derived::abnormal]], [[Derived::subnormal]], [[Derived::supernormal]], [[Derived::paranormal]], [[Derived::contranormal]], [[Derived::malnormal]], [[Derived::Pseudonormal]], [[Derived::orthonormal]], [[Derived::seminormal]], [[Derived::quasinormal]] ==Mathematics== {{:Normal (mathematics)}} ==Physics== {{:Normal (physics)}} ==Chemistry== {{:Normal (chemistry)}} ==Economics== {{:Normal (economics)}} 95f9a8195151110bea13545f26ff2102154f632a 0 50 103 2008-06-09T14:26:11Z Vipul 2 New page: <noinclude>[[Status::Semi-basic definition| ]][[Topic::Point-set topology| ]][[Primary wiki::Topospaces| ]]</noinclude> '''Perfectly normal space''': A normal space where every closed sub... wikitext text/x-wiki <noinclude>[[Status::Semi-basic definition| ]][[Topic::Point-set topology| ]][[Primary wiki::Topospaces| ]]</noinclude> '''Perfectly normal space''': A normal space where every closed subset is an intersection of countably many open subsets containing it. Primary subject wiki entry: [[Topospaces:Perfectly normal space]] 084fa81dc06e1416d95ac2e3865a913dbe42a4a9 0 7 104 40 2008-06-09T14:26:27Z Vipul 2 wikitext text/x-wiki ===In group theory=== {{:Perfect group}} ===In topology=== {{:Perfect space}} {{:Perfectly normal space}} Perfect set: A set in a metric space that has no isolated points. ===In number theory=== [[Find link::Perfect power]], for instance, perfect square or perfect cube: A perfect <math>n^{th}</math> power is an integer that occurs as the <math>n^{th}</math> power of an integer. [[Find link::Perfect number]]: A natural number that equals the sum of all its proper divisors. ===In field theory=== [[Find link::Perfect field]]: A field that either has characteristic zero, or has <math>p</math> and <math>x \mapsto x^p</math> is a surjective map. ===In graph theory=== [[Find link::Perfect graph]]: A graph with the property that for every induced subgraph, the chromatic number equals the clique number. [[Find link::Perfect matching]]: A matching in a bipartite graph such that every element on one side gets matched to exactly one element on the other side. ===In measure theory=== {{:Perfect measure}} f0cd742760a3c1c846e4c3c7b6b6e2df4dd4f82d 108 104 2008-06-09T14:36:08Z Vipul 2 wikitext text/x-wiki ===In group theory=== {{:Perfect group}} ===In topology=== {{:Perfect space}} {{:Perfectly normal space}} Perfect set: A set in a metric space that has no isolated points. ===In number theory=== {{:Perfect power}} {{:Perfect number}} ===In field theory=== {{:Perfect field}} ===In graph theory=== {{:Perfect graph}} {{:Perfect matching}} ===In measure theory=== {{:Perfect measure}} 596beb7c3702e49c0a7e8acb46d44c8ce340d322 0 51 105 2008-06-09T14:32:08Z Vipul 2 New page: <noinclude>[[Status::Basic definition| ]][[Topic::Field theory| ]]</noinclude> '''Perfect field''': A field that either has characteristic zero, or has characteristic <math>p</math> and f... wikitext text/x-wiki <noinclude>[[Status::Basic definition| ]][[Topic::Field theory| ]]</noinclude> '''Perfect field''': A field that either has characteristic zero, or has characteristic <math>p</math> and for which the map <math>x \mapsto x^p</math> is a surjective map. Equivalently, it is a field such that every algebraic extension field for it is separable. No relevant subject wiki entry. Also located at: [[Mathworld:PerfectField]], [[Planetmath:PerfectField]] e8f4191461768a2a8e058c46ea7946c3dddd6466 0 52 106 2008-06-09T14:34:28Z Vipul 2 New page: <noinclude>[[Status::Semi-basic definition| ]][[Topic::Graph theory| ]]</noinclude> '''Perfect graph''': A graph with the property that for every induced subgraph, the chromatic number eq... wikitext text/x-wiki <noinclude>[[Status::Semi-basic definition| ]][[Topic::Graph theory| ]]</noinclude> '''Perfect graph''': A graph with the property that for every induced subgraph, the chromatic number equals the clique number. '''Term variations''': Strongly perfect graph No relevant subject wiki entry. Also located at: [[Wikipedia:Perfect graph]], [[Mathworld:PerfectGraph]] 431f103fdbb98e728e6116839079ebf1c1a6bc98 0 53 107 2008-06-09T14:35:39Z Vipul 2 New page: <noinclude>[[Status::Basic definition| ]][[Topic::Graph theory| ]]</noinclude> '''Perfect matching''': A matching in a bipartite graph such that every element on one side gets matched to ... wikitext text/x-wiki <noinclude>[[Status::Basic definition| ]][[Topic::Graph theory| ]]</noinclude> '''Perfect matching''': A matching in a bipartite graph such that every element on one side gets matched to exactly one element on the other side. No relevant subject wiki entry. 730fbe786fdc085882f167f28d7a303fa05aaeb2 0 54 109 2008-06-09T14:42:21Z Vipul 2 New page: <noinclude>[[Status::Basic definition| ]][[Topic::Number theory| ]]</noinclude> '''Perfect number''': A natural number that equals the sum of all its proper (positive) divisors. No prima... wikitext text/x-wiki <noinclude>[[Status::Basic definition| ]][[Topic::Number theory| ]]</noinclude> '''Perfect number''': A natural number that equals the sum of all its proper (positive) divisors. No primary subject wiki entry. Also located at: [[Wikipedia:Perfect number]], [[Mathworld:PerfectNumber]], [[Planetmath:PerfectNumber]] 049d242652a7c6abaf1ecf26f594218ef83fe977 0 55 110 2008-06-09T14:47:11Z Vipul 2 New page: <noinclude>[[Status::Basic definition| ]][[Topic:Number theory| ]]</noinclude> '''Perfect power''': A natural number expressible as <math>a^n</math>, where <math>a,n</math> are natural nu... wikitext text/x-wiki <noinclude>[[Status::Basic definition| ]][[Topic:Number theory| ]]</noinclude> '''Perfect power''': A natural number expressible as <math>a^n</math>, where <math>a,n</math> are natural numbers and <math>n > 1</math> For <math>n = 2</math>, termed a '''perfect square'''. For <math>n = 3</math>, termed a '''perfect cube'''. No relevant subject wiki entry. Also located at: [[Wikipedia:Perfect power]] 69c33ecc278459920f959824f2afecdff17fe034 111 110 2008-06-09T14:47:45Z Vipul 2 wikitext text/x-wiki <noinclude>[[Status::Basic definition| ]][[Topic:Number theory| ]]</noinclude> '''Perfect power''': A natural number expressible as <math>a^n</math>, where <math>a,n</math> are natural numbers and <math>n > 1</math> For <math>n = 2</math>, termed a '''perfect square'''. For <math>n = 3</math>, termed a '''perfect cube'''. No relevant subject wiki entry. Also located at: [[Wikipedia:Perfect power]], [[Mathworld:PerfectPower]] e1218b8d8ab4d562ae12862d8d9ecd643a189782 112 111 2008-06-09T14:48:04Z Vipul 2 wikitext text/x-wiki <noinclude>[[Status::Basic definition| ]][[Topic::Number theory| ]]</noinclude> '''Perfect power''': A natural number expressible as <math>a^n</math>, where <math>a,n</math> are natural numbers and <math>n > 1</math> For <math>n = 2</math>, termed a '''perfect square'''. For <math>n = 3</math>, termed a '''perfect cube'''. No relevant subject wiki entry. Also located at: [[Wikipedia:Perfect power]], [[Mathworld:PerfectPower]] 2b190f02610f0ed68510c00de32a60f2330ec223 0 31 114 66 2008-06-09T20:11:55Z Vipul 2 wikitext text/x-wiki Main forms: ''perfect'', [[word type::adjective]]. Related forms: ''perfectness'' (extent to which something is perfect), ''perfection'' (being perfect) Typical use: * The best; something that cannot be improved further, something that fits the requirements exactly. Similar words: [[similar::ideal]], [[similar::optimal]] * Something pure, unsullied, without any tainting influences. Similar words: [[similar::pure]], [[similar::ideal]], [[similar::isolated]] ==Economics== {{:Perfect (economics)}} ==Mathematics== {{:Perfect (mathematics)}} 49c38837cb5052a3f600eacdad261c9214580b05 130 114 2008-06-09T21:10:54Z Vipul 2 wikitext text/x-wiki Main forms: ''perfect'', [[word type::adjective]]. Related forms: ''perfectness'' (extent to which something is perfect), ''perfection'' (being perfect) Typical use: * The best; something that cannot be improved further, something that fits the requirements exactly. Similar words: [[similar::ideal]], [[similar::optimal]] * Something pure, unsullied, without any tainting influences. Similar words: [[similar::pure]], [[similar::ideal]], [[similar::isolated]] ==Economics== {{:Perfect (economics)}} ==Mathematics== {{:Perfect (mathematics)}} ==Physics== {{:Perfect (physics)}} 531ef2fe3505e22cce91abe6ad03bca3feba092d 0 56 115 2008-06-09T20:17:34Z Vipul 2 New page: Main form: ''regular'', [[word type::adjective]] Related forms: ''regularity'' (extent to which something is regular, or whether or not something is regular), ''regularize/regularise'' (t... wikitext text/x-wiki Main form: ''regular'', [[word type::adjective]] Related forms: ''regularity'' (extent to which something is regular, or whether or not something is regular), ''regularize/regularise'' (to make regular) Typical use: * Something ordinary, and most likely to occur, as opposed to something special or rare. Similar words: [[similar::generic]], [[similar::typical]], [[similar::ordinary]] * Something desirable, smooth, with good behavior. Derived terms: [[derived::irregular]], [[derived::semiregular]] ==Mathematics== {{:Regular (mathematics)}} 94d6d880919c19816f65b3ceb44fd8b52b873a3c 0 57 116 2008-06-09T20:20:23Z Vipul 2 New page: ===In topology=== {{:Regular space}} {{:Regular covering}} ===In differential geometry=== {{:Regular value}} ===In group theory/representation theory=== {{:Regular group action}} wikitext text/x-wiki ===In topology=== {{:Regular space}} {{:Regular covering}} ===In differential geometry=== {{:Regular value}} ===In group theory/representation theory=== {{:Regular group action}} 4c6498e8381214bc650e8d8f2b19779dc8b569c1 117 116 2008-06-09T20:21:54Z Vipul 2 wikitext text/x-wiki ===In topology=== {{:Regular space}} {{:Regular covering}} ===In differential geometry=== {{:Regular value}} ===In group theory/representation theory=== {{:Regular group action}} ===In geometry=== {{:Regular polygon}} {{:Regular polyhedron}} {{:Regular polytope}} ===In number theory=== {{:Regular prime}} ===In axiomatic set theory=== {{:Axiom of regularity}} f7be4fd88d2ae84163eef4d33f1355be5eb309bc 121 117 2008-06-09T20:35:11Z Vipul 2 wikitext text/x-wiki ===In topology=== {{:Regular space}} {{:Regular covering}} ===In differential geometry=== {{:Regular value}} ===In group theory/representation theory=== {{:Regular group action}} ===In geometry=== {{:Regular polygon}} {{:Regular polyhedron}} {{:Regular polytope}} ===In commutative algebra=== {{:Regular ring}} ===In algebraic geometry=== {{:Regular map}} ===In number theory=== {{:Regular prime}} ===In axiomatic set theory=== {{:Regular cardinal}} ===In measure theory=== {{:Regular measure}} ===In category theory=== {{:Regular category}} 45b52015f307a71d552104aad59437b4ee5993da 143 121 2008-07-04T21:15:46Z Vipul 2 /* In group theory/representation theory */ wikitext text/x-wiki ===In topology=== {{:Regular space}} {{:Regular covering}} ===In differential geometry=== {{:Regular value}} ===In the theory of monoids and semigroups=== {{:Regular semigroup}} ===In group theory/representation theory=== {{:Regular group action}} ===In geometry=== {{:Regular polygon}} {{:Regular polyhedron}} {{:Regular polytope}} ===In commutative algebra=== {{:Regular ring}} ===In algebraic geometry=== {{:Regular map}} ===In number theory=== {{:Regular prime}} ===In axiomatic set theory=== {{:Regular cardinal}} ===In measure theory=== {{:Regular measure}} ===In category theory=== {{:Regular category}} b595743e6d06c827a9f57357cbdbf9e3db0a8226 0 58 119 2008-06-09T20:27:49Z Vipul 2 New page: {{:Normal good}} wikitext text/x-wiki {{:Normal good}} dfd0cc211c989e8af3f7aee00cbb188a6cf3b8f4 0 3 120 100 2008-06-09T20:29:11Z Vipul 2 wikitext text/x-wiki The term ''normal'' in mathematics is used in the following broad senses: * To denote something upright or perpendicular * To denote something that is as it ''should be''. In this sense, normal means ''good'' or ''desirable'' rather than ''typical'' It is typically used as a limiting adjective, and in a binary sense. In other words, an object in a certain class is ''normal'' if it satisfies certain conditions, and every object in that class is either normal or not normal. The noun form is ''normality''. Thus, normality can be viewed as a property in various contexts. ===In group theory=== {{:Normal subgroup}} ===In topology=== {{:Normal space}} ===In linear algebra=== {{:Normal form (linear algebra)}} {{:Normal matrix}} {{:Normal operator}} ===In field theory=== {{:Normal field extension}} ===In commutative algebra=== {{:Normal domain}} {{:Normal ring}} ===In differential geometry=== Normal to a curve in the plane: The line perpendicular to the tangent to the curve at a point. {{:Normal bundle}} {{:Normal coordinate system}} ===In category theory=== {{:Normal monomorphism}} {{:Normal epimorphism}} ===In probability/statistics=== {{:Normal distribution}} ===In axiomatic set theory=== {{:Normal function}} {{:Normal measure}} ===Others=== {{:Normal number}} 11b1f813b04d8ab423a22fc23881ab05bab31fa6 0 59 122 2008-06-09T20:38:35Z Vipul 2 New page: The term ''natural'' in mathematics is loosely used to mean ''God-given'', ''independent of choices'', and ''as might be suggested''. However, these generic uses have led to some very prec... wikitext text/x-wiki The term ''natural'' in mathematics is loosely used to mean ''God-given'', ''independent of choices'', and ''as might be suggested''. However, these generic uses have led to some very precise definitions for the term, in various contexts. ===In category theory=== {{:Natural transformation}} {{:Natural isomorphism}} ===In number theory=== {{:Natural number}} c734f1b076854530f7dd9691017803a196c2686c 0 60 123 2008-06-09T21:00:58Z Vipul 2 New page: Main form: ''free'', [[word type::adjective]]. Used both in a descriptive and in a limiting sense. Related forms: ''freedom'' (the state of being free), ''freeness'' (whether or not somet... wikitext text/x-wiki Main form: ''free'', [[word type::adjective]]. Used both in a descriptive and in a limiting sense. Related forms: ''freedom'' (the state of being free), ''freeness'' (whether or not something is free; used in the scientific contexts) Typical use: * ''Free from'' or ''free of'': Indicating the absence of constraints, relations, burdens, impositions, costs. Similar words: [[similar::unfettered]], [[similar::unconstrained]] * ''Free to'': Having the ability and permission to perform certain actions. ==Mathematics== {{:Free (mathematics)}} ==Economics== {{:Free (economics)}} ==Chemistry== {{:Free (chemistry)}} ==Physics== {{:Free (physics)}} d5c876c8252500c702c7acbc56a57fd3912b66b1 0 40 124 79 2008-06-09T21:02:00Z Vipul 2 wikitext text/x-wiki Main form: ''simple'', [[word type::adjective]] Related forms: ''simplicity'' (whether or not something is simple, extent to which something is simple) ==Mathematics== {{:Simple (mathematics)}} 62a30dc1c7e797c77bc3a76e111404b5d77ba2c9 0 61 125 2008-06-09T21:06:08Z Vipul 2 New page: In mathematics, ''free'' is typically used in the ''free from relations/constraints'' sense, rather than in the ''free to act'' sense. ===In universal algebra=== {{:Free algebra}} ===In... wikitext text/x-wiki In mathematics, ''free'' is typically used in the ''free from relations/constraints'' sense, rather than in the ''free to act'' sense. ===In universal algebra=== {{:Free algebra}} ===In group theory=== {{:Free group}} {{:Free group action}} ===In commutative algebra/noncommutative algebra=== {{:Free associative algebra}} {{:Free module}} c0e3a502532859d73afe80a89d81870dd25f3ec3 0 62 126 2008-06-09T21:07:42Z Vipul 2 New page: {{:Free market}} {{:Free trade}} {{:Free good}} wikitext text/x-wiki {{:Free market}} {{:Free trade}} {{:Free good}} f313f9c7331f0d9a78ef761f106fc8d753c2da72 0 63 127 2008-06-09T21:08:32Z Vipul 2 New page: {{:Free electron}} {{:Free radical}} wikitext text/x-wiki {{:Free electron}} {{:Free radical}} cc9e0d6b4be3e7eaeb2d878e3964b7b67ce4467a 0 64 128 2008-06-09T21:09:08Z Vipul 2 New page: {{:Free fall}} wikitext text/x-wiki {{:Free fall}} fc72e111e9855f1f30e033912131d6103daace19 0 65 129 2008-06-09T21:10:22Z Vipul 2 New page: {{:Perfect conductor}} wikitext text/x-wiki {{:Perfect conductor}} 996ebf50fc51c9857544e2772061193b6af38215 0 66 131 2008-06-11T11:51:54Z Vipul 2 New page: <noinclude>[[Status::Semi-basic definition| ]][[Topic:Group theory| ]][[Primary wiki::Groupprops| ]]</noinclude> '''Free group''': A group freely generated by a set. The elements of the g... wikitext text/x-wiki <noinclude>[[Status::Semi-basic definition| ]][[Topic:Group theory| ]][[Primary wiki::Groupprops| ]]</noinclude> '''Free group''': A group freely generated by a set. The elements of the group are simply words whose letters are elements of the set and their inverses, with multiplication of words being by concatenation, and where the only rules for simplifying words are the cancellation of an element and its inverse when placed adjacent to each other. Primary subject wiki entry: [[Groupprops:Free group]] Also located at: [[Wikipedia:Free group]], [[Mathworld:FreeGroup]], [[Planetmath:FreeGroup]] 422ad0639c03ee84495f3a6f8c322eee7dbf3133 132 131 2008-06-11T11:52:17Z Vipul 2 wikitext text/x-wiki <noinclude>[[Status::Semi-basic definition| ]][[Topic::Group theory| ]][[Primary wiki::Groupprops| ]]</noinclude> '''Free group''': A group freely generated by a set. The elements of the group are simply words whose letters are elements of the set and their inverses, with multiplication of words being by concatenation, and where the only rules for simplifying words are the cancellation of an element and its inverse when placed adjacent to each other. Primary subject wiki entry: [[Groupprops:Free group]] Also located at: [[Wikipedia:Free group]], [[Mathworld:FreeGroup]], [[Planetmath:FreeGroup]] af9d934fc9adcb485ad444c80c3a4d95028d85d8 108 67 133 2008-06-15T16:59:16Z Vipul 2 New page: Mathworld is a free, online mathematics reference work. {{quote|'''SLOGANS''': ''Wolfram Mathworld, the web's most extensive mathematics resource''<br>''Eric Weisstein's World of Mathemat... wikitext text/x-wiki Mathworld is a free, online mathematics reference work. {{quote|'''SLOGANS''': ''Wolfram Mathworld, the web's most extensive mathematics resource''<br>''Eric Weisstein's World of Mathematics''<br>''A free resource from Wolfram Research built with Mathematica technology, created, developed and nurtured by Eric Weisstein with contributions from the mathematics community''}} ==Use Mathworld== ===Start Mathworld=== Mathworld is online, or Internet-based. It can be accessed using a Web browser and an Internet connection: http://mathworld.wolfram.com An older version of Mathworld are also available in book form from CRC Press, ISBN 1584883472. ===Search Mathworld=== * http://mathworld.wolfram.com/search/ to search Mathworld. The search box can also be accessed from any Mathworld page. * Add "site:mathworld.wolfram.com" to a query to search Mathworld using an external search engine such as Google, Yahoo! or Windows Live. ===Article naming and reaching a given article=== Articles in Mathworld are named using CamelCase conventions: for terms that have multiple words, the multiple words are shrunk into a single word and the first letter of each original word is capitalized. Thus, for instance, "prime number" becomes "PrimeNumber". The Mathworld URL for a term is given by: http://mathworld.wolfram.com/CamelCaseName.html where '''CamelCaseName''' is replaced by the actual name. For instance: http://mathworld.wolfram.com/Triangle.html http://mathworld.wolfram.com/PrimeNumber.html http://mathworld.wolfram.com/NormalSubgroup.html http://mathworld.wolfram.com/GammaFunction.html To go to an article, either type the full URL or use Mathworld's site search. ===Organization by topic=== The left-hand column gives a list of major mathematical topics into which Mathworld content is organized. Each topic is divided into subtopics, which may in turn be further divided into subtopics. The lowest-level subtopics contain lists of articles. A given article may be in one or more subtopic. At the top of each article, the subtopic trees to which the article belongs, are shown. Moreover, a given topic may be a subtopic of one or more topics. The page listing subtopics/articles of a topic is: http://mathworld.wolfram.com/topics/TopicName.html where '''TopicName''' is replaced by the actual name of the topic. ==More about Mathworld for users== See also: http://mathworld.wolfram.com/about/ and http://mathworld.wolfram.com/about/faq.html Mathworld offers a number of features to augment its main site. It runs on [[Resource:Mathematica|Mathematica]], a software for mathematics-based computation and graphing. Some of the features that Mathworld offers: * Encyclopedia-like entries on various topics (this is the primary offering) * [http://mathworld.wolfram.com/classroom/ Mathworld Classroom]: This gives short entries on topics, such as a definition, a few examples, the expected level of difficulty of the topic, prerequisites. The classroom entry on a Mathworld topic is linked to from its main entry. * [http://mathworld.wolfram.com/topics/InteractiveEntries.html Interactive Entries], using graphics applets. The interactive entries on a Mathworld topic are linked to from its main entry. ==Mathworld as a free resource== ===Free access=== All content on Mathworld can be accessed for free (i.e., without any access or subscription fees). ===No freedom to copy or mirror=== Content in Mathworld is ''not'' under an open-content license. The copyright to content lies with the authors and contributors (contributors to a particular article are listed on top of the article). Others do ''not'' have the right to copy, mirror or reproduce content on the website. However, people are encouraged to cite and link to Mathworld and take printouts for personal use, with proper attribution. ==Management and organization== ===Parent organization=== Mathworld is operated by Eric Weisstein, a researcher at Wolfram Research, Inc. The funding and support for Mathworld comes from Wolfram Research, Inc. ===Software tools=== Mathworld is organized based on Mathematica, a mathematics computation and graphing tool. Mathematica is used to generate mathematical symbols and formulae used in the pages, as well as for the interactive demonstrations and elaborate pictures. 41664f65c4b32516ff894f993ec466c12f253e6c 134 133 2008-06-15T16:59:42Z Vipul 2 /* Start Mathworld */ wikitext text/x-wiki Mathworld is a free, online mathematics reference work. {{quote|'''SLOGANS''': ''Wolfram Mathworld, the web's most extensive mathematics resource''<br>''Eric Weisstein's World of Mathematics''<br>''A free resource from Wolfram Research built with Mathematica technology, created, developed and nurtured by Eric Weisstein with contributions from the mathematics community''}} ==Use Mathworld== ===Start Mathworld=== Mathworld is online, or Internet-based. It can be accessed using a Web browser and an Internet connection: http://mathworld.wolfram.com An older version of Mathworld is also available in book form from CRC Press, ISBN 1584883472. ===Search Mathworld=== * http://mathworld.wolfram.com/search/ to search Mathworld. The search box can also be accessed from any Mathworld page. * Add "site:mathworld.wolfram.com" to a query to search Mathworld using an external search engine such as Google, Yahoo! or Windows Live. ===Article naming and reaching a given article=== Articles in Mathworld are named using CamelCase conventions: for terms that have multiple words, the multiple words are shrunk into a single word and the first letter of each original word is capitalized. Thus, for instance, "prime number" becomes "PrimeNumber". The Mathworld URL for a term is given by: http://mathworld.wolfram.com/CamelCaseName.html where '''CamelCaseName''' is replaced by the actual name. For instance: http://mathworld.wolfram.com/Triangle.html http://mathworld.wolfram.com/PrimeNumber.html http://mathworld.wolfram.com/NormalSubgroup.html http://mathworld.wolfram.com/GammaFunction.html To go to an article, either type the full URL or use Mathworld's site search. ===Organization by topic=== The left-hand column gives a list of major mathematical topics into which Mathworld content is organized. Each topic is divided into subtopics, which may in turn be further divided into subtopics. The lowest-level subtopics contain lists of articles. A given article may be in one or more subtopic. At the top of each article, the subtopic trees to which the article belongs, are shown. Moreover, a given topic may be a subtopic of one or more topics. The page listing subtopics/articles of a topic is: http://mathworld.wolfram.com/topics/TopicName.html where '''TopicName''' is replaced by the actual name of the topic. ==More about Mathworld for users== See also: http://mathworld.wolfram.com/about/ and http://mathworld.wolfram.com/about/faq.html Mathworld offers a number of features to augment its main site. It runs on [[Resource:Mathematica|Mathematica]], a software for mathematics-based computation and graphing. Some of the features that Mathworld offers: * Encyclopedia-like entries on various topics (this is the primary offering) * [http://mathworld.wolfram.com/classroom/ Mathworld Classroom]: This gives short entries on topics, such as a definition, a few examples, the expected level of difficulty of the topic, prerequisites. The classroom entry on a Mathworld topic is linked to from its main entry. * [http://mathworld.wolfram.com/topics/InteractiveEntries.html Interactive Entries], using graphics applets. The interactive entries on a Mathworld topic are linked to from its main entry. ==Mathworld as a free resource== ===Free access=== All content on Mathworld can be accessed for free (i.e., without any access or subscription fees). ===No freedom to copy or mirror=== Content in Mathworld is ''not'' under an open-content license. The copyright to content lies with the authors and contributors (contributors to a particular article are listed on top of the article). Others do ''not'' have the right to copy, mirror or reproduce content on the website. However, people are encouraged to cite and link to Mathworld and take printouts for personal use, with proper attribution. ==Management and organization== ===Parent organization=== Mathworld is operated by Eric Weisstein, a researcher at Wolfram Research, Inc. The funding and support for Mathworld comes from Wolfram Research, Inc. ===Software tools=== Mathworld is organized based on Mathematica, a mathematics computation and graphing tool. Mathematica is used to generate mathematical symbols and formulae used in the pages, as well as for the interactive demonstrations and elaborate pictures. d705714a87b66cf498f7396fab794d7a9c318e65 135 134 2008-06-15T17:02:26Z Vipul 2 wikitext text/x-wiki Mathworld is a free, online mathematics reference work. {{quotation|'''SLOGANS''': ''Wolfram Mathworld, the web's most extensive mathematics resource''<br>''Eric Weisstein's World of Mathematics''<br>''A free resource from Wolfram Research built with Mathematica technology, created, developed and nurtured by Eric Weisstein with contributions from the mathematics community''}} ==Use Mathworld== ===Start Mathworld=== Mathworld is online, or Internet-based. It can be accessed using a Web browser and an Internet connection: http://mathworld.wolfram.com An older version of Mathworld is also available in book form from CRC Press, ISBN 1584883472. ===Search Mathworld=== * http://mathworld.wolfram.com/search/ to search Mathworld. The search box can also be accessed from any Mathworld page. * Add "site:mathworld.wolfram.com" to a query to search Mathworld using an external search engine such as Google, Yahoo! or Windows Live. ===Article naming and reaching a given article=== Articles in Mathworld are named using CamelCase conventions: for terms that have multiple words, the multiple words are shrunk into a single word and the first letter of each original word is capitalized. Thus, for instance, "prime number" becomes "PrimeNumber". The Mathworld URL for a term is given by: http://mathworld.wolfram.com/CamelCaseName.html where '''CamelCaseName''' is replaced by the actual name. For instance: http://mathworld.wolfram.com/Triangle.html http://mathworld.wolfram.com/PrimeNumber.html http://mathworld.wolfram.com/NormalSubgroup.html http://mathworld.wolfram.com/GammaFunction.html To go to an article, either type the full URL or use Mathworld's site search. ===Organization by topic=== The left-hand column gives a list of major mathematical topics into which Mathworld content is organized. Each topic is divided into subtopics, which may in turn be further divided into subtopics. The lowest-level subtopics contain lists of articles. A given article may be in one or more subtopic. At the top of each article, the subtopic trees to which the article belongs, are shown. Moreover, a given topic may be a subtopic of one or more topics. The page listing subtopics/articles of a topic is: http://mathworld.wolfram.com/topics/TopicName.html where '''TopicName''' is replaced by the actual name of the topic. ==More about Mathworld for users== See also: http://mathworld.wolfram.com/about/ and http://mathworld.wolfram.com/about/faq.html Mathworld offers a number of features to augment its main site. It runs on [[Resource:Mathematica|Mathematica]], a software for mathematics-based computation and graphing. Some of the features that Mathworld offers: * Encyclopedia-like entries on various topics (this is the primary offering) * [http://mathworld.wolfram.com/classroom/ Mathworld Classroom]: This gives short entries on topics, such as a definition, a few examples, the expected level of difficulty of the topic, prerequisites. The classroom entry on a Mathworld topic is linked to from its main entry. * [http://mathworld.wolfram.com/topics/InteractiveEntries.html Interactive Entries], using graphics applets. The interactive entries on a Mathworld topic are linked to from its main entry. ==Mathworld as a free resource== ===Free access=== All content on Mathworld can be accessed for free (i.e., without any access or subscription fees). ===No freedom to copy or mirror=== Content in Mathworld is ''not'' under an open-content license. The copyright to content lies with the authors and contributors (contributors to a particular article are listed on top of the article). Others do ''not'' have the right to copy, mirror or reproduce content on the website. However, people are encouraged to cite and link to Mathworld and take printouts for personal use, with proper attribution. ==Management and organization== ===Parent organization=== Mathworld is operated by Eric Weisstein, a researcher at Wolfram Research, Inc. The funding and support for Mathworld comes from Wolfram Research, Inc. ===Software tools=== Mathworld is organized based on Mathematica, a mathematics computation and graphing tool. Mathematica is used to generate mathematical symbols and formulae used in the pages, as well as for the interactive demonstrations and elaborate pictures. 48314fd1d3cdc1da7a079ea712aa6ff3169b84fb 108 68 136 2008-06-15T17:07:44Z Vipul 2 New page: Planetmath is a free, online, open-content, collaborative mathematics resource. While primarily an encyclopedia, it also contains books, expositions, papers and forums. {{quotation|'''SLO... wikitext text/x-wiki Planetmath is a free, online, open-content, collaborative mathematics resource. While primarily an encyclopedia, it also contains books, expositions, papers and forums. {{quotation|'''SLOGAN''': ''Math for the people, by the people''}} ==Use Planetmath== ===Start Planetmath=== Planetmath is online, or Internet-based. It can be accessed using a Web browser and an Internet connection: http://planetmath.org Planetmath dumps can also be downloaded at: http://aux.planetmath.org/snapshots/ ===Search Planetmath=== Planetmath offers a search box in the upper right corner. One can also search Planetmath by adding "site:planetmath.org" to a search query on a search engine like Google, Yahoo! or Windows Live. ===Article names and reaching a given article=== Articles in Planetmath are named using CamelCase conventions. However, these conventions are not rigidly followed for all article names. 413b9fdc4d8ed0856abf4c24e4fb9ed50f503c32 137 136 2008-06-15T17:15:11Z Vipul 2 wikitext text/x-wiki Planetmath is a free, online, open-content, collaborative mathematics resource. While primarily an encyclopedia, it also contains books, expositions, papers and forums. {{quotation|'''SLOGAN''': ''Math for the people, by the people''}} ==Use Planetmath== ===Start Planetmath=== Planetmath is online, or Internet-based. It can be accessed using a Web browser and an Internet connection: http://planetmath.org Planetmath dumps can also be downloaded at: http://aux.planetmath.org/snapshots/ ===Search Planetmath=== Planetmath offers a search box in the upper right corner. One can also search Planetmath by adding "site:planetmath.org" to a search query on a search engine like Google, Yahoo! or Windows Live. ===Article names and reaching a given article=== Articles in Planetmath are named using CamelCase conventions. However, these conventions are not rigidly followed for all article names. In the CamelCase convention, a term consisting of multiple words is shrunk to a single word, with the first letter of each original word capitalized. Thus, "prime number" becomes "PrimeNumber", and the planetmath URL for it is: http://planetmath.org/encyclopedia/PrimeNumber.html bb9d2ba1cea31b31c6a1e12a2e46136eb5d01b8b 108 29 138 65 2008-06-19T19:45:14Z Vipul 2 /* Compensation model */ wikitext text/x-wiki Wikipedia is a free, online, open-content, collaborative, multilingual encyclopedia project. {{quotation|'''SLOGANS''': ''Wikipedia, the free encyclopedia''<br>''Wikipedia, the free encyclopedia that anyone can edit''<br>''Imagine a world in which every single human being can freely share in the sum of all knowledge. That's our commitment.''}} ==Use Wikipedia== ===Start Wikipedia=== Wikipedia is online, or Internet-based. It can be accessed using a web browser and an Internet connection. Main website: http://www.wikipedia.org Some major language websites: * English: http://en.wikipedia.org * French: http://fr.wikipedia.org * German: http://de.wikipedia.org * Polish: http://pl.wikipedia.org A complete list of language Wikipedias is available at: http://meta.wikimedia.org/wiki/List_of_Wikipedias ===Search Wikipedia=== # http://www.wikipedia.org allows search in any of the language Wikipedias # In any of the language Wikipedias, there is a search box in the left column that allows any search # http://en.wikipedia.org/Special:Search is the English Wikipedia's internal site search # If using Mozilla Firefox as the browser, you can add Wikipedia (English) to the list of Search Engines using https://addons.mozilla.org/ ===Article names and reaching a given article=== Wikipedia content is organized into articles. A Wikipedia article on a topic is named exactly by that topic. The URL for a given Wikipedia article in the English Wikipedia is given by: http://en.wikipedia.org/wiki/Name where Name denotes the name of the article. Thus, for instance, the English Wikipedia article on India is titled [[wikipedia:India|India]], and the full URL is given by: http://en.wikipedia.org/wiki/India To go to an article with a particular title, do any of the following: # Type the full URL: http://en.wikipedia.org/wiki/Name # Use the Wikipedia Search Engine for Firefox and type the name of the topic in the search bar # Open Wikipedia and type the name of the article in the search bar, and press Go (or enter) ==More about Wikipedia for users== See also: [[Wikipedia:Wikipedia:Introduction|The English Wikipedia's introduction page]] ===Criteria for Wikipedia content=== Some important Wikipedia policies for content (nutshells as stated on the Wikipedia guideline pages): # [[Wikipedia:Wikipedia:Neutral point of view|Neutral Point of View policy]]: All Wikipedia articles and other encyclopedic content must be written from a neutral point of view, representing significant views fairly, proportionately and without bias. # [[Wikipedia:Wikipedia:No original research|No original research]]: #* Wikipedia does not publish original thought: all material in Wikipedia must be attributable to a reliable, published source. #* Articles may not contain any new analysis or synthesis of published material that serves to advance a position not clearly advanced by the sources. # [[Wikipedia:Wikipedia:Notability|Notability]]: If a topic has received significant coverage in reliable secondary sources that are independent of the subject, it is presumed to be notable. # [[Wikipedia:Wikipedia:What Wikipedia is not|What Wikipedia is not]]: Gives a list of things Wikipedia is not suitable for. # [[Wikipedia:Wikipedia:Verifiability|Verifiability policy]]: Material challenged or likely to be challenged, and all quotations, must be attributed to a reliable, published source. These criteria are taken as guidelines, and not every page on Wikipedia may conform to the criteria at any given time. ===Using Wikipedia=== Some pieces of information put up by Wikipedia for Wikipedia users: # [[Wikipedia:Wikipedia:Ten things you may not know about Wikipedia|Ten things about Wikipedia]]: Important information about Wikipedia # [[Wikipedia:Wikipedia:Researching with Wikipedia|Researching with Wikipedia]]: Information and suggestions about how to use Wikipedia for research # [[Wikipedia:Wikipedia:Basic navigation|Navigating and finding information on Wikipedia]] ==Wikipedia as a free resource== ===Free access=== All content on Wikipedia can be accessed for free (i.e., without any access or subscription fees). ===Copyright and reuse=== Any contributions made to Wikipedia are automatically considered to be released under the GNU Free Documentation License ([[license::GFDL]]), version 1.2. The GFDL is a ''copyleft'', ''open-content'' license, whose key feature is that it allows reuse without permission from the original author, as long as such reuse is also licensed under a similar license. Main reference pages: # [[Wikipedia:Wikipedia:Copyrights]] explains Wikipedia's copyright policy # [[Wikipedia:Wikipedia:Text of the GNU Free Documentation License]] is the full text of the license used by Wikipedia ==Management and organization== General information in this direction is available at: http://en.wikipedia.org/wiki/Wikipedia:About ===Parent organization=== Wikipedia is managed by the [[parent organization::Wikimedia Foundation]], a non-profit organization. The Wikimedia Foundation is responsible both for the servers that host Wikipedia content, and for maintaining and improving MediaWiki, the underlying software for Wikipedia. Main reference pages: # http://www.wikimedia.org is the Wikimedia Foundation's central landing page (listing Wikipedia and other sister projects) # http://www.wikimediafoundation.org is the Wikimedia Foundation's wiki # http://meta.wikimedia.org is a wiki run by the Wikimedia Foundation to coordinate the Wikimedia Foundation's projects ===Software tools=== Wikipedia runs on [[software::MediaWiki]], a software designed for web-base collaborative authoring of articles. MediaWiki is ''free software'', and is licensed under the GNU General Public License, Version 2.0. MediaWiki was originally written by Magnus Manske, specifically for Wikipedia. Its development is handled by software professionals working for the Wikimedia Foundation. However, MediaWiki is free software and can be both used and modified by anybody. A number of extensions to MediaWiki have been developed by people outside the Wikimedia Foundation. MediaWiki is also used as a software in a number of other collaborative software-based tools, such as the Subject Wikis Reference Guide. # http://www.mediawiki.org is the MediaWiki website. It has the MediaWiki manual, links for downloading the software, as well as a listing of extensions developed for MediaWiki # http://www.wikiindex.org gives a list of wikis (not comprehensive) ===Financial model=== The Wikimedia Foundation (Wikipedia's parent organization) is a non-profit organization (registered under Section 501 (c)(3)) founded in St. Petersburg, Florida, United States in 2002 and headquartered in San Francisco, California, United States. The financial model of the Wikimedia Foundation primarily rests on voluntary donation. # http://wikimediafoundation.org/wiki/Donate The donate page ==Production model== ===Collaborative content creation=== Wikipedia uses a ''strongly collaborative'' model of content creation, powered by a wiki software called MediaWiki. Some of the key features of the model used on Wikipedia: # Anybody can register an account on Wikipedia, without necessarily revealing his/her real name and identity # Anybody can start an article on a topic (as of now, only logged-in users can start new articles) if no article on the topic exists # Once an article is created, anybody (and not just the original author of the article) can edit the article # Every revision of the article is stored on the wiki, and different revisions can be viewed and compared using the "history" tab of the article # People seeking to collaborate on a given article can discuss material related to the article on its "discussion page" or "talk page", which can be accessed using the "discussion" tab of the article # A person can nominate an article for deletion on grounds that it violates Wikipedia policy. The issue is then discussed on a separate page, till a consensus is reached either way Relevant links: # http://en.wikipedia.org/w/index.php?title=Special:UserLogin&type=signup is the page for account creation on English Wikipedia # [[Wikipedia:Help:Editing|Wikipedia Editing help]] ===Compensation model=== Wikipedia does not, as a rule, compensate people for writing articles. In particular: # There is no monetary compensation or reimbursement for writing articles on Wikipedia # No credits for an article are given on the article page itself. The authorship of a given article may be seen using the "history" tab or the "credits" option. However, the information revealed by the history may be incomplete because anonymous users may have made certain edits, and even the registered users may not have used their real names and identities ===Security model=== Wikipedia relies on soft security to combat problems such as vandalism and bias in articles. Dedicated volunteers patrol the wiki to locate acts of vandalism, or content that otherwise violates Wikipedia policy. User accounts or IP addresses associated with such editing may be banned from editing Wikipedia. ==Related resources== ===Competitors=== * [[Competitor::Resource:Encyclopædia Britannica]]: Managed by Britannica Corporation, this has been one of the world's leading encyclopedia resources for many centuries. * [[Competitor::Resource:Citizendium]] ===Sister projects=== * [[Sister::Resource:Wikibooks]] * [[Sister::Resource:Wiktionary]] * [[Sister::Resource:Wikiverse]] * [[Sister::Resource:Wikisource]] * [[Sister::Resource:Wikiversity]] * [[Sister::Resource:Wikinews]] * [[Sister::Resource:Wikiquote]] * [[Sister::Resource:Wikispecies]] f2174324b923603791329722bb8b3ceb2f0cf5c4 108 69 139 2008-06-19T20:23:13Z Vipul 2 New page: JSTOR is an online archive of published academic/scholarly journal material. {{quote|'''SLOGANS''': ''Trusted archive for scholarship''}} ==Use JSTOR== ===Start JSTOR=== JSTOR is onlin... wikitext text/x-wiki JSTOR is an online archive of published academic/scholarly journal material. {{quote|'''SLOGANS''': ''Trusted archive for scholarship''}} ==Use JSTOR== ===Start JSTOR=== JSTOR is online, or Internet-based. It can be accessed using a web browser and Internet connection, at: http://www.jstor.org ===Search JSTOR=== # http://www.jstor.org/action/showBasicSearch is JSTOR's basic search. It allows search filtered by discipline. # http://www.jstor.org/action/showAdvancedSearch is JSTOR's advanced search, which allows field-specific search (full-text, article title, author, abstract, caption) and also allows filtering by discipline/journal. # http://www.jstor.org/action/showArticleLocator is JSTOR's article locator, where an article can be retrieved using the article title and journal. # JSTOR content can also be searched using [[Resource:Google Scholar|Google Scholar]]. JSTOR offers search help and tutorials: # http://www.jstor.org/page/info/help/search/index.jsp is JSTOR's search help page. # http://www.jstor.org/page/info/about/archives/tutorials.jsp offers links to tutorial videos. ==More about JSTOR for users== 4b8bc7ad04c0dc9c97950b4b004fc7596e729441 140 139 2008-06-19T22:52:22Z Vipul 2 wikitext text/x-wiki JSTOR is an online archive of published academic/scholarly journal material. {{quote|'''SLOGANS''': ''Trusted archive for scholarship''}} ==Use JSTOR== ===Start JSTOR=== JSTOR is online, or Internet-based. It can be accessed using a web browser and Internet connection, at: http://www.jstor.org ===Search JSTOR=== # http://www.jstor.org/action/showBasicSearch is JSTOR's basic search. It allows search filtered by discipline. # http://www.jstor.org/action/showAdvancedSearch is JSTOR's advanced search, which allows field-specific search (full-text, article title, author, abstract, caption) and also allows filtering by discipline/journal. # http://www.jstor.org/action/showArticleLocator is JSTOR's article locator, where an article can be retrieved using the article title and journal. # JSTOR content can also be searched using [[Resource:Google Scholar|Google Scholar]]. JSTOR offers search help and tutorials: # http://www.jstor.org/page/info/help/search/index.jsp is JSTOR's search help page. # http://www.jstor.org/page/info/about/archives/tutorials.jsp offers links to tutorial videos. ==More about JSTOR for users== ===JSTOR content=== More information at: http://www.jstor.org/page/info/about/archives/index.jsp JSTOR archives content from journals across the world, in a large range of academic topics. Content is often available starting from the very first issue and volume of the journal. The oldest content on JSTOR dates back to 1665. Archival of content is based on an agreement between JSTOR and the content provider (usually, the publisher of an academic journal). Recent issues (such as those in the last 3-5 years) are typically not available through JSTOR. This time lag is termed the [http://www.jstor.org/page/info/about/archives/journals/movingWall.jsp Moving Wall] and varies wih the journal. ==JSTOR as a resource== ===Access to JSTOR articles=== More information at: http://www.jstor.org/page/info/participate/new/index.jsp Access to articles archived on JSTOR is by subsciption only. Subscriptions are typically taken by institutions, rather than individuals (though individuals can also subscribe). Institutions subscribing to JSTOR content must pay certain fees depending on the nature of the institution. People within a participating institution can access all JSTOR content. Full JSTOR articles are available in PDF and TIFF formats, and can also be viewed online page by page. ===Usage restrictions=== http://www.jstor.org/page/info/about/policies/terms.jsp Use of articles available through JSTOR is subject to their terms and conditions, which includes that the articles are not re-sold for commercial purposes, and that the copyright notices are not destroyed in any further copies circulated. ==Management and organization== General information in this regard is available at: http://www.jstor.org/page/info/about/organization/index.jsp JSTOR is run by a non-profit organization with the same name. The organization's stated aim is to help libraries offer scholars a wide range of scholarly content in an easily accessible electronic form, including back issues of journals. 513d18ad01ac87eb0f2e12d08cee7058d5256b05 141 140 2008-06-19T22:57:35Z Vipul 2 wikitext text/x-wiki JSTOR is an online archive of published academic/scholarly journal material. {{quotation|'''SLOGANS''': ''Trusted archive for scholarship''}} ==Use JSTOR== ===Start JSTOR=== JSTOR is online, or Internet-based. It can be accessed using a web browser and Internet connection, at: http://www.jstor.org ===Search JSTOR=== # http://www.jstor.org/action/showBasicSearch is JSTOR's basic search. It allows search filtered by discipline. # http://www.jstor.org/action/showAdvancedSearch is JSTOR's advanced search, which allows field-specific search (full-text, article title, author, abstract, caption) and also allows filtering by discipline/journal. # http://www.jstor.org/action/showArticleLocator is JSTOR's article locator, where an article can be retrieved using the article title and journal. # JSTOR content can also be searched using [[Resource:Google Scholar|Google Scholar]]. JSTOR offers search help and tutorials: # http://www.jstor.org/page/info/help/search/index.jsp is JSTOR's search help page. # http://www.jstor.org/page/info/about/archives/tutorials.jsp offers links to tutorial videos. ==More about JSTOR for users== ===JSTOR content=== More information at: http://www.jstor.org/page/info/about/archives/index.jsp JSTOR archives content from journals across the world, in a large range of academic topics. Content is often available starting from the very first issue and volume of the journal. The oldest content on JSTOR dates back to 1665. Archival of content is based on an agreement between JSTOR and the content provider (usually, the publisher of an academic journal). Recent issues (such as those in the last 3-5 years) are typically not available through JSTOR. This time lag is termed the [http://www.jstor.org/page/info/about/archives/journals/movingWall.jsp Moving Wall] and varies wih the journal. ==JSTOR as a resource== ===Access to JSTOR articles=== More information at: http://www.jstor.org/page/info/participate/new/index.jsp Access to articles archived on JSTOR is by subsciption only. Subscriptions are typically taken by institutions, rather than individuals (though individuals can also subscribe). Institutions subscribing to JSTOR content must pay certain fees depending on the nature of the institution. People within a participating institution can access all JSTOR content. Full JSTOR articles are available in PDF and TIFF formats, and can also be viewed online page by page. ===Usage restrictions=== http://www.jstor.org/page/info/about/policies/terms.jsp Use of articles available through JSTOR is subject to their terms and conditions, which includes that the articles are not re-sold for commercial purposes, and that the copyright notices are not destroyed in any further copies circulated. ==Management and organization== General information in this regard is available at: http://www.jstor.org/page/info/about/organization/index.jsp JSTOR is run by a non-profit organization with the same name. The organization's stated aim is to help libraries offer scholars a wide range of scholarly content in an easily accessible electronic form, including back issues of journals. b92a4ed441a16a6569ffb02c63f98af434313bc3 108 70 142 2008-06-19T23:12:29Z Vipul 2 New page: Google Scholar is a full-text search tool for published academic/scholarly journal material. ==Use Google Scholar== ===Start Google Scholar=== Google Scholar is Internet-based. It can b... wikitext text/x-wiki Google Scholar is a full-text search tool for published academic/scholarly journal material. ==Use Google Scholar== ===Start Google Scholar=== Google Scholar is Internet-based. It can be accessed using a browser and an Internet connection at: http://scholar.google.com 2f7a97d9a3488075545087c3f10fc8c13fbd47d6 0 71 144 2008-07-04T21:16:36Z Vipul 2 New page: '''Regular element in a semigroup''': An element <math>a</math> such that there exists <math>b</math> satisfying <math>aba = a</math>. '''Regular semigroup''': A semigroup where every ele... wikitext text/x-wiki '''Regular element in a semigroup''': An element <math>a</math> such that there exists <math>b</math> satisfying <math>aba = a</math>. '''Regular semigroup''': A semigroup where every element is regular. aff62413b85e20ccc486468be85ab8ba8a25c774 106 72 145 2008-07-22T22:12:52Z Vipul 2 New page: Each subject wiki has certain kinds of articles called ''definition articles'' or ''terminology articles''. Each such article is about a term being defined, and the essential component of ... wikitext text/x-wiki Each subject wiki has certain kinds of articles called ''definition articles'' or ''terminology articles''. Each such article is about a term being defined, and the essential component of any such article is the '''definition''' of that term. The definition forms a separate section of the article, and this section is designed keeping a number of objectives in mind. ==Goals of the definition section== ===Potential audience for a definition section=== Some people know the term and are looking for its definition. This could include: * People encountering the term for the first time * People who've seen the term before, and know what it means, but want a precise definition * People who already know one definition, but want to know/read multiple definitions On the other hand, there could also be people who know the definition already, and are looking for the precise term having that definition. ===What the definition should provide=== A definition should be ''clear'', ''succinct'', ''actionable'', and ''interesting''. * It must clearly answer the question: ''what is this'' * If the definition provides a differentiation or augmentation of an existing concept (for instance, a particular breed of dog, or a particular property of groups, or a particular type of number) then it should give a clear differentiating criterion. * It should give an idea of what related terms and facts are and how one can explore the notion better * It should be amenable to reverse search ==Challenges in creating a definition section== ===The challenge of ingredients and dependencies=== To define one term often requires other terms as ingredients. For instance: * The definition of a prime number relies on the definitions of natural number and divisor. * The typical definition of a [[groupprops:finitely generated group|finitely generated group]] relies on the definitions of [[groupprops:group|group]], [[groupprops:generating set of a group|generating set of a group]], and, of course, the notion of ''finiteness''. * The typical definition of Hausdorff space relies on the definition of topological space. To cope with this challenge, we need to decide what ingredients are ''taken for granted'' (i.e., their definitions are not included) and what ingredients are to be defined along with the main term we're defining. The way chosen in subject wikis is to have a certain '''primitive''' definition -- a definition that has as few dependencies as is possible while maintaining the conceptual integrity. Then, we create further definitions that are shorter and simpler, but rely on concepts beyond just the primitives. ===The challenge of multiple definitions=== Important terms can be defined from multiple perspectives. In fact, the equivalence of different definitions often gives an insight into why the concept is important. Multiple definitions are both a boon and a challenge. Conventional wisdom suggests that giving a barrage of definitions can create a cognitive overload that makes it hard for people to grasp a new ideas. On the other hand, theories of learning, association and memory also suggest that the more ''hooks'' we have on an idea, the easier it is to understand. In addition to genuinely different definitions, there are also equivalent definitions obtained by a slight rewording and change in the defining ingredients. For instance, as discussed in the previous subsection, a primitive definition may be made more compact and conceptually elegant by introducing further defining ingredients. In this case, both the primitive and the compacted definition are stated as equivalent definitions. This again has advantages for definition cognition and definition understanding: in the effort to try to reconcile various rewordings of the definition, somebody trying to learn and understnad the definition gets a better understanding of what the term means. ==Formats of definition== ===Quick phrases=== Quick phrases are to be seen at the top of definition sections in some articles. These are phrases which one can use to remember the term being defined in a very easy way. For instance, a quick phrase for a [[prime number]] is ''number with no nontrivial factorization''. For quick phrases, conceptual ease matters more than precise meaning, so some of the quick phrases may not make precise mathematical sense. ===Symbol-free definition=== A symbol-free definition is a definition that avoids that use of algebraic symbols and notation, relying instead on constructs of natural language such as pronouns and prepositions. For instance, here's a symbol-free definition of [[permutable subgroup]]: {{quotation|A [[subgroup]] of a [[group]] is termed '''permutable''' if its [[product of subgroups|product]] with every subgroup of the group is a subgroup, or equivalently, if it commutes with every subgroup.}} Here, symbols for the group and the two subgroups ar enot introduced. Rather, constructs of natural language, in particular, the pronoun ''it'', are used. Sometimes 1-2 symbols may be introduced in a symbol-free definition to avoid ambiguous pronouns. Some advantages of symbol-free definitions: # More effective for reverse search i.e. locating the term from the definition. # Might be easier to read if you already know the definition and are trying to recall it (because of less demand on working memory) # Easier to keep in mind, memorize, and act upon in more diverse situations where the notation and symbols differ # More effective linking with other concepts and ideas ===Definition with symbols=== This gives the definition with explicit introduction of symbols for every quantified object. For instance the definition with symbols of the property of being a [[permutable subgroup]] is: {{quotation|A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''permutable''' if for every subgroup <math>K</math>, <math>HK=KH</math>.}} The definition with symbols has a number of advantages, for instance: * It may be better for first-time reading * It may be more actionable * It may require less knowledge of the subject or of terminology in the subject ==Multiple definitions== ===Equivalent definitions as a list=== Often, the same term may have multiple equivalent definitions. In this case, all the equivalent forms are stated as bullet points. The same format is repeated in the symbol-free definition and in the definition with symbols. For instance, the symbol-free definition of [[normal subgroup]]: A [[subgroup]] of a [[group]] is said to be '''normal''' if it satisfies the following equivalent conditions: # It is the [[kernel]] of a [[homomorphism]] from the group. # It is invariant under all [[inner automorphism]]s. Thus, normality is the [[invariance property]] with respect to the property of an automorphism being inner. This definition also motivates the term ''invariant subgroup'' for normal subgroup (which was used earlier). # It equals each of its [[Defining ingredient::conjugate subgroups|conjugates]] in the whole group. This definition also motivates the term ''self-conjugate subgroup'' for normal subgroup (which was used earlier). # Its [[left coset]]s are the same as its [[right coset]]s (that is, it commutes with every element of the group) # It is a union of [[conjugacy class]]es. The definition with symbols of [[normal subgroup]]: A subgroup <math>N</math> of a group <math>G</math> is said to be '''normal''' in <math>G</math> (in symbols, <math>N \triangleleft G</math> or <math>G \triangleright N</math>{{subgroup-notation-page}}) if the following equivalent conditions hold: # There is a [[homomorphism]] <math>\phi</math> from <math>G</math> to another group such that the [[kernel]] of <math>\phi</math> is precisely <math>N</math>. # For all <math>g</math> in <math>G</math>, <math>gNg^{-1} \subseteq N</math>. More explicitly, for all <math>g \in G, h \in N</math>, we have <math>ghg^{-1} \in N</math>. # For all <math>g</math> in <math>G</math>, <math>gNg^{-1} = N</math>. # For all <math>g</math> in <math>G</math>, <math>gN = Ng</math>. # <math>N</math> is a union of [[Defining ingredient::conjugacy class]]es We try to follow these conventions: * When multiple definitions are given, the same order is maintained for the definitions in the symbol-free format and the definition with symbols format. * The multiple definitions should preferably be numbered, making it easier to refer to the numbering * The definitions are ordered in a way that makes the equivalence between them as self-evident as possible. ===Separate definitions as separate subsections=== Sometimes, if the equivalent definitions are long and intricate or each requires their own machinery, they may be developed in separate subsections. The title of each subsection gives some ''title'' to the definition. For instance, the two definitions of group are labeled as the '''textbook definition''' and the '''universal algebraic definition'''. ===Equivalence of multiple definitions=== Usually, there is a separate section within the definition part explaining the equivalence of definitions, or giving a link to a page that gives a ''full proof'' of this equivalence. ==Defining ingredients== ===Linking style=== Whenever a term is being defined using other terms, put links to those terms. For instance, when defining a group property, begin with: A [[group]] is said to be ... if ... Typically, the symbol-free definition will not attempt to define other terms referenced in the definition. The definition with symbols, because it gives a more explicit description, may also give a clearer idea of other terms. For instance, in the symbol-free definition of normality, the second point states that a subgroup is normal if it is invariant under all [[inner automorphism]]s. In the definition with symbols, the explicit form of inner automorphisms is given. ===Locating defining ingredients=== The wiki markup contains information about what terms are used as ingredients in defining a particular term (using semantic MediaWiki). At the bottom of the article, a box called ''Facts'' gives a list of defining ingredients. Clicking on the magnifying glass icon next to any of these ingredients gives a list of all the other terms that use the same defining ingredient. ===Links to survey articles clarifying the definition=== For definitions of important terms, the definition section may begin with a box containing links to survey articles clarifying the definition. ==Other comments== The definition section may also contain a few other comments, in so far as these comments play a direct role in clarifying the definition. Examples are typically reserved for another section, and under special circumstances, there is a separate section titled ''Importance''. 31d7e45421f0e1a0dee55358bfc062c26a7838c8 146 145 2008-07-22T22:34:12Z Vipul 2 /* Challenges in creating a definition section */ wikitext text/x-wiki Each subject wiki has certain kinds of articles called ''definition articles'' or ''terminology articles''. Each such article is about a term being defined, and the essential component of any such article is the '''definition''' of that term. The definition forms a separate section of the article, and this section is designed keeping a number of objectives in mind. ==Goals of the definition section== ===Potential audience for a definition section=== Some people know the term and are looking for its definition. This could include: * People encountering the term for the first time * People who've seen the term before, and know what it means, but want a precise definition * People who already know one definition, but want to know/read multiple definitions On the other hand, there could also be people who know the definition already, and are looking for the precise term having that definition. ===What the definition should provide=== A definition should be ''clear'', ''succinct'', ''actionable'', and ''interesting''. * It must clearly answer the question: ''what is this'' * If the definition provides a differentiation or augmentation of an existing concept (for instance, a particular breed of dog, or a particular property of groups, or a particular type of number) then it should give a clear differentiating criterion. * It should give an idea of what related terms and facts are and how one can explore the notion better * It should be amenable to reverse search ==Challenges in creating a definition section== ===The challenge of ingredients and dependencies=== To define one term often requires other terms as ingredients. For instance: * The definition of a prime number relies on the definitions of natural number and divisor. * The typical definition of a [[groupprops:finitely generated group|finitely generated group]] relies on the definitions of [[groupprops:group|group]], [[groupprops:generating set of a group|generating set of a group]], and, of course, the notion of ''finiteness''. * The typical definition of Hausdorff space relies on the definition of topological space. To cope with this challenge, we need to decide what ingredients are ''taken for granted'' (i.e., their definitions are not included) and what ingredients are to be defined along with the main term we're defining. The way chosen in subject wikis is to have a certain '''primitive''' definition -- a definition that has as few dependencies as is possible while maintaining the conceptual integrity. Then, we create further definitions that are shorter and simpler, but rely on concepts beyond just the primitives. ===The challenge of multiple definitions=== Important terms can be defined from multiple perspectives. In fact, the equivalence of different definitions often gives an insight into why the concept is important. Multiple definitions are both a boon and a challenge. Conventional wisdom suggests that giving a barrage of definitions can create a cognitive overload that makes it hard for people to grasp a new ideas. On the other hand, theories of learning, association and memory also suggest that the more ''hooks'' we have on an idea, the easier it is to understand. In addition to genuinely different definitions, there are also equivalent definitions obtained by a slight rewording and change in the defining ingredients. For instance, as discussed in the previous subsection, a primitive definition may be made more compact and conceptually elegant by introducing further defining ingredients. In this case, both the primitive and the compacted definition are stated as equivalent definitions. This again has advantages for definition cognition and definition understanding: in the effort to try to reconcile various rewordings of the definition, somebody trying to learn and understnad the definition gets a better understanding of what the term means. ===The challenge of symbols and expressions=== For new definitions, specially complicated ones, notation is often necessary, or at any rate, helpful. Thus, we often start definitions like: A ... is a set <math>K</math> with .... Symbols and expressions can be useful to set up a definition, particularly when the number of variables involved, or the nature of operations, is too large for the nouns and pronouns and constructs of natural language. However, they also have a crucial disadvantage: they may not carry conceptual clarity. In many situations, the mind relates more clearly to natural language than to variables and formulas (we've learned natural language since preschool, but algebra is usually introduced in middle school). Also, symbols are often at odds with the use of further ingredients or concepts. The subject wikis meet this challenge through a somewhat unusual method: most definitions carry with them two sections, the '''symbol-free definition''', which seeks to use natural language only without any variables or symbols, and the '''definition with symbols''', which expresses everything in symbols (so whatever objects are referred to are given symbols). ===The challenge of stickiness=== Definitions can often be dull and dry, and the tremendous conceptual compactification they represent may not be obvious to people just starting. A definition in isolation, thus, may not be too sticky. The subject wikis try to remedy this in a number of ways. The most commonly implemented way is the use of '''Quick Phrases'''. On some definition pages, a '''Quick Phrases''' box appears right on top of the definition section, with phrases that could be used to compactly remember the definition, or the essential meaning of the term. Accuracy is sometimes sacrificed for memorability or analogical usefulness in the quick phrases. ===The challenge of misinterpretation and confusion=== Definitions, read in isolation, are prone to being poorly understood, misinterpreted, and wrongly applied. All these cannot be remedied simply by a better definition. Other techniques, including the use of examples, variations, analogies, formalisms for definitions, and further exploration and analysis of the concept, are necessary to ensure a proper and useful understanding of the term being defined. Nonetheless, certain things are done on the subject wikis to reduce the chances of confusion: * '''Redundancy''' and '''multiple perspectives''': The same definition is written in many different ways, both with and without symbols, as well as with quick phrases * Links to '''definition understanding''' pages, '''funky definitions''' pages (pages with preposterous, weird, and sometimes circular definitions for the term) * Links to '''definition equivalence''' pages for proofs of the equivalence of definitions ===The challenge of motivation=== Probably the biggest challenge, that definitions on subject wikis fail to satisfy, is the challenge of motivation. ''Why'' is this definition important? ''Why'' were things defined this way, rather than that? Conventional wisdom suggests that every definition should be preceded by examples. This is often excellent pedagogy; however, in a structure of this kind, there is no natural notion of ''order'', and illustrative examples that build up to the definition may take up too much space on the definition page, and may not be very useful to people who want a quick review of the definition. On the other hand, the linked structure of the wiki makes it possible to do the examples elsewhere, but link to them, allowing people to zoom in. Some of the special features in the definition section to cater to the challenge of motivation are: * Links to definition understanding pages. These pages often explore the definition and various aspects of the definition in detail. * The definition section is preceded by a '''History''' section that may give information, links from the historical perspective. The definition section is succeeded by an '''Examples''' section that explores examples of various kinds (learn more at [[Subwiki:Examples]]). ==Formats of definition== ===Quick phrases=== Quick phrases are to be seen at the top of definition sections in some articles. These are phrases which one can use to remember the term being defined in a very easy way. For instance, a quick phrase for a [[prime number]] is ''number with no nontrivial factorization''. For quick phrases, conceptual ease matters more than precise meaning, so some of the quick phrases may not make precise mathematical sense. ===Symbol-free definition=== A symbol-free definition is a definition that avoids that use of algebraic symbols and notation, relying instead on constructs of natural language such as pronouns and prepositions. For instance, here's a symbol-free definition of [[permutable subgroup]]: {{quotation|A [[subgroup]] of a [[group]] is termed '''permutable''' if its [[product of subgroups|product]] with every subgroup of the group is a subgroup, or equivalently, if it commutes with every subgroup.}} Here, symbols for the group and the two subgroups ar enot introduced. Rather, constructs of natural language, in particular, the pronoun ''it'', are used. Sometimes 1-2 symbols may be introduced in a symbol-free definition to avoid ambiguous pronouns. Some advantages of symbol-free definitions: # More effective for reverse search i.e. locating the term from the definition. # Might be easier to read if you already know the definition and are trying to recall it (because of less demand on working memory) # Easier to keep in mind, memorize, and act upon in more diverse situations where the notation and symbols differ # More effective linking with other concepts and ideas ===Definition with symbols=== This gives the definition with explicit introduction of symbols for every quantified object. For instance the definition with symbols of the property of being a [[permutable subgroup]] is: {{quotation|A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''permutable''' if for every subgroup <math>K</math>, <math>HK=KH</math>.}} The definition with symbols has a number of advantages, for instance: * It may be better for first-time reading * It may be more actionable * It may require less knowledge of the subject or of terminology in the subject ==Multiple definitions== ===Equivalent definitions as a list=== Often, the same term may have multiple equivalent definitions. In this case, all the equivalent forms are stated as bullet points. The same format is repeated in the symbol-free definition and in the definition with symbols. For instance, the symbol-free definition of [[normal subgroup]]: A [[subgroup]] of a [[group]] is said to be '''normal''' if it satisfies the following equivalent conditions: # It is the [[kernel]] of a [[homomorphism]] from the group. # It is invariant under all [[inner automorphism]]s. Thus, normality is the [[invariance property]] with respect to the property of an automorphism being inner. This definition also motivates the term ''invariant subgroup'' for normal subgroup (which was used earlier). # It equals each of its [[Defining ingredient::conjugate subgroups|conjugates]] in the whole group. This definition also motivates the term ''self-conjugate subgroup'' for normal subgroup (which was used earlier). # Its [[left coset]]s are the same as its [[right coset]]s (that is, it commutes with every element of the group) # It is a union of [[conjugacy class]]es. The definition with symbols of [[normal subgroup]]: A subgroup <math>N</math> of a group <math>G</math> is said to be '''normal''' in <math>G</math> (in symbols, <math>N \triangleleft G</math> or <math>G \triangleright N</math>{{subgroup-notation-page}}) if the following equivalent conditions hold: # There is a [[homomorphism]] <math>\phi</math> from <math>G</math> to another group such that the [[kernel]] of <math>\phi</math> is precisely <math>N</math>. # For all <math>g</math> in <math>G</math>, <math>gNg^{-1} \subseteq N</math>. More explicitly, for all <math>g \in G, h \in N</math>, we have <math>ghg^{-1} \in N</math>. # For all <math>g</math> in <math>G</math>, <math>gNg^{-1} = N</math>. # For all <math>g</math> in <math>G</math>, <math>gN = Ng</math>. # <math>N</math> is a union of [[Defining ingredient::conjugacy class]]es We try to follow these conventions: * When multiple definitions are given, the same order is maintained for the definitions in the symbol-free format and the definition with symbols format. * The multiple definitions should preferably be numbered, making it easier to refer to the numbering * The definitions are ordered in a way that makes the equivalence between them as self-evident as possible. ===Separate definitions as separate subsections=== Sometimes, if the equivalent definitions are long and intricate or each requires their own machinery, they may be developed in separate subsections. The title of each subsection gives some ''title'' to the definition. For instance, the two definitions of group are labeled as the '''textbook definition''' and the '''universal algebraic definition'''. ===Equivalence of multiple definitions=== Usually, there is a separate section within the definition part explaining the equivalence of definitions, or giving a link to a page that gives a ''full proof'' of this equivalence. ==Defining ingredients== ===Linking style=== Whenever a term is being defined using other terms, put links to those terms. For instance, when defining a group property, begin with: A [[group]] is said to be ... if ... Typically, the symbol-free definition will not attempt to define other terms referenced in the definition. The definition with symbols, because it gives a more explicit description, may also give a clearer idea of other terms. For instance, in the symbol-free definition of normality, the second point states that a subgroup is normal if it is invariant under all [[inner automorphism]]s. In the definition with symbols, the explicit form of inner automorphisms is given. ===Locating defining ingredients=== The wiki markup contains information about what terms are used as ingredients in defining a particular term (using semantic MediaWiki). At the bottom of the article, a box called ''Facts'' gives a list of defining ingredients. Clicking on the magnifying glass icon next to any of these ingredients gives a list of all the other terms that use the same defining ingredient. ===Links to survey articles clarifying the definition=== For definitions of important terms, the definition section may begin with a box containing links to survey articles clarifying the definition. ==Other comments== The definition section may also contain a few other comments, in so far as these comments play a direct role in clarifying the definition. Examples are typically reserved for another section, and under special circumstances, there is a separate section titled ''Importance''. 983f5e44cdad55c1cff6737e876d67deca61119a 106 73 147 2008-07-22T22:50:20Z Vipul 2 New page: Examples are a crucial aid to learning and understanding ideas, and an '''Examples''' section is to be found in most near-complete definition articles in subject wikis, as well as in some ... wikitext text/x-wiki Examples are a crucial aid to learning and understanding ideas, and an '''Examples''' section is to be found in most near-complete definition articles in subject wikis, as well as in some fact articles. ==Goals of the examples section== a510406ccb1d88fd0044d230a152f0b0ec141a00 148 147 2008-07-22T23:23:41Z Vipul 2 wikitext text/x-wiki Examples are a crucial aid to learning and understanding ideas, and an '''Examples''' section is to be found in most near-complete definition articles in subject wikis, as well as in some fact articles. ==Goals of the examples section== The example section should provide the following: * '''Extremes testing''': These are the extreme examples of a definition or fact. For instance, an extreme example of a set might be the empty set or one-point set. An extreme example of a number might be 0 or 1. Extreme examples, even when they seem trivial and uninteresting, are important because they give a baseline understanding of the definition. Another goal of extremes is to clear misconceptions in an understanding of the definition. For instance, a person reading the definition of [[topospaces:Hausdorff space]] may naively assume that the empty space and the one-point space cannot qualify to be Hausdorff. * '''Representative range''': The examples section should provide a reasonably representative range of situations. One criterion for representativeness of examples of a collection of objects is that any statement true for every example is highly likely to be true for everything in the collection. * '''Litmus test for defining criteria''': Going through each example and justifying why it is a definition, should help a learner reinforce every aspect of the definition. A person who has read and understood every example should have acquired competence in checking the defining criteria in arbitrary situations. * '''Non-examples''': Non-examples are also often useful -- they provide a contrast with the examples and help reinforce what is special in the examples that is ''not'' true in the non-examples. fa7571219425c44642ac97f6f1f6004f2428e60a 108 74 149 2008-08-04T15:07:11Z Vipul 2 New page: Google search is a free, online web search tool. ==Use Google search== ===Start Google search=== Google search is online, or Internet-based. It can be accessed at: http://www.google.co... wikitext text/x-wiki Google search is a free, online web search tool. ==Use Google search== ===Start Google search=== Google search is online, or Internet-based. It can be accessed at: http://www.google.com Google search can also be accessed through a mobile phone, at: http://m.google.com/search Google search can also be accessed through the search bar (typically located to the right of the URL bar) on search engines such as Mozilla Firefox and Internet Explorer. ===Initiate a search query=== # Type the query you want to search in the search textbox, and press the ''Enter'' or ''Return'' key. # Google will return a page with a ranked list of result pages (termed the ''Search Engine Rankings Page''). Each result that Googe thinks is relevant to the query is presented in the form of the page title, followed by a few lines from the page that contain the search terms, followed by the page URL. # Visit the pages that you consider relevant. Alternatively, you may want to modify the search terms. ==Search syntax== ===Default syntax: all words treated with AND operator=== If a series of words is fed to Google search, Google by default looks for pages that have ''all'' the words, not necessarily next to each other. For instance, the search query: <nowiki>red blue green</nowiki> searches for pages that have all the terms ''red'', ''blue'' and ''green''. This is equivalent to the search query: <nowiki>red+blue+green</nowiki> Google search may ignore very common words like ''a'', ''an'', and ''it''. The order in which terms are entered may have an effect on the order in which results are displayed. For instance: <nowiki>red blue green</nowiki> and: <nowiki>red green blue</nowiki> return results in somewhat different order. ===Phrase search using quotes=== To search for a phrase, quotes need to be put around that phrase. For instance: <nowiki>"a bird in the hand is worth two in the bush"</nowiki> searches for ''precisely'' that phrase. Phrase searches are case-insensitive, and Google automatically looks for conversions between whitespace and hyphenation. ===The OR operator=== To make Google search for pages that have one term ''or'' another, the syntax uses the OR operator, for instance: <nowiki>red blue OR green</nowiki> searches either for pages that contain the term ''red'', and in addition, contain either the term ''blue'' or the term ''green''. ===Negation=== It is also possible to make Google search for pages that do ''not'' contain a particular search term. Each search term that is to be negated should be preceded by a - sign, for instance: <nowiki>red -blue green</nowiki> returns pages that contain the terms ''red'' and ''green'' but do ''not'' contain the term ''blue''. ==Other operators== ===Site-specific search=== Google allows users to search only within a certain site. For this, the search query should contain: <nowiki>site:sitename</nowiki> The sitename should ''not'' include the http:// part. For instance, to search on the subwiki.org site, the syntax is: <nowiki>red blue green site:subwiki.org</nowiki> The search can also be specific to subdomains, and to directories within sites, for instance: <nowiki>group site:groupprops.subwiki.org</nowiki> ===Wildcard search=== Google allows wildcard search, for instance: <nowiki>"the real * is in the *"</nowiki> This looks for phrases where the two *s could be replaced by arbitrary words (not necessarily the same word). There is no mechanism to search for matching wildcard strings (i.e., for insisting that the wildcards at different places be the same word). ===Filetype search=== Google also allows users to specify the filetype of search results. For this, the search query should contain: <nowiki>filetype:typename</nowiki> For instance, to search for PDF documents on "small world": <nowiki>"small world" filetype:pdf</nowiki> ==Other options== Not all options used for Google search are accessible in the query box. For some options, it is necessary to go to Google's advanced search page: http://www.google.com/advanced_search b9bc45dabaafc600ed590ec27b677972b7b07490 150 149 2008-08-04T15:08:46Z Vipul 2 /* Search syntax */ wikitext text/x-wiki Google search is a free, online web search tool. ==Use Google search== ===Start Google search=== Google search is online, or Internet-based. It can be accessed at: http://www.google.com Google search can also be accessed through a mobile phone, at: http://m.google.com/search Google search can also be accessed through the search bar (typically located to the right of the URL bar) on search engines such as Mozilla Firefox and Internet Explorer. ===Initiate a search query=== # Type the query you want to search in the search textbox, and press the ''Enter'' or ''Return'' key. # Google will return a page with a ranked list of result pages (termed the ''Search Engine Rankings Page''). Each result that Googe thinks is relevant to the query is presented in the form of the page title, followed by a few lines from the page that contain the search terms, followed by the page URL. # Visit the pages that you consider relevant. Alternatively, you may want to modify the search terms. ==Search syntax== ===Default syntax: all words treated with AND operator=== If a series of words is fed to Google search, Google by default looks for pages that have ''all'' the words, not necessarily next to each other. For instance, the search query: <pre>red blue green</pre> searches for pages that have all the terms ''red'', ''blue'' and ''green''. This is equivalent to the search query: <pre>red+blue+green</pre> Google search may ignore very common words like ''a'', ''an'', and ''it''. The order in which terms are entered may have an effect on the order in which results are displayed. For instance: <pre>red blue green</pre> and: <pre>red green blue</pre> return results in somewhat different order. ===Phrase search using quotes=== To search for a phrase, quotes need to be put around that phrase. For instance: <pre>"a bird in the hand is worth two in the bush"</pre> searches for ''precisely'' that phrase. Phrase searches are case-insensitive, and Google automatically looks for conversions between whitespace and hyphenation. ===The OR operator=== To make Google search for pages that have one term ''or'' another, the syntax uses the OR operator, for instance: <pre>red blue OR green</pre> searches either for pages that contain the term ''red'', and in addition, contain either the term ''blue'' or the term ''green''. ===Negation=== It is also possible to make Google search for pages that do ''not'' contain a particular search term. Each search term that is to be negated should be preceded by a - sign, for instance: <pre>red -blue green</pre> returns pages that contain the terms ''red'' and ''green'' but do ''not'' contain the term ''blue''. ==Other operators== ===Site-specific search=== Google allows users to search only within a certain site. For this, the search query should contain: <nowiki>site:sitename</nowiki> The sitename should ''not'' include the http:// part. For instance, to search on the subwiki.org site, the syntax is: <nowiki>red blue green site:subwiki.org</nowiki> The search can also be specific to subdomains, and to directories within sites, for instance: <nowiki>group site:groupprops.subwiki.org</nowiki> ===Wildcard search=== Google allows wildcard search, for instance: <nowiki>"the real * is in the *"</nowiki> This looks for phrases where the two *s could be replaced by arbitrary words (not necessarily the same word). There is no mechanism to search for matching wildcard strings (i.e., for insisting that the wildcards at different places be the same word). ===Filetype search=== Google also allows users to specify the filetype of search results. For this, the search query should contain: <nowiki>filetype:typename</nowiki> For instance, to search for PDF documents on "small world": <nowiki>"small world" filetype:pdf</nowiki> ==Other options== Not all options used for Google search are accessible in the query box. For some options, it is necessary to go to Google's advanced search page: http://www.google.com/advanced_search c99059be8d2bf4f2e032fd7a9abfca466ca058f4 108 74 151 150 2008-08-04T15:09:41Z Vipul 2 /* Other operators */ wikitext text/x-wiki Google search is a free, online web search tool. ==Use Google search== ===Start Google search=== Google search is online, or Internet-based. It can be accessed at: http://www.google.com Google search can also be accessed through a mobile phone, at: http://m.google.com/search Google search can also be accessed through the search bar (typically located to the right of the URL bar) on search engines such as Mozilla Firefox and Internet Explorer. ===Initiate a search query=== # Type the query you want to search in the search textbox, and press the ''Enter'' or ''Return'' key. # Google will return a page with a ranked list of result pages (termed the ''Search Engine Rankings Page''). Each result that Googe thinks is relevant to the query is presented in the form of the page title, followed by a few lines from the page that contain the search terms, followed by the page URL. # Visit the pages that you consider relevant. Alternatively, you may want to modify the search terms. ==Search syntax== ===Default syntax: all words treated with AND operator=== If a series of words is fed to Google search, Google by default looks for pages that have ''all'' the words, not necessarily next to each other. For instance, the search query: <pre>red blue green</pre> searches for pages that have all the terms ''red'', ''blue'' and ''green''. This is equivalent to the search query: <pre>red+blue+green</pre> Google search may ignore very common words like ''a'', ''an'', and ''it''. The order in which terms are entered may have an effect on the order in which results are displayed. For instance: <pre>red blue green</pre> and: <pre>red green blue</pre> return results in somewhat different order. ===Phrase search using quotes=== To search for a phrase, quotes need to be put around that phrase. For instance: <pre>"a bird in the hand is worth two in the bush"</pre> searches for ''precisely'' that phrase. Phrase searches are case-insensitive, and Google automatically looks for conversions between whitespace and hyphenation. ===The OR operator=== To make Google search for pages that have one term ''or'' another, the syntax uses the OR operator, for instance: <pre>red blue OR green</pre> searches either for pages that contain the term ''red'', and in addition, contain either the term ''blue'' or the term ''green''. ===Negation=== It is also possible to make Google search for pages that do ''not'' contain a particular search term. Each search term that is to be negated should be preceded by a - sign, for instance: <pre>red -blue green</pre> returns pages that contain the terms ''red'' and ''green'' but do ''not'' contain the term ''blue''. ==Other operators== ===Site-specific search=== Google allows users to search only within a certain site. For this, the search query should contain: <pre>site:sitename</pre> The sitename should ''not'' include the http:// part. For instance, to search on the subwiki.org site, the syntax is: <pre>red blue green site:subwiki.org</pre> The search can also be specific to subdomains, and to directories within sites, for instance: <pre>group site:groupprops.subwiki.org</pre> ===Wildcard search=== Google allows wildcard search, for instance: <pre>"the real * is in the *"</pre> This looks for phrases where the two *s could be replaced by arbitrary words (not necessarily the same word). There is no mechanism to search for matching wildcard strings (i.e., for insisting that the wildcards at different places be the same word). ===Filetype search=== Google also allows users to specify the filetype of search results. For this, the search query should contain: <pre>filetype:typename</pre> For instance, to search for PDF documents on "small world": <pre>"small world" filetype:pdf</pre> ==Other options== Not all options used for Google search are accessible in the query box. For some options, it is necessary to go to Google's advanced search page: http://www.google.com/advanced_search ca992d8dd4d8cda2ecd4fdafc170b69697a6b7d6 108 75 152 2008-08-04T15:10:35Z Vipul 2 Redirecting to [[Resource:Google search]] wikitext text/x-wiki #redirect [[Resource:Google search]] 3e2ed3b49db382b5ad4547eb5ca82eef8b83bad2 0 76 153 2008-08-04T15:31:06Z Vipul 2 New page: <noinclude>[[Status::Basic definition| ]][[Topic::Point-set topology| ]][[Primary wiki::Topospaces| ]]</noinclude> '''Regular space''': A topological space is termed regular if all points ... wikitext text/x-wiki <noinclude>[[Status::Basic definition| ]][[Topic::Point-set topology| ]][[Primary wiki::Topospaces| ]]</noinclude> '''Regular space''': A topological space is termed regular if all points are closed sets (the <math>T_1</math> assumption), and, given a point and a closed set not containing it, there are disjoint open sets containing the point and closed set respectively. In some definitions, the <math>T_1</math> assumption is skipped. Related terms: regularity (the property of a topological space being regular) Primary subject wiki entry: [[Topospaces:Regular space]] Also located at: [[Wikipedia:Regular space]], [[Planetmath:RegularSpace]], [[Mathworld:RegularSpace]] b509238d0556eda0533c63758b7ff1a05b001a51 0 77 154 2008-08-04T15:47:03Z Vipul 2 New page: <noinclude>[[Topic::Algebraic number theory| ]]</noinclude> '''Regular prime''': A regular prime is a prime number <math>p</math> that does not divide the class number of the cyclotomic fi... wikitext text/x-wiki <noinclude>[[Topic::Algebraic number theory| ]]</noinclude> '''Regular prime''': A regular prime is a prime number <math>p</math> that does not divide the class number of the cyclotomic field obtained by adjoining <math>p^{th}</math> roots of unity to the field of rational numbers. No subject wiki entry. Also located at [[Wikipedia:Regular prime]], [[Mathworld:RegularPrime]], [[Planetmath:RegularPrime]] 9629542c36aae16cc4e4f6ece02c0b700d38b4bf 0 78 155 2008-08-04T15:49:41Z Vipul 2 New page: '''Regular polygon''': A polygon in the Euclidean plane is termed regular if all its sides have equal length and all its angles (the internal angles at its vertices) have equal measure. N... wikitext text/x-wiki '''Regular polygon''': A polygon in the Euclidean plane is termed regular if all its sides have equal length and all its angles (the internal angles at its vertices) have equal measure. No subject wiki entry. Also located at [[Wikipedia:Regular polygon]], [[Mathworld:RegularPolygon]], [[Planetmath:RegularPolygon]] d3d22307ada7159e423726b5bc44ee9b180cd2aa 0 79 156 2008-08-04T15:55:12Z Vipul 2 New page: <noinclude>[[Topic::Economics| ]][[Status::Basic definition| ]]</noinclude> '''Normal good''': A good for which demand increases with an increase in (real) income (other things remaining t... wikitext text/x-wiki <noinclude>[[Topic::Economics| ]][[Status::Basic definition| ]]</noinclude> '''Normal good''': A good for which demand increases with an increase in (real) income (other things remaining the same). No subject wiki entry. Also located at [[Wikipedia:Normal good]] 8cef6556a62f3d391895f6f6140ca861013cda3b 157 156 2008-08-04T15:55:39Z Vipul 2 wikitext text/x-wiki <noinclude>[[Topic::Economics| ]][[Status::Basic definition| ]]</noinclude> '''Normal good''': A good for which demand increases with an increase in (real) income (other things remaining the same). No subject wiki entry. Also located at [[Wikipedia:Normal good]] b1565448eafe9d9d6f728001c81a6eb7e9fce1d8 0 80 158 2008-08-04T15:58:46Z Vipul 2 New page: '''Ordinary good''': A good for which demand increases with a drop in price (other things remaining the same). Opposite: [[Giffen good]] No subject wiki entry found. Also located at: [[... wikitext text/x-wiki '''Ordinary good''': A good for which demand increases with a drop in price (other things remaining the same). Opposite: [[Giffen good]] No subject wiki entry found. Also located at: [[Wikipedia::Ordinary good]] 66057fc8eb747cd56460ced847f3a4b3b23ebdba 159 158 2008-08-04T15:59:00Z Vipul 2 wikitext text/x-wiki '''Ordinary good''': A good for which demand increases with a drop in price (other things remaining the same). Opposite: [[Giffen good]] No subject wiki entry found. Also located at: [[Wikipedia:Ordinary good]] ad322551c297b9ce911980cff0ecdee548c3c847 0 81 160 2008-08-04T16:04:04Z Vipul 2 New page: {{:Normal form (formal language theory)}} {{:Normal form (database theory)}} {{:Normal form (game theory)}} wikitext text/x-wiki {{:Normal form (formal language theory)}} {{:Normal form (database theory)}} {{:Normal form (game theory)}} 3916f0b5c82edd1db7e3f0ff866e222c8ea7e4a0 162 160 2008-08-04T16:05:40Z Vipul 2 wikitext text/x-wiki ===In formal language theory=== {{:Normal form (formal language theory)}} ===In database theory=== {{:Normal form (database theory)}} ===In game theory=== {{:Normal form (game theory)}} 8637f08fea636cd093d294f95d8191d506ef7df5 0 6 161 118 2008-08-04T16:04:14Z Vipul 2 wikitext text/x-wiki Main form: ''normal'', [[word type::adjective]]. Used both descriptively (adding detail) and restrictively (restricting the subject). In sciences, typically used in a binary sense (something is either normal or is not normal), whereas in the social sciences and daily parlance, typically used to describe position on a wider scale. Related forms: ''normality'' (how normal something is), ''normalcy'' (the condition of being normal), ''normalize''/''normalise'' (make normal) Same roots: [[Root related::norm]] Typical use: * Something typical, expected, or standard. Similar words: [[Similar::ordinary]], [[Similar::typical]], [[Similar::standard]], [[Similar::average]] * Something good or desirable, or something one optimistically hopes for. Similar words: [[Similar::conventional]], [[Similar::standard]], [[Similar::canonical]] * Upright or perpendicular. [[Similar::orthogonal]], [[Similar::perpendicular]] Opposite words: [[Opposite::abnormal]], [[Opposite::paranormal]] Derived words: [[Derived::conormal]], [[Derived::abnormal]], [[Derived::subnormal]], [[Derived::supernormal]], [[Derived::paranormal]], [[Derived::contranormal]], [[Derived::malnormal]], [[Derived::Pseudonormal]], [[Derived::orthonormal]], [[Derived::seminormal]], [[Derived::quasinormal]] ==Mathematics== {{:Normal (mathematics)}} ==Computer science== {{:Normal (computer science)}} ==Physics== {{:Normal (physics)}} ==Chemistry== {{:Normal (chemistry)}} ==Economics== {{:Normal (economics)}} e6954adbdd27c7e78c1354588494794f8fe8f033 190 161 2008-08-05T22:00:11Z Vipul 2 wikitext text/x-wiki Main form: ''normal'', [[word type::adjective]]. Used both descriptively (adding detail) and restrictively (restricting the subject). In sciences, typically used in a binary sense (something is either normal or is not normal), whereas in the social sciences and daily parlance, typically used to describe position on a wider scale. Related forms: ''normality'' (how normal something is), ''normalcy'' (the condition of being normal), ''normalize''/''normalise'' (make normal) Same roots: [[Root related::norm]] Typical use: * Something customary, typical, routine, expected, or standard. Similar words: [[Similar::ordinary]], [[Similar::typical]], [[Similar::standard]], [[Similar::average]], [[Similar::expected]], [[Similar::routine]] * Something good or desirable, or something one optimistically hopes for. Similar words: [[Similar::conventional]], [[Similar::standard]], [[Similar::canonical]] * Upright or perpendicular. [[Similar::orthogonal]], [[Similar::perpendicular]] Opposite words: [[Opposite::abnormal]], [[Opposite::paranormal]], [[Opposite::strange]] Derived words: [[Derived::conormal]], [[Derived::abnormal]], [[Derived::subnormal]], [[Derived::supernormal]], [[Derived::paranormal]], [[Derived::contranormal]], [[Derived::malnormal]], [[Derived::Pseudonormal]], [[Derived::orthonormal]], [[Derived::seminormal]], [[Derived::quasinormal]] ==Mathematics== {{:Normal (mathematics)}} ==Computer science== {{:Normal (computer science)}} ==Physics== {{:Normal (physics)}} ==Chemistry== {{:Normal (chemistry)}} ==Economics== {{:Normal (economics)}} 1ebd3a2d93e8c0c8abc9db6ab30522ee064b25b7 0 82 163 2008-08-04T16:07:06Z Vipul 2 New page: ===In formal language theory=== {{:Regular language}} wikitext text/x-wiki ===In formal language theory=== {{:Regular language}} 0b15bc1d3eb55841717982b997ed1667dfa93cba 165 163 2008-08-04T16:10:26Z Vipul 2 wikitext text/x-wiki ===In formal language theory=== {{:Regular language}} {{:Regular expression}} ===In coding theory=== {{:Regular code}} f0c507378c911e13cbaf21ec4d100e41f0e58ac4 0 57 164 143 2008-08-04T16:07:49Z Vipul 2 wikitext text/x-wiki ===In topology=== {{:Regular space}} {{:Regular covering}} ===In differential geometry=== {{:Regular value}} ===In the theory of monoids and semigroups=== {{:Regular semigroup}} ===In group theory/representation theory=== {{:Regular group action}} ===In geometry=== {{:Regular polygon}} {{:Regular polyhedron}} {{:Regular polytope}} ===In commutative algebra=== {{:Regular ring}} ===In algebraic geometry=== {{:Regular map}} ===In number theory=== {{:Regular prime}} ===In graph theory=== {{:Regular graph}} ===In axiomatic set theory=== {{:Regular cardinal}} ===In measure theory=== {{:Regular measure}} ===In category theory=== {{:Regular category}} c1843f1cf08796f2257f662c6380ac80f74f553c 200 164 2008-08-05T22:55:51Z Vipul 2 /* In commutative algebra */ wikitext text/x-wiki ===In topology=== {{:Regular space}} {{:Regular covering}} ===In differential geometry=== {{:Regular value}} ===In the theory of monoids and semigroups=== {{:Regular semigroup}} ===In group theory/representation theory=== {{:Regular group action}} ===In geometry=== {{:Regular polygon}} {{:Regular polyhedron}} {{:Regular polytope}} ===In commutative algebra=== {{:Regular ring}} {{:Regular sequence}} ===In algebraic geometry=== {{:Regular map}} ===In number theory=== {{:Regular prime}} ===In graph theory=== {{:Regular graph}} ===In axiomatic set theory=== {{:Regular cardinal}} ===In measure theory=== {{:Regular measure}} ===In category theory=== {{:Regular category}} de56f10527b04a2ef8b875d3bf245396c14decbe 0 26 166 47 2008-08-04T16:14:14Z Vipul 2 wikitext text/x-wiki ===In general/inorganic chemistry=== {{:Periodic table group}} ===In organic chemistry=== {{:Functional group}} 01e4f3a541c6e29acc8f682e09a414c9e63423af 0 24 167 45 2008-08-04T16:17:27Z Vipul 2 wikitext text/x-wiki The term ''group'' loosely refers to a collection of many similar items that behave in a cohesive manner. In particular subjects, the usage tends to emphasize the cohesiveness and group identity. Related forms: ''grouping'' (putting things together in a group) Typical usage: * As a collection of similar things. Similar terms: [[similar::set]], [[similar::cluster]], [[similar::collection]], [[similar::class]] * As a cohesive unit that works in synchrony or harmony. Similar terms: [[similar::team]], [[similar::organization]] ==In mathematics== {{:Group (mathematics)}} ==In chemistry== {{:Group (chemistry)}} ==In sociology== {{:Group (sociology)}} 4bcaacecb9cca182ad134b9bf9225ab5ffda924a 168 167 2008-08-04T16:20:32Z Vipul 2 wikitext text/x-wiki The term ''group'' loosely refers to a collection of many similar items that behave in a cohesive manner. In particular subjects, the usage tends to emphasize the cohesiveness and group identity. Related forms: ''grouping'' (putting things together in a group) Typical usage: * As a collection of similar things. Similar terms: [[similar::set]], [[similar::cluster]], [[similar::collection]] * As a collection obtained as one piece of a classification. Similar terms: [[similar::family]], [[similar::class]], [[similar::genus]], [[similar::species]] * As a cohesive unit that works in synchrony or harmony. Similar terms: [[similar::team]], [[similar::organization]] ==In mathematics== {{:Group (mathematics)}} ==In chemistry== {{:Group (chemistry)}} ==In sociology== {{:Group (sociology)}} af9ef96a108c34534a02b28b1a4fbad49276fc8c 0 83 169 2008-08-04T16:22:26Z Vipul 2 New page: ===In set theory=== {{:Proper subset}} ===In point-set topology=== {{:Proper map}} {{:Proper space}} ===In algebraic geometry=== {{:Proper morphism}} wikitext text/x-wiki ===In set theory=== {{:Proper subset}} ===In point-set topology=== {{:Proper map}} {{:Proper space}} ===In algebraic geometry=== {{:Proper morphism}} c73ac55cef7748093ae45bab97cb311a02ef08fb 0 84 170 2008-08-04T16:23:24Z Vipul 2 New page: ==In mathematics== {{:Proper (mathematics)}} wikitext text/x-wiki ==In mathematics== {{:Proper (mathematics)}} 55f73db65005a7b47aeede55f5fd73eddc09fb83 0 85 171 2008-08-04T16:24:37Z Vipul 2 New page: ==In mathematics== {{:Normal form (mathematics)}} ==In computer science== {{:Normal form (computer science)}} wikitext text/x-wiki ==In mathematics== {{:Normal form (mathematics)}} ==In computer science== {{:Normal form (computer science)}} 77041fcc35323ab3ba4bd53ffd1e7a772212fe98 0 86 172 2008-08-04T16:24:54Z Vipul 2 New page: ===In linear algebra=== {{:Normal form (linear algebra)}} wikitext text/x-wiki ===In linear algebra=== {{:Normal form (linear algebra)}} f7fb0d93eacc8cfa708bb09ee755da91d967916a 0 87 173 2008-08-04T16:26:07Z Vipul 2 New page: ===In formal language theory=== {{:Normal form (formal language theory)}} ===In Boolean arithmetic=== {{:Normal form (Boolean arithmetic)}} ===In database theory=== {{:Normal form (da... wikitext text/x-wiki ===In formal language theory=== {{:Normal form (formal language theory)}} ===In Boolean arithmetic=== {{:Normal form (Boolean arithmetic)}} ===In database theory=== {{:Normal form (database theory)}} ===In game theory=== {{:Normal form (game theory)}} c189fbd0647005304b10a03936ac5c6b4ac1ed73 0 88 174 2008-08-04T16:32:16Z Vipul 2 New page: ===In programming=== {{:Extreme programming}} wikitext text/x-wiki ===In programming=== {{:Extreme programming}} 59cf146fa572898ee0cefb79ae641546be2f095e 0 89 175 2008-08-04T16:32:44Z Vipul 2 New page: ==In computer science== {{:Extreme (computer science)}} wikitext text/x-wiki ==In computer science== {{:Extreme (computer science)}} 0c02c97e3564576be83e25ff55da0af62e8f1988 0 90 176 2008-08-04T16:34:47Z Vipul 2 New page: ==In mathematics== {{:Ordinary (mathematics)}} ==In economics== {{:Ordinary (economics)}} wikitext text/x-wiki ==In mathematics== {{:Ordinary (mathematics)}} ==In economics== {{:Ordinary (economics)}} bf8470ab9f229d72357400204808b74e5f35969c 177 176 2008-08-04T16:38:25Z Vipul 2 wikitext text/x-wiki ==Mathematics== {{:Ordinary (mathematics)}} ==Economics== {{:Ordinary (economics)}} d8ce6432d5ade60e199400d3447b49688d48780c 191 177 2008-08-05T22:00:25Z Vipul 2 wikitext text/x-wiki Main form: ''ordinary'', [[word type::adjective]]. Used restrictively (restricting the subject) or emphatically (to emphasize the subject, without actually adding any detail). In sciences, typically used in a binary sense (something is either normal or is not normal), whereas in the social sciences and daily parlance, typically used to describe position on a wider scale. Typical use: * Something customary, typical, expected, routine, or average. [[Similar::typical]], [[Similar::standard]], [[Similar::average]], [[Similar::routine]], [[Similar::normal]], [[Similar::expected]] * Something mundane or boring, lacking in special qualities. Opposite words: [[Opposite::special]], [[Opposite::extraordinary]] Derived words: [[Derived::Extraordinary]] ==Mathematics== {{:Ordinary (mathematics)}} ==Economics== {{:Ordinary (economics)}} df051d0c97d6923255a062477d8184d5b65a3c75 0 91 178 2008-08-04T16:39:08Z Vipul 2 New page: ==Mathematics== {{:Choice (mathematics)}} wikitext text/x-wiki ==Mathematics== {{:Choice (mathematics)}} 6908053fe8d569902b35243e97362c40027216d5 0 92 179 2008-08-04T16:43:00Z Vipul 2 New page: ==Philosophy of science and social science== {{:Open problem}} ==Mathematics== {{:Open (mathematics)}} ==Computer science== {{:Open (computer science)}} wikitext text/x-wiki ==Philosophy of science and social science== {{:Open problem}} ==Mathematics== {{:Open (mathematics)}} ==Computer science== {{:Open (computer science)}} e0df66ffece8a7d75ac9f14e6593d4024f0e4d44 106 72 180 146 2008-08-05T12:29:39Z Vipul 2 wikitext text/x-wiki Each subject wiki has certain kinds of articles called ''definition articles'' or ''terminology articles''. Each such article is about a term being defined, and the essential component of any such article is the '''definition''' of that term. The definition forms a separate section of the article, and this section is designed keeping a number of objectives in mind. In this article, we use the technical word ''definiendum'' for the term being defined, though it may also be used more loosely for the concept, idea, or notion behind the term. ==Goals of the definition section== ===Potential audience for a definition section=== Some people know the definiendum and are looking for its definition. This could include: * People encountering the definiendum for the first time. * People who've seen the definiendum before, and know what it means, but want a precise definition. * People who already know one definition, but want to know/read multiple definitions. On the other hand, there could also be people who know the definition already, and are looking for the precise term having that definition. ===What the definition should provide=== A definition should be ''clear'', ''succinct'', ''actionable'', and ''interesting''. * It must clearly answer the question: ''what is this''. * If the definition provides a differentiation or augmentation of an existing concept (for instance, a particular breed of dog, or a particular property of groups, or a particular type of number) then it should give a clear differentiating criterion. * It should give an idea of what related terms and facts are and how one can explore the notion better. * It should be amenable to reverse search. ==Challenges in presenting intensional definitions== Most definitions in mathematics are ''intensional definitions''. A mathematical object is defined by specifying necessary and sufficient conditions for what it means to be such an object. Intensional definitions are also found for some simple terms outside mathematics. Intensional definitions are in sharp contrast with extensional definitions, that define a concept through a range of representative examples. They also differ, for instance, from sense definitions, that define an object in terms of how it appears to the senses. The main feature of definitions in mathematics, and of intensional definitions in general, is that the definition is ''de facto'' always true, and a complete description. Thus, it is not usually necessary to justify this definition against prior notions of what the term or concept means. This sharply contrasts with definitions involving terms that are already in daily use and carry strong and diverse connotations in people's minds: terms like ''society'', ''environment'', ''terrorism'', and ''patriotism''. ===The challenge of ingredients and dependencies=== To define one term often requires other terms as ingredients. For instance: * The definition of a prime number relies on the definitions of natural number and divisor. * The typical definition of a [[groupprops:finitely generated group|finitely generated group]] relies on the definitions of [[groupprops:group|group]], [[groupprops:generating set of a group|generating set of a group]], and, of course, the notion of ''finiteness''. * The typical definition of Hausdorff space relies on the definition of topological space. To cope with this challenge, we need to decide what ingredients are ''taken for granted'' (i.e., their definitions are not included) and what ingredients are to be defined along with the main term we're defining. The way chosen in subject wikis is to have a certain '''primitive''' definition -- a definition that has as few dependencies as is possible while maintaining the conceptual integrity. Then, we create further definitions that are shorter and simpler, but rely on concepts beyond just the primitives. ===The challenge of multiple definitions=== Important terms can be defined from multiple perspectives. In fact, the equivalence of different definitions often gives an insight into why the concept is important. Multiple definitions are both a boon and a challenge. Conventional wisdom suggests that giving a barrage of definitions can create a cognitive overload that makes it hard for people to grasp a new idea. On the other hand, theories of learning, association and memory also suggest that the more ''hooks'' we have on an idea, the easier it is to understand. In addition to genuinely different definitions, there are also equivalent definitions obtained by a slight rewording and change in the defining ingredients. For instance, as discussed in the previous subsection, a primitive definition may be made more compact and conceptually elegant by introducing further defining ingredients. In this case, both the primitive and the compacted definition are stated as equivalent definitions. This again has advantages for definition cognition and definition understanding: in the effort to try to reconcile various rewordings of the definition, somebody trying to learn and understand the definition gets a better understanding of what the term means. ===The challenge of symbols and expressions=== For new definitions, specially complicated ones, notation is often necessary, or at any rate, helpful. Thus, we often start definitions like: A ... is a set <math>K</math> with .... Symbols and expressions can be useful to set up a definition, particularly when the number of variables involved, or the nature of operations, is too large for the nouns and pronouns and constructs of natural language. However, they also have a crucial disadvantage: they may not carry conceptual clarity. In many situations, the mind relates more clearly to natural language than to variables and formulas (we've learned natural language since preschool, but algebra is usually introduced in middle school). Also, symbols are often at odds with the use of further ingredients or concepts. The subject wikis meet this challenge through a somewhat unusual method: most definitions carry with them two sections, the '''symbol-free definition''', which seeks to use natural language only without any variables or symbols, and the '''definition with symbols''', which expresses everything in symbols (so whatever objects are referred to are given symbols). ===The challenge of stickiness=== Definitions can often be dull and dry, and the tremendous conceptual compactification they represent may not be obvious to people just starting. A definition in isolation, thus, may not be too sticky. The subject wikis try to remedy this in a number of ways. The most commonly implemented way is the use of '''Quick Phrases'''. On some definition pages, a '''Quick Phrases''' box appears right on top of the definition section, with phrases that could be used to compactly remember the definition, or the essential meaning of the term. Accuracy is sometimes sacrificed for memorability or analogical usefulness in the quick phrases. ===The challenge of misinterpretation and confusion=== Definitions, read in isolation, are prone to being poorly understood, misinterpreted, and wrongly applied. All these cannot be remedied simply by a better definition. Other techniques, including the use of examples, variations, analogies, formalisms for definitions, and further exploration and analysis of the concept, are necessary to ensure a proper and useful understanding of the term being defined. Nonetheless, certain things are done on the subject wikis to reduce the chances of confusion: * '''Redundancy''' and '''multiple perspectives''': The same definition is written in many different ways, both with and without symbols, as well as with quick phrases * Links to '''definition understanding''' pages, '''funky definitions''' pages (pages with preposterous, weird, and sometimes circular definitions for the term) * Links to '''definition equivalence''' pages for proofs of the equivalence of definitions ===The challenge of motivation=== Probably the biggest challenge, that definitions on subject wikis fail to satisfy, is the challenge of motivation. ''Why'' is this definition important? ''Why'' were things defined this way, rather than that? Conventional wisdom suggests that every definition should be preceded by examples. This is often excellent pedagogy; however, in a structure of this kind, there is no natural notion of ''order'', and illustrative examples that build up to the definition may take up too much space on the definition page, and may not be very useful to people who want a quick review of the definition. On the other hand, the linked structure of the wiki makes it possible to do the examples elsewhere, but link to them, allowing people to zoom in. Some of the special features in the definition section to cater to the challenge of motivation are: * Links to definition understanding pages. These pages often explore the definition and various aspects of the definition in detail. * The definition section is preceded by a '''History''' section that may give information, links from the historical perspective. The definition section is succeeded by an '''Examples''' section that explores examples of various kinds (learn more at [[Subwiki:Examples]]). ==Genus-differentia definitions== Many definitions are of the ''genus-differentia'' kind. These define new terms by providing differentiating criteria within an existing genus. For instance: * The definition of ''prime number'' starts with the ''genus'' (generic term) of natural numbers and provides the ''differentiating criterion'' (differentiae specificae) of being prime. * The definition of ''skyscraper'' starts with the ''genus'' of buildings and provides the ''differentiating criterion of being tall. * The definition of ''e-book'' starts with the ''genus'' of book and provides the ''differentiating criterion'' of being in an electronic format. Genus-differentia definitions are of different kinds, and each operates somewhat differently: # [[Subwiki:Property definition|Property definition]]s: These are most common in precise fields like mathematics where practically all definitions are intensional. Given a collection of objects, or a ''context space'', a property on that context space is something that everything in the context space either satisfies or does ''not'' satisfy. Thus, a property picks out a subcollection of the context space comprising those things that ''do'' satisfy the property. # [[Subwiki:Modifier definition|Modifier definition]]s: These are found both in precise fields like mathematics and in other fields. Here, the differentia may not strictly lie within the genus, but rather, may modify it somewhat. For instance, ''ecoterrorism'' is obtained by modifying ''terrorism'', but may not itself be a form of terrorism. # Relationships of physical containment: Here, the definiendum is physically contained inside the general term. For instance, the general term might be the city of London while the definiendum might be Bloomsbury, a district of London). # Taxonomical classification: For instance, the Linnaean taxonomy for living beings. ==Formats of definition== ===Quick phrases=== Quick phrases are to be seen at the top of definition sections in some articles. These are phrases which one can use to remember the term being defined in a very easy way. For instance, a quick phrase for a [[prime number]] is ''number with no nontrivial factorization''. For quick phrases, conceptual ease matters more than precise meaning, so some of the quick phrases may not make precise mathematical sense. ===Symbol-free definition=== A symbol-free definition is a definition that avoids that use of algebraic symbols and notation, relying instead on constructs of natural language such as pronouns and prepositions. For instance, here's a symbol-free definition of [[permutable subgroup]]: {{quotation|A [[subgroup]] of a [[group]] is termed '''permutable''' if its [[product of subgroups|product]] with every subgroup of the group is a subgroup, or equivalently, if it commutes with every subgroup.}} Here, symbols for the group and the two subgroups ar enot introduced. Rather, constructs of natural language, in particular, the pronoun ''it'', are used. Sometimes 1-2 symbols may be introduced in a symbol-free definition to avoid ambiguous pronouns. Some advantages of symbol-free definitions: # More effective for reverse search i.e. locating the term from the definition. # Might be easier to read if you already know the definition and are trying to recall it (because of less demand on working memory) # Easier to keep in mind, memorize, and act upon in more diverse situations where the notation and symbols differ # More effective linking with other concepts and ideas ===Definition with symbols=== This gives the definition with explicit introduction of symbols for every quantified object. For instance the definition with symbols of the property of being a [[permutable subgroup]] is: {{quotation|A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''permutable''' if for every subgroup <math>K</math>, <math>HK=KH</math>.}} The definition with symbols has a number of advantages, for instance: * It may be better for first-time reading * It may be more actionable * It may require less knowledge of the subject or of terminology in the subject ==Multiple definitions== ===Equivalent definitions as a list=== Often, the same term may have multiple equivalent definitions. In this case, all the equivalent forms are stated as bullet points. The same format is repeated in the symbol-free definition and in the definition with symbols. For instance, the symbol-free definition of [[normal subgroup]]: A [[subgroup]] of a [[group]] is said to be '''normal''' if it satisfies the following equivalent conditions: # It is the [[kernel]] of a [[homomorphism]] from the group. # It is invariant under all [[inner automorphism]]s. Thus, normality is the [[invariance property]] with respect to the property of an automorphism being inner. This definition also motivates the term ''invariant subgroup'' for normal subgroup (which was used earlier). # It equals each of its [[Defining ingredient::conjugate subgroups|conjugates]] in the whole group. This definition also motivates the term ''self-conjugate subgroup'' for normal subgroup (which was used earlier). # Its [[left coset]]s are the same as its [[right coset]]s (that is, it commutes with every element of the group) # It is a union of [[conjugacy class]]es. The definition with symbols of [[normal subgroup]]: A subgroup <math>N</math> of a group <math>G</math> is said to be '''normal''' in <math>G</math> (in symbols, <math>N \triangleleft G</math> or <math>G \triangleright N</math>{{subgroup-notation-page}}) if the following equivalent conditions hold: # There is a [[homomorphism]] <math>\phi</math> from <math>G</math> to another group such that the [[kernel]] of <math>\phi</math> is precisely <math>N</math>. # For all <math>g</math> in <math>G</math>, <math>gNg^{-1} \subseteq N</math>. More explicitly, for all <math>g \in G, h \in N</math>, we have <math>ghg^{-1} \in N</math>. # For all <math>g</math> in <math>G</math>, <math>gNg^{-1} = N</math>. # For all <math>g</math> in <math>G</math>, <math>gN = Ng</math>. # <math>N</math> is a union of [[Defining ingredient::conjugacy class]]es We try to follow these conventions: * When multiple definitions are given, the same order is maintained for the definitions in the symbol-free format and the definition with symbols format. * The multiple definitions should preferably be numbered, making it easier to refer to the numbering * The definitions are ordered in a way that makes the equivalence between them as self-evident as possible. ===Separate definitions as separate subsections=== Sometimes, if the equivalent definitions are long and intricate or each requires their own machinery, they may be developed in separate subsections. The title of each subsection gives some ''title'' to the definition. For instance, the two definitions of group are labeled as the '''textbook definition''' and the '''universal algebraic definition'''. ===Equivalence of multiple definitions=== Usually, there is a separate section within the definition part explaining the equivalence of definitions, or giving a link to a page that gives a ''full proof'' of this equivalence. ==Defining ingredients== ===Linking style=== Whenever a term is being defined using other terms, put links to those terms. For instance, when defining a group property, begin with: A [[group]] is said to be ... if ... Typically, the symbol-free definition will not attempt to define other terms referenced in the definition. The definition with symbols, because it gives a more explicit description, may also give a clearer idea of other terms. For instance, in the symbol-free definition of normality, the second point states that a subgroup is normal if it is invariant under all [[inner automorphism]]s. In the definition with symbols, the explicit form of inner automorphisms is given. ===Locating defining ingredients=== The wiki markup contains information about what terms are used as ingredients in defining a particular term (using semantic MediaWiki). At the bottom of the article, a box called ''Facts'' gives a list of defining ingredients. Clicking on the magnifying glass icon next to any of these ingredients gives a list of all the other terms that use the same defining ingredient. ===Links to survey articles clarifying the definition=== For definitions of important terms, the definition section may begin with a box containing links to survey articles clarifying the definition. ==Other comments== The definition section may also contain a few other comments, in so far as these comments play a direct role in clarifying the definition. Examples are typically reserved for another section, and under special circumstances, there is a separate section titled ''Importance''. 8f7f47adcd64b4a8a5789a979e1b6914cdfaae9b 8 93 181 2008-08-05T16:43:13Z Vipul 2 New page: * navigation ** mainpage|mainpage ** recentchanges-url|recentchanges ** randompage-url|randompage * subject wikis ** Groupprops:Main Page|Groupprops ** Topospaces:Main Page|Topospaces ** C... wikitext text/x-wiki * navigation ** mainpage|mainpage ** recentchanges-url|recentchanges ** randompage-url|randompage * subject wikis ** Groupprops:Main Page|Groupprops ** Topospaces:Main Page|Topospaces ** Commalg:Main Page|Commalg ** Diffgeom:Main Page|Diffgeom ** Measure:Main Page|Measure ** Noncommutative:Main Page|Noncommutative ** Companal:Main Page|Companal ** Cattheory:Main Page|Cattheory 8b97574d2f4adb326f941e1d76fd2f87ec666d56 106 94 182 2008-08-05T17:17:16Z Vipul 2 New page: ''Formalisms'' are formal expression systems, usually involving symbols, that may be used to define a term, state a fact, or describe a relationship between existing ideas. Formalisms rang... wikitext text/x-wiki ''Formalisms'' are formal expression systems, usually involving symbols, that may be used to define a term, state a fact, or describe a relationship between existing ideas. Formalisms range from mathematically and technically rigorous specifications to loose analogical formalisms. Each subject wiki has its own collection of formalisms. This article discusses, with examples and illustrations, the general principles behind the use of formalisms. ==How formalisms arise== ===Formalisms as a common pattern=== The idea behind formalisms for defining terms is that, often, a lot of closely related definitions have a similar structure, with some inputs to the structure changed. For instance, many definitions of group properties have the form: ''a group where every subgroup has property <math>p</math>''. Many definitions of topological space properties, closely related to compactness, have the form: ''a topological space where every open cover satisfying property <math>p</math> has a refinement having property <math>q</math>''. A formalism for definitions of this structure helps ''unite'' all these definitions and provides insight into how to manipulate the common features across all the terms. The advantages are: * It allows for grouping together of terms expressible using the same formalism * It helps in reverse search * It helps provide a better understanding of the relation between different terms * It sometimes suggests general tools for reasoning that can be applied whenever something is expressible in the given formalism, allowing for a better understanding of individual proofs. ===Formalisms as a language at the right level of generality=== While some formalisms are simply a way of recognizing and acknowledging existing patterns, other formalisms involve a more proactive approach in ''creating'' formalisms. These may involve certain special language tools within the subject that specialize in expressing a diverse range of ideas, while at the same time cannot be used to express ''everything''. For instance, the languages of model theory, category theory, and universal algebra provide some general frameworks in which a large number (though far from all) of definitions, facts and relationships can be stated. ==Presenting formalisms== ===Formalisms in definition articles=== In articles about terminology (called ''terminology articles'' or ''definition articles''), there may be a separate section title '''Formalisms'''. This section is usually after the '''History''', '''Definition''', and '''Examples''' sections. One subsection is devoted to each formalism. The heading of the subsection usually states the name of the formalism, or the ''operator'' used to obtain that formalism. Just below the heading is a box stating the general name of the formalism or operator, along with a link to more information about the formalism and to other terms expressible using the formalism. This is followed by a precise explanation of how the given term is expressed using the formalism. ===Formalisms in proof articles=== Proofs involving terms expressible using a certain formalism can occasionally be simplified using general techniques for manipulating expressions in that formalism. Usually, a standard proof is given in any case, and an ''alternative'' proof in terms of the formalism is provided in a separate section or subsection. This proof clearly translates the statement in terms of the formalism and explains the manipulation rules being used. 31b62ff7c0caf9828412c6698588def5eaed136c 106 95 183 2008-08-05T17:27:24Z Vipul 2 New page: Each subject wiki has certain kinds of articles called ''definition articles'' or ''terminology articles''. Each such article is about a term being defined. Definition articles form an ext... wikitext text/x-wiki Each subject wiki has certain kinds of articles called ''definition articles'' or ''terminology articles''. Each such article is about a term being defined. Definition articles form an extremely important part of every subject wiki. This article discusses the overall design and organization of definition articles on subject wikis. 385aed56560ec429bdefdf60db9e67414dbbd587 184 183 2008-08-05T18:16:29Z Vipul 2 wikitext text/x-wiki Each subject wiki has certain kinds of articles called ''definition articles'' or ''terminology articles''. Each such article is about a term being defined. Definition articles form an extremely important part of every subject wiki. This article discusses the overall design and organization of definition articles on subject wikis. We'll use the word ''definiendum'' to describe the kind of term being defined. ==Article-tagging templates== There are boxes at the top of many definition articles on subject wikis. These boxes give general information about the definiendum, such as the ''type'' of term it is. Each such box is generated by an ''article-tagging template''. {{further|[[Subwiki:Definition article-tagging template]]}} ==Sections of the definition article== ===History section=== {{further|[[Subwiki:History section in definition articles]]}} This is not a mandatory section, but when it exists, it is usually at the beginning of the article, ''before'' the definition section. The history section provides a ''brief'' overview of the origin of the term and of the notion, along with the name of the person who came up with the term. This section ''might'' also contain further information about the evolution of the term or concept since it was introduced. ''Links from this section'': If more detailed information on the history of the concept or term is available on a separate page, a link to that page is provided. ===Definition section=== {{further|[[Subwiki:Definition]]}} This section is mandatory for all definition articles. The definition section, titled '''Definition''', contains ''at least one'' definition of the definiendum. It may contain multiple equivalent definitions, and may also contain the same definition stated in different ways (for instance, in words and in symbols). Some definition sections have a ''Quick phrases'' box on top with quick, easy-to-remember phrases that capture the meaning of the term. ''Links from this section'': The section links to pages explaining the equivalence of multiple definitions, or pages proving facts implicit in the formulation of the definition. It also links to survey articles that parse and study the definition carefully and to survey articles that consider further alternative definitions. ===Examples section=== {{further|[[Subwiki:Examples]]}} This section is recommended for all definition articles, though it is currently not present in too many of them. The section may be divided into subsections containing different kinds of examples. The aim is for examples to provide a representative range, cover the extremes, and provide a litmus test for definition understanding. Non-examples may also be included to contrast with examples. ===Formalisms section=== {{further|[[Subwiki:Formalisms]]}} ===Relation section=== {{further|[[Subwiki:Relations in definition article]]}} ===Facts section or more refined section choices=== {{fillin}} e6aabd8cbcb7f20298a6aaf93c3d916e530615bb 10 96 185 2008-08-05T18:17:00Z Vipul 2 New page: :''Further information: {{{1}}}'' wikitext text/x-wiki :''Further information: {{{1}}}'' a6cdd390093a926037e0f67ea5a62740a4520be6 10 97 186 2008-08-05T18:19:50Z Vipul 2 New page: ''Fill this in later'' wikitext text/x-wiki ''Fill this in later'' 7f59663a4bcf61a8ba35251b148e48567c5d3b52 106 98 187 2008-08-05T20:44:50Z Vipul 2 New page: '''Property-theoretic categorization''' is part of the broader property-theoretic paradigm of organization followed on some subject wikis, particularly those in mathematics. Properties ar... wikitext text/x-wiki '''Property-theoretic categorization''' is part of the broader property-theoretic paradigm of organization followed on some subject wikis, particularly those in mathematics. Properties are a more precise formulation of the general idea of genus and differentia. ==Properties== A ''property'' over a collection of objects is something which every object either ''has'' (or ''satisfies'') or ''does not have'' (or ''does not satisfy''). Examples from elementary mathematics: * Being prime is a ''property'' over the collection of natural numbers. Every natural number either ''has'' the property of being prime, or ''does not have'' the property of being prime. * Being positive is a ''property'' over the collection of real numbers. Every real number either ''is'' positive or is ''not'' positive. * Being isosceles is a ''property'' over the collection of triangles. Every triangle either ''is'' isosceles or ''is not'' isosceles. Examples from higher mathematics: * Being Abelian is a ''property'' over the collection of groups. Every group either ''is'' Abelian or ''is not'' Abelian. * Being compact is a ''property'' over the collection of topological spaces. Every topological space either ''is'' compact or ''is not'' compact. The collection of objects over which the property is being evaluted is here termed the ''context space'' for the property. ===Categorization of properties=== All the properties over the same context space are put in a category, and the category is labeled by the context space. For instance all properties over the context space of groups, are called group properties, and are put in a category titled ''Group properties''. A generic element here is termed a ''group property''. There are two formats for naming the collection of properties over a given context space: * context space name, followed by the word ''properties'': For instance, ''group properties'', ''subgroup properties''. * ''properties of'', followed by context space name: For instance, ''properties of topological spaces''. There is no distinction between the two ways of naming, and one may be chosen over the other for reasons of greater ease of use. The categories that list all properties over a given context space are termed ''property categories'', and a list of all such categories is found in the category titled ''Properties'' in the subject wiki. For instance, [[Groupprops:Category:Properties]] lists all the property categories on the Group Properties Wiki. ===Further subcategories of the property category=== The category of ''all'' properties over a given context space may often be overwhelmingly huge and diverse. In some cases, this category is divided into subcategories. However, the following general convention is observed: ''all properties are listed both in the parent category and in the subcategories''. Subcategories are created in many ways: * Based on metaproperties: properties the properties may or may not satisfy. * Based on variation, analogy or opposites relationships with an existing pivotal property: For instance, [[Topospaces:Category:Variations of normality]] lists properties of topological spaces obtained by varying the property of being a normal space. * Based on meta-criteria like degree of importance, implementation in software packages, and historical development. ==Binary relations== {{fillin}} ==Metaproperties== Metaproperties are properties whose context space is itself a property space. In other words, they are properties evaluated for properties. ae6d5d2a657900b423e5759208b6e04f7ec8294b 188 187 2008-08-05T21:32:17Z Vipul 2 wikitext text/x-wiki '''Property-theoretic categorization''' is part of the broader property-theoretic paradigm of organization followed on some subject wikis, particularly those in mathematics. Properties are a more precise formulation of the general idea of genus and differentia. On a subject wiki using property-theoretic categorization, a complete listing of property-theoretic ''supercategories'' is available at the category titled ''Property-theoretic categories''. For instance, on the Groupprops wiki, this page is [[Groupprops:Category:Property-theoretic categories]]. ==Properties== A ''property'' over a collection of objects is something which every object either ''has'' (or ''satisfies'') or ''does not have'' (or ''does not satisfy''). Examples from elementary mathematics: * Being prime is a ''property'' over the collection of natural numbers. Every natural number either ''has'' the property of being prime, or ''does not have'' the property of being prime. * Being positive is a ''property'' over the collection of real numbers. Every real number either ''is'' positive or is ''not'' positive. * Being isosceles is a ''property'' over the collection of triangles. Every triangle either ''is'' isosceles or ''is not'' isosceles. Examples from higher mathematics: * Being Abelian is a ''property'' over the collection of groups. Every group either ''is'' Abelian or ''is not'' Abelian. * Being compact is a ''property'' over the collection of topological spaces. Every topological space either ''is'' compact or ''is not'' compact. The collection of objects over which the property is being evaluted is here termed the ''context space'' for the property. ===Categorization of properties=== All the properties over the same context space are put in a category, and the category is labeled by the context space. For instance all properties over the context space of groups, are called group properties, and are put in a category titled ''Group properties''. A generic element here is termed a ''group property''. There are two formats for naming the collection of properties over a given context space: * context space name, followed by the word ''properties'': For instance, ''group properties'', ''subgroup properties''. * ''properties of'', followed by context space name: For instance, ''properties of topological spaces''. There is no distinction between the two ways of naming, and one may be chosen over the other for reasons of greater ease of use. The categories that list all properties over a given context space are termed ''property categories'', and a list of all such categories is found in the category titled ''Properties'' in the subject wiki. For instance, [[Groupprops:Category:Properties]] lists all the property categories on the Group Properties Wiki. ===Further subcategories of the property category=== The category of ''all'' properties over a given context space may often be overwhelmingly huge and diverse. In some cases, this category is divided into subcategories. However, the following general convention is observed: ''all properties are listed both in the parent category and in the subcategories''. Subcategories are created in many ways: * Based on metaproperties: properties the properties may or may not satisfy. * Based on variation, analogy or opposites relationships with an existing pivotal property: For instance, [[Topospaces:Category:Variations of normality]] lists properties of topological spaces obtained by varying the property of being a normal space. * Based on meta-criteria like degree of importance, implementation in software packages, and historical development. ==Binary relations== {{fillin}} ==Metaproperties== Metaproperties are properties whose context space is itself a property space. In other words, they are properties evaluated for properties. Here are some examples: * A property of natural numbers is termed ''multiplicatively closed'' if whenever <math>a,b</math> are natural numbers satisfying the property, then <math>ab</math> also satisfies the property. * A property of groups is termed ''direct product-closed'' if the direct product of a family of groups, each with the property, also has the property. ===Categorization of metaproperties=== The metaproperties over a given context space (i.e., the properties that can be evaluated for properties over that context space) are put in a category. For instance, [[Groupprops:Category:Group metaproperties]] is the category of all group metaproperties. A generic element of this category is termed a ''group metaproperty''. There are two formats for naming the collection of properties over a given context space: * context space name, followed by the word ''metaproperties'': For instance, ''group metaproperties'', ''subgroup metaproperties''. * ''metaproperties of'', followed by context space name: For instance, ''metaproperties of topological spaces''. The categories that list all metaproperties over a given context space are termed ''metaproperty categories'' and the list of all metaproperty categories is available at a category called ''Metaproperties''. For instance, the metaproperty category for groups is [[Groupprops:Category:Metaproperties]]. ==Property modifiers== 2644e9dd91962ff1a1306c6330e94a45ae64e71e 106 99 189 2008-08-05T21:45:08Z Vipul 2 New page: Each subject wiki has certain kinds of articles called ''fact articles''. These articles state a known or established fact. Further sections in the article are highly variable and depend b... wikitext text/x-wiki Each subject wiki has certain kinds of articles called ''fact articles''. These articles state a known or established fact. Further sections in the article are highly variable and depend both on the nature and organization of the subject wiki, and the nature of the specific fact. ==Article-tagging templates== There are boxes at the top of many fact articles on subject wikis. Each such box is generated by an ''article-tagging template''. {{further|[[Subwiki:Fact article-tagging template]]}} ==Sections of the fact article== ===Statement=== {{further|[[Subwiki:Statement section in fact articles]]}} This is a mandatory section. It clearly states the fact, possibly in many different formats (for instance, with symbols, and without symbols). The goal of this section is that people who simply want to know the fact should be satisfied with reading the '''Statement''' section. ===Related facts=== This is an optional section, giving other facts closely related to the given fact. The relation may be in form of statement, method of proof, or nature of implications. ===Definitions used=== This is an optional section, giving the (relevantly chosen) definitions of terms used in the statement of the fact. ''All'' definitions of these terms need not be given; the definitions that are most directly pertinent to the statement should be provided. ===Facts used=== This is an optional section, listing those facts that are used in the proof or demonstration of truth of the given fact. ===Applications=== This section gives a brief description of how and where the fact is applied. ===Proof=== {{further|[[Subwiki:Proof]]}} This section gives the proof. The format of the proof may depend on the precise subject wiki, though there are general recommended guidelines. 1076222ca3fe5ef7cba50b7174f5673b538a9d09 0 100 192 2008-08-05T22:03:10Z Vipul 2 New page: In mathematics, ''full'' is typically used in the sense of ''surjective'': a map that takes on its full image. ===In category theory=== {{:Full functor}} {{:Full subcategory}} wikitext text/x-wiki In mathematics, ''full'' is typically used in the sense of ''surjective'': a map that takes on its full image. ===In category theory=== {{:Full functor}} {{:Full subcategory}} 91414456dbbdb8e23dfdf1641cd7fff6c831b307 0 101 193 2008-08-05T22:05:48Z Vipul 2 New page: ===In sheaf theory=== {{:Soft sheaf}} wikitext text/x-wiki ===In sheaf theory=== {{:Soft sheaf}} c4a47aa4fda9af16734046c12a496260d2732211 0 102 194 2008-08-05T22:08:04Z Vipul 2 New page: ===In group theory=== {{:Small cancellation group}} wikitext text/x-wiki ===In group theory=== {{:Small cancellation group}} 3707d66dbcb1d5a165de73f8683cb76459c5b113 0 103 195 2008-08-05T22:08:42Z Vipul 2 New page: ===In metric space theory=== {{:Short map}} wikitext text/x-wiki ===In metric space theory=== {{:Short map}} 5987625a0652d450347b0bdc0a9596ccb6cd5f77 0 104 196 2008-08-05T22:09:59Z Vipul 2 New page: ===Others=== {{:Happy number}} wikitext text/x-wiki ===Others=== {{:Happy number}} 84251276ed9b885d54ff0a3e76ffb1ced0b5b4da 0 105 197 2008-08-05T22:25:44Z Vipul 2 New page: Main form: ''circle'', [[word type::noun]]. Alternative form: ''circle'', [[word type::verb]] Related forms: ''circular'' (adjective), ''circularity'' (noun), ''encircle'' (verb), Typica... wikitext text/x-wiki Main form: ''circle'', [[word type::noun]]. Alternative form: ''circle'', [[word type::verb]] Related forms: ''circular'' (adjective), ''circularity'' (noun), ''encircle'' (verb), Typical use: * A closed curve in the plane. * A closed curve in the plane comprising all points at a fixed positive distance from a given point in the same plane. * A group of persons sharing a common interest or working together. Similar words: [[Similar::cycle]], [[Similar::ring]], [[Similar::round]] Derived words: [[derived::pseudocircle]] ==Mathematics== {{:Circle (mathematics)}} ==Physics== {{:Circle (physics)}} 4dde08e411c8be9dd78e417787a61269263e4ea6 198 197 2008-08-05T22:27:54Z Vipul 2 wikitext text/x-wiki Main form: ''circle'', [[word type::noun]]. Alternative form: ''circle'', [[word type::verb]] Related forms: ''circular'' (adjective), ''circularity'' (noun), ''encircle'' (verb), Typical use: * A closed curve in the plane. * A closed curve in the plane comprising all points at a fixed positive distance from a given point in the same plane. * A group of persons sharing a common interest or working together. Similar words: [[Similar::cycle]], [[Similar::ring]], [[Similar::round]], [[Similar::loop]] Derived words: [[derived::pseudocircle]], [[derived::semicircle]] ==Mathematics== {{:Circle (mathematics)}} ==Physics== {{:Circle (physics)}} 7a377b5a55b07e0f432c8fac2659988d372b7cc2 0 106 199 2008-08-05T22:53:02Z Vipul 2 New page: ==Mathematics== {{:Principal (mathematics)}} ==Economics== {{:Principal (economics)}} wikitext text/x-wiki ==Mathematics== {{:Principal (mathematics)}} ==Economics== {{:Principal (economics)}} df4501c4b490f0f38d6d9d09b20340e2b172dfdf 0 57 201 200 2008-08-05T22:57:23Z Vipul 2 /* In differential geometry */ wikitext text/x-wiki ===In topology=== {{:Regular space}} {{:Regular covering}} ===In differential geometry=== {{:Regular value}} ===In the theory of differential equations=== {{:Regular singular point}} ===In the theory of monoids and semigroups=== {{:Regular semigroup}} ===In group theory/representation theory=== {{:Regular group action}} ===In geometry=== {{:Regular polygon}} {{:Regular polyhedron}} {{:Regular polytope}} ===In commutative algebra=== {{:Regular ring}} {{:Regular sequence}} ===In algebraic geometry=== {{:Regular map}} ===In number theory=== {{:Regular prime}} ===In graph theory=== {{:Regular graph}} ===In axiomatic set theory=== {{:Regular cardinal}} ===In measure theory=== {{:Regular measure}} ===In category theory=== {{:Regular category}} 59594e8490b6cc08c48e0d6fb860a3f47dea8e17 202 201 2008-08-05T22:58:44Z Vipul 2 wikitext text/x-wiki ===In topology=== {{:Regular space}} {{:Regular covering}} {{:Regular CW-complex}} ===In differential geometry=== {{:Regular value}} ===In the theory of differential equations=== {{:Regular singular point}} ===In the theory of monoids and semigroups=== {{:Regular semigroup}} ===In group theory/representation theory=== {{:Regular group action}} ===In geometry=== {{:Regular polygon}} {{:Regular polyhedron}} {{:Regular polytope}} ===In commutative algebra=== {{:Regular ring}} {{:Regular sequence}} {{:von Neumann-regular ring}} ===In algebraic geometry=== {{:Regular map}} ===In number theory=== {{:Regular prime}} ===In graph theory=== {{:Regular graph}} ===In axiomatic set theory=== {{:Regular cardinal}} ===In measure theory=== {{:Regular measure}} ===In category theory=== {{:Regular category}} dc83638ef67d7856a16ba93dd0594899c1af35ef 106 107 203 2008-08-08T21:52:00Z Vipul 2 New page: In all subject wikis, articles form the fundamental ''unit of knowledge''. Each article is treated as a separate, independent unit of knowlede. Articles are strongly linked to each other b... wikitext text/x-wiki In all subject wikis, articles form the fundamental ''unit of knowledge''. Each article is treated as a separate, independent unit of knowlede. Articles are strongly linked to each other but all inter-article dependencies are loose. ==Goal of an article== The goal of each article is to: * Answer certain specific questions clearly and comprehensively. * Provide a vantage point for exploration of related ideas and for answers to related questions. The two goals together mean that each article should be very clear about what questions it provides comprehensive answers to, and what questions it provides relevant links for. ==Types of articles== ===Definition articles=== {{further|[[Subwiki:Definition article]], [[Subwiki:Definition]]}} Most subject wikis have articles defining terms. These articles are called definition articles or terminology articles. A definition article must contain the definition, and, depending on the type of term and the organizational paradigms of the subject wiki, it may contain other sections giving further context to the term. Definition articles follow a brief and terse style. Their main purpose is to serve as vantage points for exploring a term and understanding related terms and facts. Definition articles do ''not'' contain proofs, explanatory arguments, exploration and motivation. Rather, they give short statements with clear explanations, linking to more complete proofs and explanation. ===Fact articles=== {{further|[[Subwiki:Fact article]], [[Subwiki:Fact]]}} Fact articles aim to state a fact clearly, and provide a proof or explanation of the fact. Where the proof is too long, involved, or unilluminating, it is outlined and separate components are linked to. Fact articles follow a brief and terse style. Their main purpose is to serve as vantage points for exploring the fact, its underlying ideas, and related terms and facts. Fact articles contain definitions of important terms used, and state facts used, but do ''not'' attempt to prove other facts being used in their proof. ===Survey articles=== {{further|[[Subwiki:Survey article]]}} Survey articles explore an idea. The idea could be based on a definition or fact or it could be based on the relation between multiple definitions and facts. Survey articles do not follow rigid guidelines like definition and fact articles, and the value they offer is considerably more variable. This is because a survey article is not usually intended to answer a ''specific question'', but rather, it is intended to address general curiosities. Unlike definition and fact articles, survey articles are not obliged to provide complete and comprehensive statements of definitions and facts used. They can provide short summaries of the definitions and facts, linking to the full article for further information. 40b221594eb678e74686613c89ccc2c7c4dc6158 106 95 204 184 2008-08-08T21:58:35Z Vipul 2 wikitext text/x-wiki Each subject wiki has certain kinds of articles called ''definition articles'' or ''terminology articles''. Each such article is about a term being defined. Definition articles form an extremely important part of every subject wiki. This article discusses the overall design and organization of definition articles on subject wikis. We'll use the word ''definiendum'' to describe the kind of term being defined. ==Goals of the definition article== ===Content goals=== The definition article should provide: * A clear definition * A representative range of examples * An idea of the different ways in which the definiendum is understood and used ===Link goals=== The definition article should provide brief information with further links to: * Articles detailing the history of the definiendum * Survey articles exploring how to understand the definition * Related terms, either related by an implication (one being stronger or weaker than the other), or by other means * Proofs of properties, conditions and results satisfied by the definiendum ==Article-tagging templates== There are boxes at the top of many definition articles on subject wikis. These boxes give general information about the definiendum, such as the ''type'' of term it is. Each such box is generated by an ''article-tagging template''. {{further|[[Subwiki:Definition article-tagging template]]}} ==Sections of the definition article== ===History section=== {{further|[[Subwiki:History section in definition articles]]}} This is not a mandatory section, but when it exists, it is usually at the beginning of the article, ''before'' the definition section. The history section provides a ''brief'' overview of the origin of the term and of the notion, along with the name of the person who came up with the term. This section ''might'' also contain further information about the evolution of the term or concept since it was introduced. ''Links from this section'': If more detailed information on the history of the concept or term is available on a separate page, a link to that page is provided. ===Definition section=== {{further|[[Subwiki:Definition]]}} This section is mandatory for all definition articles. The definition section, titled '''Definition''', contains ''at least one'' definition of the definiendum. It may contain multiple equivalent definitions, and may also contain the same definition stated in different ways (for instance, in words and in symbols). Some definition sections have a ''Quick phrases'' box on top with quick, easy-to-remember phrases that capture the meaning of the term. ''Links from this section'': The section links to pages explaining the equivalence of multiple definitions, or pages proving facts implicit in the formulation of the definition. It also links to survey articles that parse and study the definition carefully and to survey articles that consider further alternative definitions. ===Examples section=== {{further|[[Subwiki:Examples]]}} This section is recommended for all definition articles, though it is currently not present in too many of them. The section may be divided into subsections containing different kinds of examples. The aim is for examples to provide a representative range, cover the extremes, and provide a litmus test for definition understanding. Non-examples may also be included to contrast with examples. ===Formalisms section=== {{further|[[Subwiki:Formalisms]]}} ===Relation section=== {{further|[[Subwiki:Relations in definition article]]}} ===Facts section or more refined section choices=== {{fillin}} 4a37a0eb557bca7fde4c308b04b295d6c0b2b72e 106 108 205 2008-08-08T22:17:13Z Vipul 2 New page: Categorization involves the use of the in-built ''Category'' Feature offered by MediaWiki software to organize the material within a subject wiki. ==General information on categories== =... wikitext text/x-wiki Categorization involves the use of the in-built ''Category'' Feature offered by MediaWiki software to organize the material within a subject wiki. ==General information on categories== ===Category as a list=== A category is a list of articles sharing a common feature. An article on that list is said to be ''in'' the category. Each category has a page, called a ''category page'', that provides the list along with general information about further resources on these pages (for instance, related categories). An article can be in multiple categories. The categories in which an article is, can be found in a box at the bottom of the article. ===Subject wikis convention: categories versus supercategories=== The MediaWiki software powering subject wiki allows categories to be in categories, and calls a category inside another category a ''subcategory''. We make use of subcategories in two ways. In addition to the categories described above, which are lists of articles, the subject wikis also have certain other categories, which are ''lists of categories''. These are called ''supercategories''. A supercategory ''only'' contains categories but does ''not'' contain any articles. Further, some ordinary categories in subject wikis have ''subcategories''. For a subcategory of an ordinary category, the following convention is used: all articles in the subcategory are ''also'' in the main category. ===Use of categories=== Categories are used for two purposes: * Indicating the ''type'' of a definition, fact, or other article. * Indicating the ''topic area'' of a definition, fact, or other article. * Indicating the ''difficulty level'' of a definition, fact, or other article. In other words, categories are used for clustering, either based on a logical notion of type, or based on other criteria. ===Categories versus semantic MediaWiki=== Categories are ''not'' used for relationships between multiple terms, facts and survey articles ''except'' when such relationships give rise to a strong cluster around a center. Thus, there may be a category titled ''variations of normality'' if there is a cluster of many variations of normality. However, there is unlikely to be a category of ''things stronger than normality'' if the relation is more symmetric and does not lead to clustering behavior. For such relations, semantic MediaWiki is used. These allow relations with different tags between properties. {{further|[[Subwiki:Semantic MediaWiki]]}} 69a1878150d918e1d810acb7c6ce47a6bfc855dc 106 109 206 2008-08-25T12:10:32Z Vipul 2 New page: [[Subwiki:Definition article|Definition articles]], or article defining new terms, have ''relation'' sections. These relation sections describe how the term being defined (henceforth, call... wikitext text/x-wiki [[Subwiki:Definition article|Definition articles]], or article defining new terms, have ''relation'' sections. These relation sections describe how the term being defined (henceforth, called the ''definiendum'') is related to other terms and concepts. ==Relation with similarly typed terms== The ''type'' of a term is the specific role that term plays. For instance, some terms may be of the ''property'' type: ''prime number'' is a property of natural numbers. Some terms may be of the ''species'' type, so ''mangifera indica'' is a species of plant. (Learn more about typing at [[Subwiki:Type-based organization]]). The relation with similarly typed terms describes how the definiendum is related to terms of the same ''type''. ===Stronger and weaker=== For properties over some context space, one can talk of ''stronger properties'' and ''weaker properties'', to describe the relation with other properties over the same context space. Thus, for instance, the property of natural numbers of being a perfect fourth power is ''stronger than'' the property of being a perfect square, while the property of being a perfect power is ''weaker than'' the property of being a perfect square. Something similar is true for extra structures placed. A ''stronger structure'' in this case is a structure that carries more information (or more constraints) than a weaker structure, and from which the weaker structure can be recovered. For instance, the structure of a Riemannian manifold is ''stronger than'' the structure of a topological manifold, and the structure of a group is ''stronger than'' the structure of a magma (a set with a binary operation). We typically say that the stronger property (resp., structure) ''implies'' (resp., gives rise to) the weaker property (resp., structure). Here's how this is presented in the case of properties: * In a section titled '''Relation with other properties''', there are subsections titled '''Stronger properties''' and '''Weaker properties'''. * In the section titled '''Stronger properties''', there are bullet points listing the stronger properties. When available or desired, each property is accompanied by a short description either of the property or of why it is stronger; there is usually also a link to a more complete proof, both of the implication, and of why the converse implication fails in general. There may also be a link to a survey article comparing the properties. * In the section titled '''Weaker properties''', there are bullet points listing the weaker properties. When available or desired, each property is accompanied by a short description either of the property or of why it is weaker; there is usually also a link to a more complete proof, both of the implication, and of why the converse implication fails in general. There may also be a link to a survey article comparing the properties. * There may also be a section titled '''Related properties''' or '''Incomparable properties''' that lists properties that are neither stronger nor weaker but are still closely related. A similar format is followed for stronger/weaker structure and for stronger notions and weaker notions in other senses. ===Conjunctions and disjunctions=== Sometimes new terms are obtained by taking the definiendum and '''AND'''ing it with something else. For instance, we can take the '''AND''' of the property of being a prime number and the property of being an odd number, and get the property of being an odd prime. We can take the '''OR''' of being a number that is 1 mod 3 and a number tha tis 2 mod 3, to get the property of being a number that is not a multiple of 3. For property articles, conjunctions and disjunctions are typically handled in separate subsections under the '''Relation with other properties''' section. The section with conjunctions is titled '''Conjunction with other properties''' while the section on disjunctions is titled '''Disjunction with other properties'''. ==Relation with terms of related but different types== Apart from being related with terms of the same type, the definiendum may also be related with terms of different types. These relations may be somewhat more complicated. ===A broad banner of related properties=== Suppose a subgroup property has some related group properties. Then, these group properties are listed in a subsection titled '''Related group properties''' under the section '''Relation with other properties'''. 2569493930cc8c68d989b607d0d2eb2da5df3707 106 110 207 2008-08-25T12:15:26Z Vipul 2 New page: Type-based organization is one of the fundamental organizing principles of subject wikis. The core idea is that the organization and structuring of individual articles, as well as the rela... wikitext text/x-wiki Type-based organization is one of the fundamental organizing principles of subject wikis. The core idea is that the organization and structuring of individual articles, as well as the relations between articles, should be based on the ''type'' of terms/facts the articles deal with. For instance, all articles that deal with cities should follow a certain broad structural outline. Here, the ''type'' -- ''city'', determines the way the article is structured as well as the way it is linked and related to other parts of the wiki. Implementing type-based organization successfully requires good mechanisms for identifying ''type'' in a strong logical sense as well as clear ideas on how the type of an article determines its structuring. 6f6dc587f0e57bd9c80c453f67da47db852271f7 209 207 2008-08-25T16:04:43Z Vipul 2 wikitext text/x-wiki Type-based organization is one of the fundamental organizing principles of subject wikis. The core idea is that the organization and structuring of individual articles, as well as the relations between articles, should be based on the ''type'' of terms/facts the articles deal with. For instance, all articles that deal with cities should follow a certain broad structural outline. Here, the ''type'' -- ''city'', determines the way the article is structured as well as the way it is linked and related to other parts of the wiki. Implementing type-based organization successfully requires good mechanisms for identifying ''type'' in a strong logical sense as well as clear ideas on how the type of an article determines its structuring. ==What is a type?== The ''type'' of a term or fact is a generic pattern into which it fits. Two things are viewed as being of the same ''type'' if the same sentence (whether a question or a statement) could be framed with the two terms used interchangeably in the sentence. Words with similar connotations as ''type'' include ''class'' and ''kind''. ===Type as genus=== A dog is a type of animal; so is a giraffe. Thus, dog and giraffe share a ''type''. The ''genus'' in this case is animal, while the specifics: dog versus giraffe, differ. To define a particular animal, one needs to provide a ''differential'' criterion for that particular animal. For instance, one may attempt to define a dog by the differentiating criteria of being ''domesticated'', 'mammalian'', and more specifically, part of the ''wolf'' family. ===Type of property=== Some terms are used, not so much to denote specific things, but rather, properties that narrow down a general class of things to a small class. For instance, the property of being a ''herbivore'' narrows down the genus of animals, by excluding those animals that rely on other animals for food. Similarly, the property of being ''widely domesticated'' narrows down the animals further; it eliminates animals like lions and tigers while including animals like dogs and cats. The property of being ''able to fly'' eliminates dogs and tigers while allowing most birds, most insects, and bats in. A ''property'' over a context space is something that everything in the context space either satisfies or does not satisfy. The ''type'' of a property is specified by its context space; thus a property that can be evaluated for animals is a property of animals. A property that can be evaluated for automobiles is a property of automobiles. ===Type of operator or process=== Certain terms define operators, processes, or mechanisms. These start from one thing (the ''source'' or ''domain'') and go to another thing (the ''target'' or ''range''). The type of an operator or process is often viewed in terms of the type of the source and the type of the target. Or, it could be viewed in terms of the way the process is carried out, or the goals of the process. For instance, the process of a lion hunting, chasing and killing a deer could be viewed under the type of ''food acquisition processes'' or under the type of ''activities of lions'' or under the type of ''activities of deer''. ===Type of relation=== A relation between two or more terms is typed by the nature of the terms, and the nature of the relation between them. For instance, a relation could be of the type ''have a common ancestor'', where two species of animals are related by a common evolutionary ancestor. 4076c54735b7931848f30b78c3772884e57f5083 212 209 2008-08-25T18:22:47Z Vipul 2 /* Type as genus */ wikitext text/x-wiki Type-based organization is one of the fundamental organizing principles of subject wikis. The core idea is that the organization and structuring of individual articles, as well as the relations between articles, should be based on the ''type'' of terms/facts the articles deal with. For instance, all articles that deal with cities should follow a certain broad structural outline. Here, the ''type'' -- ''city'', determines the way the article is structured as well as the way it is linked and related to other parts of the wiki. Implementing type-based organization successfully requires good mechanisms for identifying ''type'' in a strong logical sense as well as clear ideas on how the type of an article determines its structuring. ==What is a type?== The ''type'' of a term or fact is a generic pattern into which it fits. Two things are viewed as being of the same ''type'' if the same sentence (whether a question or a statement) could be framed with the two terms used interchangeably in the sentence. Words with similar connotations as ''type'' include ''class'' and ''kind''. ===Type as genus=== A dog is a type of animal; so is a giraffe. Thus, dog and giraffe share a ''type''. The ''genus'' in this case is animal, while the specifics: dog versus giraffe, differ. To define a particular animal, one needs to provide a ''differential'' criterion for that particular animal. For instance, one may attempt to define a dog by the differentiating criteria of being ''domesticated'', ''mammalian'', and more specifically, part of the ''wolf'' family. ===Type of property=== Some terms are used, not so much to denote specific things, but rather, properties that narrow down a general class of things to a small class. For instance, the property of being a ''herbivore'' narrows down the genus of animals, by excluding those animals that rely on other animals for food. Similarly, the property of being ''widely domesticated'' narrows down the animals further; it eliminates animals like lions and tigers while including animals like dogs and cats. The property of being ''able to fly'' eliminates dogs and tigers while allowing most birds, most insects, and bats in. A ''property'' over a context space is something that everything in the context space either satisfies or does not satisfy. The ''type'' of a property is specified by its context space; thus a property that can be evaluated for animals is a property of animals. A property that can be evaluated for automobiles is a property of automobiles. ===Type of operator or process=== Certain terms define operators, processes, or mechanisms. These start from one thing (the ''source'' or ''domain'') and go to another thing (the ''target'' or ''range''). The type of an operator or process is often viewed in terms of the type of the source and the type of the target. Or, it could be viewed in terms of the way the process is carried out, or the goals of the process. For instance, the process of a lion hunting, chasing and killing a deer could be viewed under the type of ''food acquisition processes'' or under the type of ''activities of lions'' or under the type of ''activities of deer''. ===Type of relation=== A relation between two or more terms is typed by the nature of the terms, and the nature of the relation between them. For instance, a relation could be of the type ''have a common ancestor'', where two species of animals are related by a common evolutionary ancestor. 879ed3d8eb84ade27b52c1809f8d344b90229030 8 93 208 181 2008-08-25T12:16:04Z Vipul 2 wikitext text/x-wiki * SEARCH * navigation ** mainpage|mainpage ** recentchanges-url|recentchanges ** randompage-url|randompage * subject wikis ** Groupprops:Main Page|Groupprops ** Topospaces:Main Page|Topospaces ** Commalg:Main Page|Commalg ** Diffgeom:Main Page|Diffgeom ** Measure:Main Page|Measure ** Noncommutative:Main Page|Noncommutative ** Companal:Main Page|Companal ** Cattheory:Main Page|Cattheory 4b6515a1ebc1d2275ecfe2052b3f7192ca53fceb 223 208 2008-09-11T16:55:24Z Vipul 2 wikitext text/x-wiki * SEARCH * navigation ** mainpage|mainpage ** recentchanges-url|recentchanges ** randompage-url|randompage * subject wikis ** Groupprops:Main Page|Groupprops ** Topospaces:Main Page|Topospaces ** Commalg:Main Page|Commalg ** Diffgeom:Main Page|Diffgeom ** Measure:Main Page|Measure ** Noncommutative:Main Page|Noncommutative ** Companal:Main Page|Companal ** Cattheory:Main Page|Cattheory * credits ** http://www.4am.co.in|4AM (tech support) 9b56db250ea38488e024fca9483998e4a54a538c 106 111 210 2008-08-25T16:39:44Z Vipul 2 New page: Property-theoretic organization is a broad organizing principle found on many subject wikis, particularly those in mathematics. It is a particular form of [[Subwiki:Type-based organization... wikitext text/x-wiki Property-theoretic organization is a broad organizing principle found on many subject wikis, particularly those in mathematics. It is a particular form of [[Subwiki:Type-based organization|type-based organization]]. 1d2490e9ddf911a2a268f486bed4a02d00126d12 106 112 211 2008-08-25T18:18:33Z Vipul 2 New page: On certain subject wikis following a paradigm of [[Subwiki:property-theoretic organization|property-theoretic organization]], many of the terms being defined are ''properties''. This artic... wikitext text/x-wiki On certain subject wikis following a paradigm of [[Subwiki:property-theoretic organization|property-theoretic organization]], many of the terms being defined are ''properties''. This article discusses the forma of definition for properties. For more information about the overall article structure for properties, refer [[Subwiki:Property definition article]]. For more information about the definition section in general, refer [[Subwiki:Definition]]. ==What is a property?== A property over a collection of objects is something that every object in the collection either ''satisfies'' or ''does not'' satisfy. Thus, the collection can be partitioned into two subcollections: those objects that ''satisfy'' the property, and those objects that ''do not satisfy'' the property. The collection of objects over which a property can be evaluated is termed the context space of the property. For instance, the property of being prime can be evaluated over natural numbers; in this case, the ''context space'' for the property of being prime is ''natural number''. Primality is thus a ''property of natural numbers''. ===Word forms for the property=== * Adjective: For instance ''prime'' is an adjective describing the property of being prime. The adjective modifies the noun for the context space, so a number that is prime is termed a ''prime number''. * Instance-noun: For instance, ''perfect square'' is a noun describing the property of being a perfect square. Here, the noun describes an object satisfying the property. * Property noun: For instance, ''primality'' is a property noun describing the property of being prime. ==Goals of the property definition== ===What the property definition should provide=== The definition should make clear: * What the context space is, over which the property is being evaluated. * How, given an object in the context space, one can determine whether or not it satisfies the property. The criteria should be ''necessary and sufficient'': they should reject everything that does not satisfy the property, and accept everything that does satisfy the property. * What other properties the given property builds upon in its definition. ==Presenting the property definition== ===The symbol-free definition=== A ''symbol-free definition'' is a definition free of symbols, and using natural language as far as possible. Symbol-free definitions often provide more useful and memorable formulations of properties, highlighting more clearly the relation with existing properties. Here are the typical formats of a symbol-free definition: # A (context space object) is (said to be)/(termed)/(called) (property name) if .... #* (''adjectival example'') A natural number is termed prime if it is greater than one and the only natural numbers dividing it are one and the number itself. #* (''instance-noun example'') A natural number is termed a perfect square if it is the square of a natural number. #* (''property noun example'') * A (property name) is a (context space object) satisfying ... #* (''adjectival example'') A prime number is a natural number greater than one for which the only natural numbers dividing it are one and the number itself. #* (''instance-noun example'') A perfect square is a natural number that can be expressed as the square of a natural number. ===Definition with symbols=== A '''definition with symbols''' is a definition that sets explicit symbols for the object for which we're providing the criterion. These symbols are used in the definition, in place of pronouns and other natural language constructs. For instance: # A (context space object) (symbol introduced) is (said to be)/(termed)/(called) (property name) if .... #* (''adjectival example'') A natural number <math>n</math> is termed prime if <math>n > 1</math> and for any natural number <math>d</math>, <math>d|n</math> if and only if <math>d = 1</math> or <math>d = n</math>. #* (''instance-noun example'') A natural number <math>n</math> is termed a perfect square if there exists a natural number <math>m</math> such that <math>n = m^2</math>. ===Multiple equivalent definitions=== In situations where there are multiple equivalent definitions, the definitions are provided as an enumerated or bulleted list, with the context space-setting happening ''before'' the enumeration. The enumeration thus looks like: A (context space object) is (said to be)/(termed)/(called) (property name) if it satisfies the following equivalent conditions: (followed by an enumeration). fb442046eb24ff83318a8e11fd2833630af4eccb 213 211 2008-08-25T18:57:05Z Vipul 2 /* Presenting the property definition */ wikitext text/x-wiki On certain subject wikis following a paradigm of [[Subwiki:property-theoretic organization|property-theoretic organization]], many of the terms being defined are ''properties''. This article discusses the forma of definition for properties. For more information about the overall article structure for properties, refer [[Subwiki:Property definition article]]. For more information about the definition section in general, refer [[Subwiki:Definition]]. ==What is a property?== A property over a collection of objects is something that every object in the collection either ''satisfies'' or ''does not'' satisfy. Thus, the collection can be partitioned into two subcollections: those objects that ''satisfy'' the property, and those objects that ''do not satisfy'' the property. The collection of objects over which a property can be evaluated is termed the context space of the property. For instance, the property of being prime can be evaluated over natural numbers; in this case, the ''context space'' for the property of being prime is ''natural number''. Primality is thus a ''property of natural numbers''. ===Word forms for the property=== * Adjective: For instance ''prime'' is an adjective describing the property of being prime. The adjective modifies the noun for the context space, so a number that is prime is termed a ''prime number''. * Instance-noun: For instance, ''perfect square'' is a noun describing the property of being a perfect square. Here, the noun describes an object satisfying the property. * Property noun: For instance, ''primality'' is a property noun describing the property of being prime. ==Goals of the property definition== ===What the property definition should provide=== The definition should make clear: * What the context space is, over which the property is being evaluated. * How, given an object in the context space, one can determine whether or not it satisfies the property. The criteria should be ''necessary and sufficient'': they should reject everything that does not satisfy the property, and accept everything that does satisfy the property. * What other properties the given property builds upon in its definition. ==Presenting the property definition== ===The symbol-free definition=== A ''symbol-free definition'' is a definition free of symbols, and using natural language as far as possible. Symbol-free definitions often provide more useful and memorable formulations of properties, highlighting more clearly the relation with existing properties. Here are the typical formats of a symbol-free definition: # A (context space object) is (said to be)/(termed)/(called) (property name) if .... #* (''adjectival example'') A natural number is termed prime if it is greater than one and the only natural numbers dividing it are one and the number itself. #* (''instance-noun example'') A natural number is termed a perfect square if it is the square of a natural number. #* (''property noun example'') # A (property name) is a (context space object) satisfying ... #* (''adjectival example'') A prime number is a natural number greater than one for which the only natural numbers dividing it are one and the number itself. #* (''instance-noun example'') A perfect square is a natural number that can be expressed as the square of a natural number. ===Definition with symbols=== A '''definition with symbols''' is a definition that sets explicit symbols for the object for which we're providing the criterion. These symbols are used in the definition, in place of pronouns and other natural language constructs. For instance: # A (context space object) (symbol introduced) is (said to be)/(termed)/(called) (property name) if .... #* (''adjectival example'') A natural number <math>n</math> is termed prime if <math>n > 1</math> and for any natural number <math>d</math>, <math>d|n</math> if and only if <math>d = 1</math> or <math>d = n</math>. #* (''instance-noun example'') A natural number <math>n</math> is termed a perfect square if there exists a natural number <math>m</math> such that <math>n = m^2</math>. ===Multiple equivalent definitions=== In situations where there are multiple equivalent definitions, the definitions are provided as an enumerated or bulleted list, with the context space-setting happening ''before'' the enumeration. The enumeration thus looks like: A (context space object) is (said to be)/(termed)/(called) (property name) if it satisfies the following equivalent conditions: (followed by an enumeration). e37dfe28e4ae3c35f6d56dfa27bffd8050ff01be 238 213 2008-12-25T11:06:55Z Vipul 2 /* Presenting the property definition */ wikitext text/x-wiki On certain subject wikis following a paradigm of [[Subwiki:property-theoretic organization|property-theoretic organization]], many of the terms being defined are ''properties''. This article discusses the forma of definition for properties. For more information about the overall article structure for properties, refer [[Subwiki:Property definition article]]. For more information about the definition section in general, refer [[Subwiki:Definition]]. ==What is a property?== A property over a collection of objects is something that every object in the collection either ''satisfies'' or ''does not'' satisfy. Thus, the collection can be partitioned into two subcollections: those objects that ''satisfy'' the property, and those objects that ''do not satisfy'' the property. The collection of objects over which a property can be evaluated is termed the context space of the property. For instance, the property of being prime can be evaluated over natural numbers; in this case, the ''context space'' for the property of being prime is ''natural number''. Primality is thus a ''property of natural numbers''. ===Word forms for the property=== * Adjective: For instance ''prime'' is an adjective describing the property of being prime. The adjective modifies the noun for the context space, so a number that is prime is termed a ''prime number''. * Instance-noun: For instance, ''perfect square'' is a noun describing the property of being a perfect square. Here, the noun describes an object satisfying the property. * Property noun: For instance, ''primality'' is a property noun describing the property of being prime. ==Goals of the property definition== ===What the property definition should provide=== The definition should make clear: * What the context space is, over which the property is being evaluated. * How, given an object in the context space, one can determine whether or not it satisfies the property. The criteria should be ''necessary and sufficient'': they should reject everything that does not satisfy the property, and accept everything that does satisfy the property. * What other properties the given property builds upon in its definition. ==Presenting the property definition== ===The symbol-free definition=== A ''symbol-free definition'' is a definition free of symbols, and using natural language as far as possible. Symbol-free definitions often provide more useful and memorable formulations of properties, highlighting more clearly the relation with existing properties. Here are the typical formats of a symbol-free definition: # A (context space object) is (said to be)/(termed)/(called) (property name) if .... #* (''adjectival example'') A natural number is termed prime if it is greater than one and the only natural numbers dividing it are one and the number itself. #* (''instance-noun example'') A natural number is termed a perfect square if it is the square of a natural number. #* (''property noun example'') A natural number is said to satisfy the property of primality if it is greater than one and on the only natural numbers dividing it are one and the number itself. # A (property name) is a (context space object) satisfying ... #* (''adjectival example'') A prime number is a natural number greater than one for which the only natural numbers dividing it are one and the number itself. #* (''instance-noun example'') A perfect square is a natural number that can be expressed as the square of a natural number. #* (''property noun example'') Primality is the property of a natural number being greater than one and not having any prime divisors other than one and the number itself. ===Definition with symbols=== A '''definition with symbols''' is a definition that sets explicit symbols for the object for which we're providing the criterion. These symbols are used in the definition, in place of pronouns and other natural language constructs. For instance: # A (context space object) (symbol introduced) is (said to be)/(termed)/(called) (property name) if .... #* (''adjectival example'') A natural number <math>n</math> is termed prime if <math>n > 1</math> and for any natural number <math>d</math>, <math>d|n</math> if and only if <math>d = 1</math> or <math>d = n</math>. #* (''instance-noun example'') A natural number <math>n</math> is termed a perfect square if there exists a natural number <math>m</math> such that <math>n = m^2</math>. #* (''property noun example'') A natural number <math>n</math> is said to satisfy the property of primality if <math>n > 1</math> and for any natural number <math>d</math>, <math>d|n</math> if and only if <math>d = 1</math> or <math>d = n</math>. ===Multiple equivalent definitions=== In situations where there are multiple equivalent definitions, the definitions are provided as an enumerated or bulleted list, with the context space-setting happening ''before'' the enumeration. The enumeration thus looks like: A (context space object) is (said to be)/(termed)/(called) (property name) if it satisfies the following equivalent conditions: (followed by an enumeration). 6cf3bdfff5f31ac537e3df1058fad99faab09dce 108 68 214 137 2008-09-06T12:31:18Z Vipul 2 wikitext text/x-wiki Planetmath is a free, online, open-content, collaborative mathematics resource. While primarily an encyclopedia, it also contains books, expositions, papers and forums. {{quotation|'''SLOGAN''': ''Math for the people, by the people''}} ==Use Planetmath== ===Start Planetmath=== Planetmath is online, or Internet-based. It can be accessed using a Web browser and an Internet connection: http://planetmath.org Planetmath dumps can also be downloaded at: http://aux.planetmath.org/snapshots/ ===Search Planetmath=== Planetmath offers a search box in the upper right corner. One can also search Planetmath by adding "site:planetmath.org" to a search query on a search engine like Google, Yahoo! or Windows Live. ===Article names and reaching a given article=== Articles in Planetmath are named using CamelCase conventions. However, these conventions are not rigidly followed for all article names. In the CamelCase convention, a term consisting of multiple words is shrunk to a single word, with the first letter of each original word capitalized. Thus, "prime number" becomes "PrimeNumber", and the planetmath URL for it is: http://planetmath.org/encyclopedia/PrimeNumber.html ==More about Planetmath for users== View [http://planetmath.org/?op=getobj&from=collab&id=35 the Planetmath FAQ] and [http://planetmath.org/?op=sitedoc further documentation]. Key features described here: * Planetmath is similar to [[Resource:Mathworld|Mathworld]], except for its greater focus on collaboration, and its open-content license. * Planetmath is a lot like Wikipedia, except in the following respects: First, its content management system is specifically geared for good display of mathematical expressions (it uses TeX/LaTeX-based input). Second, every object is "owned" by a person, rather than being collectively owned, and the system for revising/correcting an object is different from that on wikis. ==Planetmath as a free resource== ===Free access=== All content on Planetmath can be accessed for free (i.e., there are no access or subscription fees). ===Copyright and reuse=== Planetmath content is released under the GNU Free Documentation License (GFDL). This is the same as the license used by [[Resource:Wikipedia|Wikipedia]], and allows people to reuse the content on Planetmath so long as they release their own modifications under the same license. View [http://planetmath.org/?op=license the license page] for more details. ==Management and organization== ===Software tools=== Planetmath runs on Noosphere, a content management system suitable for collaborative mathematics-rich projects. View [http://aux.planetmath.org/noosphere/ Planetmath's Noosphere description] for more details. ===Financial model=== Planetmath is run by a US-based non-profit organization, Planetmath.org, Ltd (registered under Section 501(c)(3)) incorporated in Alexandria, Virginia. The financial model rests primarily on donations: # http://aux.planetmath.org/doc/donate.html The donate page http://aux.planetmath.org/doc/donate.html ==Production model== ===Collaborative content creation=== Content is created collaboratively on Planetmath: anybody can start an article. However, there is an important difference between Planetmath and content creation mechanisms like Wikipedia. On Planetmath, every article is ''owned'' by somebody (the person who creates it) and this person acts as a gatekeeper for the article, moderating suggestions by other people for improving the article. Planetmath also allows people to use a wiki model in content creation. ===Compensation model=== Planetmath does not provide monetary compensation for writing articles. However, the person starting an article is given ''ownership'' over it and his/her username is displayed at the bottom of the article. 874439e9c35509da1658f4c6a61f334f5f92d53e 108 113 215 2008-09-06T12:47:21Z Vipul 2 New page: MIT OpenCourseWare (OCW) is a free, open and web-based publication of MIT course content. {{quote|'''SLOGAN''': ''Unlocking knowledge, empowering minds''}} ==Use MIT OpenCourseWare== ==... wikitext text/x-wiki MIT OpenCourseWare (OCW) is a free, open and web-based publication of MIT course content. {{quote|'''SLOGAN''': ''Unlocking knowledge, empowering minds''}} ==Use MIT OpenCourseWare== ===Start MIT OpenCourseWare=== MIT OpenCourseWare is online, or Internet-based. It can be started using a web browser and Internet connection: http://ocw.mit.edu ===Search MIT OpenCourseWare=== There is a search box accessible from all MIT OpenCourseWare pages. More advanced search is available at: http://ocw.mit.edu/OcwWeb/search/AdvancedSearch.htm This also includes search through the transcripts of video lectures for those video lectures where transcripts have been prepared. ===Course lists=== The left sidebar on the main page offers links to view lists of all courses, view lists of popular courses, view lists of courses with audio/video. It also offers options for browsing course lists by subject. ==More about MIT OCW for users== * [http://ocw.mit.edu/OcwWeb/web/help/overview/index.htm A site overview]: Gives a description on navigating the OCW site. * [http://ocw.mit.edu/OcwWeb/web/help/start/index.htm Getting started]: A page giving information on how to get started with MIT OpenCourseWare. * [http://ocw.mit.edu/OcwWeb/web/help/faq1/index.htm FAQ on getting started] * [http://ocw.mit.edu/OcwWeb/web/help/faq2/index.htm FAQ on using OCW materials] * [http://ocw.mit.edu/OcwWeb/web/help/faq3/index.htm FAQ on Intellectual Property] * [http://ocw.mit.edu/OcwWeb/web/help/faq4/index.htm FAQ on Technology] * [http://ocw.mit.edu/OcwWeb/web/help/faq5/index.htm FAQ on Highlights for High School] * [http://ocw.mit.edu/OcwWeb/web/help/cite/index.htm Cite/attribute content] * [http://ocw.mit.edu/OcwWeb/web/help/tech/index.htm Technical requirements] ==MIT OpenCourseWare as a free resource== ===Free access=== All content on MIT OpenCourseWare can be accessed for free (i.e., there are no access or subscription fees). ===Copyright and reuse=== Unless otherwise specified, all content in MIT OpenCourseWare is considered released under the [http://creativecommons.org/licenses/by-nc-sa/3.0/us/ CC-by-NC-SA license]. This license allows free reuse and remixing, as long as all derivative works are also released under a compatible license. The original copyright is retained largely by the faculty, though in some cases, copyright is retained by MIT. All uses beyond the scope of the license require permission from the copyright holders. More information available at [http://ocw.mit.edu/OcwWeb/web/help/faq2/index.htm FAQ on using OCW materials] and [http://ocw.mit.edu/OcwWeb/web/help/faq3/index.htm FAQ on Intellectual Property]. d9e64e44297f775493719b247e10f8924e92d412 217 215 2008-09-06T12:53:25Z Vipul 2 wikitext text/x-wiki MIT OpenCourseWare (OCW) is a free, open and web-based publication of MIT course content. {{quotation|'''SLOGAN''': ''Unlocking knowledge, empowering minds''}} ==Use MIT OpenCourseWare== ===Start MIT OpenCourseWare=== MIT OpenCourseWare is online, or Internet-based. It can be started using a web browser and Internet connection: http://ocw.mit.edu ===Search MIT OpenCourseWare=== There is a search box accessible from all MIT OpenCourseWare pages. More advanced search is available at: http://ocw.mit.edu/OcwWeb/search/AdvancedSearch.htm This also includes search through the transcripts of video lectures for those video lectures where transcripts have been prepared. ===Course lists=== The left sidebar on the main page offers links to view lists of all courses, view lists of popular courses, view lists of courses with audio/video. It also offers options for browsing course lists by subject. ==More about MIT OCW for users== * [http://ocw.mit.edu/OcwWeb/web/help/overview/index.htm A site overview]: Gives a description on navigating the OCW site. * [http://ocw.mit.edu/OcwWeb/web/help/start/index.htm Getting started]: A page giving information on how to get started with MIT OpenCourseWare. * [http://ocw.mit.edu/OcwWeb/web/help/faq1/index.htm FAQ on getting started] * [http://ocw.mit.edu/OcwWeb/web/help/faq2/index.htm FAQ on using OCW materials] * [http://ocw.mit.edu/OcwWeb/web/help/faq3/index.htm FAQ on Intellectual Property] * [http://ocw.mit.edu/OcwWeb/web/help/faq4/index.htm FAQ on Technology] * [http://ocw.mit.edu/OcwWeb/web/help/faq5/index.htm FAQ on Highlights for High School] * [http://ocw.mit.edu/OcwWeb/web/help/cite/index.htm Cite/attribute content] * [http://ocw.mit.edu/OcwWeb/web/help/tech/index.htm Technical requirements] ==MIT OpenCourseWare as a free resource== ===Free access=== All content on MIT OpenCourseWare can be accessed for free (i.e., there are no access or subscription fees). ===Copyright and reuse=== Unless otherwise specified, all content in MIT OpenCourseWare is considered released under the [http://creativecommons.org/licenses/by-nc-sa/3.0/us/ CC-by-NC-SA license]. This license allows free reuse and remixing, as long as all derivative works are also released under a compatible license. The original copyright is retained largely by the faculty, though in some cases, copyright is retained by MIT. All uses beyond the scope of the license require permission from the copyright holders. More information available at [http://ocw.mit.edu/OcwWeb/web/help/faq2/index.htm FAQ on using OCW materials] and [http://ocw.mit.edu/OcwWeb/web/help/faq3/index.htm FAQ on Intellectual Property]. 7009d041e0f2f6b5f959a8043ad39eadfe248c8b 218 217 2008-09-06T12:57:15Z Vipul 2 wikitext text/x-wiki MIT OpenCourseWare (OCW) is a free, open and web-based publication of MIT course content. {{quotation|'''SLOGAN''': ''Unlocking knowledge, empowering minds''}} ==Use MIT OpenCourseWare== ===Start MIT OpenCourseWare=== MIT OpenCourseWare is online, or Internet-based. It can be started using a web browser and Internet connection: http://ocw.mit.edu ===Search MIT OpenCourseWare=== There is a search box accessible from all MIT OpenCourseWare pages. More advanced search is available at: http://ocw.mit.edu/OcwWeb/search/AdvancedSearch.htm This also includes search through the transcripts of video lectures for those video lectures where transcripts have been prepared. ===Course lists=== The left sidebar on the main page offers links to view lists of all courses, view lists of popular courses, view lists of courses with audio/video. It also offers options for browsing course lists by subject. ==More about MIT OCW for users== * [http://ocw.mit.edu/OcwWeb/web/help/overview/index.htm A site overview]: Gives a description on navigating the OCW site. * [http://ocw.mit.edu/OcwWeb/web/help/start/index.htm Getting started]: A page giving information on how to get started with MIT OpenCourseWare. * [http://ocw.mit.edu/OcwWeb/web/help/faq1/index.htm FAQ on getting started] * [http://ocw.mit.edu/OcwWeb/web/help/faq2/index.htm FAQ on using OCW materials] * [http://ocw.mit.edu/OcwWeb/web/help/faq3/index.htm FAQ on Intellectual Property] * [http://ocw.mit.edu/OcwWeb/web/help/faq4/index.htm FAQ on Technology] * [http://ocw.mit.edu/OcwWeb/web/help/faq5/index.htm FAQ on Highlights for High School] * [http://ocw.mit.edu/OcwWeb/web/help/cite/index.htm Cite/attribute content] * [http://ocw.mit.edu/OcwWeb/web/help/tech/index.htm Technical requirements] ==MIT OpenCourseWare as a free resource== ===Free access=== All content on MIT OpenCourseWare can be accessed for free (i.e., there are no access or subscription fees). ===Copyright and reuse=== Unless otherwise specified, all content in MIT OpenCourseWare is considered released under the [http://creativecommons.org/licenses/by-nc-sa/3.0/us/ CC-by-NC-SA license]. This license allows free reuse and remixing, as long as all derivative works are also released under a compatible license. The original copyright is retained largely by the faculty, though in some cases, copyright is retained by MIT. All uses beyond the scope of the license require permission from the copyright holders. More information available at [http://ocw.mit.edu/OcwWeb/web/help/faq2/index.htm FAQ on using OCW materials] and [http://ocw.mit.edu/OcwWeb/web/help/faq3/index.htm FAQ on Intellectual Property]. ==Related resources== * [[Competitor::Resource:Yale OpenYaleCourses]] * [[Competitor::Resource:Berkeley Webcast]] * [[Competitor::Resource:Utah State University OpenCourseWare]] * [[Competitor::Resource:Tufts University OpenCourseWare]] * [[Competitor::Resource:Notre Dame OpenCourseWare]] 4e2d5be862af3d2f0ed73ac84eb3f2c92663ad8e 219 218 2008-09-06T13:03:23Z Vipul 2 /* MIT OpenCourseWare as a free resource */ wikitext text/x-wiki MIT OpenCourseWare (OCW) is a free, open and web-based publication of MIT course content. {{quotation|'''SLOGAN''': ''Unlocking knowledge, empowering minds''}} ==Use MIT OpenCourseWare== ===Start MIT OpenCourseWare=== MIT OpenCourseWare is online, or Internet-based. It can be started using a web browser and Internet connection: http://ocw.mit.edu ===Search MIT OpenCourseWare=== There is a search box accessible from all MIT OpenCourseWare pages. More advanced search is available at: http://ocw.mit.edu/OcwWeb/search/AdvancedSearch.htm This also includes search through the transcripts of video lectures for those video lectures where transcripts have been prepared. ===Course lists=== The left sidebar on the main page offers links to view lists of all courses, view lists of popular courses, view lists of courses with audio/video. It also offers options for browsing course lists by subject. ==More about MIT OCW for users== * [http://ocw.mit.edu/OcwWeb/web/help/overview/index.htm A site overview]: Gives a description on navigating the OCW site. * [http://ocw.mit.edu/OcwWeb/web/help/start/index.htm Getting started]: A page giving information on how to get started with MIT OpenCourseWare. * [http://ocw.mit.edu/OcwWeb/web/help/faq1/index.htm FAQ on getting started] * [http://ocw.mit.edu/OcwWeb/web/help/faq2/index.htm FAQ on using OCW materials] * [http://ocw.mit.edu/OcwWeb/web/help/faq3/index.htm FAQ on Intellectual Property] * [http://ocw.mit.edu/OcwWeb/web/help/faq4/index.htm FAQ on Technology] * [http://ocw.mit.edu/OcwWeb/web/help/faq5/index.htm FAQ on Highlights for High School] * [http://ocw.mit.edu/OcwWeb/web/help/cite/index.htm Cite/attribute content] * [http://ocw.mit.edu/OcwWeb/web/help/tech/index.htm Technical requirements] ==MIT OpenCourseWare as a free resource== ===Free access=== All content on MIT OpenCourseWare can be accessed for free (i.e., there are no access or subscription fees). ===Copyright and reuse=== Unless otherwise specified, all content in MIT OpenCourseWare is considered released under the [[license::CC-by-NC-SA]] license [http://creativecommons.org/licenses/by-nc-sa/3.0/us/ (read license here)]. This license allows free reuse and remixing, as long as all derivative works are also released under a compatible license. The original copyright is retained largely by the faculty, though in some cases, copyright is retained by MIT. All uses beyond the scope of the license require permission from the copyright holders. More information available at [http://ocw.mit.edu/OcwWeb/web/help/faq2/index.htm FAQ on using OCW materials] and [http://ocw.mit.edu/OcwWeb/web/help/faq3/index.htm FAQ on Intellectual Property]. ==Related resources== * [[Competitor::Resource:Yale OpenYaleCourses]] * [[Competitor::Resource:Berkeley Webcast]] * [[Competitor::Resource:Utah State University OpenCourseWare]] * [[Competitor::Resource:Tufts University OpenCourseWare]] * [[Competitor::Resource:Notre Dame OpenCourseWare]] 309a0b1c880111865840c17d4f40728b7197bfb9 108 114 216 2008-09-06T12:47:48Z Vipul 2 Redirecting to [[Resource:MIT OpenCourseWare]] wikitext text/x-wiki #redirect [[Resource:MIT OpenCourseWare]] 4c173e2fa4de4548649f40a870cf4ac27ed6a055 6 115 220 2008-09-11T16:43:37Z Vipul 2 wikitext text/x-wiki da39a3ee5e6b4b0d3255bfef95601890afd80709 8 21 221 41 2008-09-11T16:44:49Z Vipul 2 wikitext text/x-wiki [[Image:Logo.jpg|thumb|75px|right]] [[Main Page|'''The Subject Wikis Reference Guide (in preparation)''']] 02b36091d509c99700e506cc3a0c77bd30be4989 222 221 2008-09-11T16:45:53Z Vipul 2 wikitext text/x-wiki [[Image:Logo.jpg|75px|right]] [[Main Page|'''The Subject Wikis Reference Guide (in preparation)''']] 9fdacd4244ea8a41b78ed12bbd7f2cb10b68acef 241 222 2008-12-25T11:51:59Z Vipul 2 wikitext text/x-wiki [[Image:Logo.jpg|thumb|75px|right|[http://www.4am.co.in Visit]]] [[Main Page|'''The Subject Wikis Reference Guide (in preparation)''']] [[Groupprops:Main Page|Groupprops, The Group Properties Wiki]], [[Topospaces:Main Page|Topospaces, The Topology Wiki]], [[Diffgeom:Main Page|Diffgeom, The Differential Geometry Wiki]] [[Commalg:Main Page|Commalg, The Commutative Algebra Wiki]], [[Cattheory:Main Page|Cattheory, The Category Theory Wiki]], [[Companal:Main Page|Companal, The Complex Analysis Wiki]] Want more? [[Ref:Subwiki list|Get a complete list here]]. b20a4326c7ad1d2cc5e77855b39bbc1b240846b5 0 116 224 2008-11-08T15:06:40Z Vipul 2 New page: <noinclude>[[Status::Semi-basic definition| ]][[Primary wiki::Groupprops| ]][[Topic::Group theory| ]]</noinclude> '''Fully characteristic subgroup''': A subgroup of a group that is invari... wikitext text/x-wiki <noinclude>[[Status::Semi-basic definition| ]][[Primary wiki::Groupprops| ]][[Topic::Group theory| ]]</noinclude> '''Fully characteristic subgroup''': A subgroup of a group that is invariant under all endomorphisms of the group. Primary subject wiki entry: [[Groupprops:Fully characteristic subgroup]] Also located at: [[Wikipedia:Fully characteristic subgroup]], [[Planetmath:FullyInvariantSubgroup]] 6c6ed46fb3ae785e553596ca87208daa1a2f93a0 0 117 225 2008-11-08T15:08:53Z Vipul 2 New page: <noinclude>[[Status::Basic definition| ]][[Primary wiki::Commalg| ]][[Topic::Commutative algebra| ]][[Topic::Noncommutative algebra| ]]</noinclude> '''Noetherian ring''': A Noetherian rin... wikitext text/x-wiki <noinclude>[[Status::Basic definition| ]][[Primary wiki::Commalg| ]][[Topic::Commutative algebra| ]][[Topic::Noncommutative algebra| ]]</noinclude> '''Noetherian ring''': A Noetherian ring is a (unital) ring in which every ideal is finitely generated. In commutative algebra, Noetherian ring stands for a ''commutative unital ring'' in which every ideal is finitely generated, while in noncommutative algebra, it stands for a possibly noncommutative unital ring in which every ideal is finitely generated. Primary subject wiki entry: [[Commalg:Noetherian ring]] Also located at: [[Wikipedia:Noetherian ring]] 5dd3da9cf934cdbc06c782ad6f7dd93ec00b9bf7 0 118 226 2008-11-08T15:13:04Z Vipul 2 New page: '''Subnormal subgroup''': A subgroup of a group such that there is an ascending chain of subgroups, starting from that subgroup to the whole group, with each member normal in the next. Suc... wikitext text/x-wiki '''Subnormal subgroup''': A subgroup of a group such that there is an ascending chain of subgroups, starting from that subgroup to the whole group, with each member normal in the next. Such an ascending chain is termed a subnormal series. A <math>k</math>-subnormal subgroup is a subgroup for which there exists a subnormal series of length at most <math>k</math>. Particular examples are <math>k = 1</math> (normal subgroup), <math>k = 2</math> (2-subnormal subgroup), <math>k = 3</math> (3-subnormal subgroup). Primary subject wiki entry: [[Groupprops:Subnormal subgroup]] Related terms: [[Groupprops:Normal subgroup]], [[Groupprops:2-subnormal subgroup]], [[Groupprops:3-subnormal subgroup]], [[Groupprops:Ascendant subgroup]], [[Groupprops:Descendant subgroup]], [[Groupprops:Serial subgroup]], [[Groupprops:Subnormal series]], [[Groupprops:Normal series]] a708baa1daa7d381310d283fe59a6457fa5eb178 227 226 2008-11-08T15:14:34Z Vipul 2 wikitext text/x-wiki '''Subnormal subgroup''': A subgroup of a group such that there is an ascending chain of subgroups, starting from that subgroup to the whole group, with each member normal in the next. Such an ascending chain is termed a subnormal series. A <math>k</math>-subnormal subgroup is a subgroup for which there exists a subnormal series of length at most <math>k</math>. Particular examples are <math>k = 1</math> (normal subgroup), <math>k = 2</math> (2-subnormal subgroup), <math>k = 3</math> (3-subnormal subgroup). Primary subject wiki entry: [[Groupprops:Subnormal subgroup]] Related terms: [[Groupprops:Normal subgroup|Normal subgroup]], [[Groupprops:2-subnormal subgroup|2-subnormal subgroup]], [[Groupprops:3-subnormal subgroup|3-subnormal subgroup]], [[Groupprops:Ascendant subgroup|ascendant subgroup]], [[Groupprops:Descendant subgroup|descendant subgroup]], [[Groupprops:Serial subgroup|serial subgroup]], [[Groupprops:Subnormal series|subnormal series]], [[Groupprops:Normal series|normal series]] 77dea5f81148dd1d64593fd360cb7bb763e2266c 0 119 228 2008-11-08T15:15:11Z Vipul 2 Redirecting to [[Subnormal subgroup]] wikitext text/x-wiki #redirect [[subnormal subgroup]] 1de849d65e1033b0b0716ec9f50502ab6f17d150 0 120 229 2008-11-08T15:15:39Z Vipul 2 Redirecting to [[Normal subgroup]] wikitext text/x-wiki #redirect [[Normal subgroup]] 5a4b2b3db5b8454f300522b4a05cd6c2ac92f9b6 12 121 230 2008-11-17T22:36:37Z Vipul 2 New page: This page gives detailed information about using MediaWiki search on subject wikis. Note that MediaWiki search in a given subject wiki searches ''only that wiki''. ==The search bar== Th... wikitext text/x-wiki This page gives detailed information about using MediaWiki search on subject wikis. Note that MediaWiki search in a given subject wiki searches ''only that wiki''. ==The search bar== The search bar appears at the top of the left-hand sidebar in every page, right below the logo (or logo region). Below the bar are two options: the "Go" button and the "Search" button. ===Search suggestions=== As you type queries into the search bar, a drop-down menu appears with a list of suggested completions. These completions include all pages whose name begins with the string typed so far. These suggestions are called "AJAX suggestions". As a logged-in user, you can disable these suggestions by clicking through "My Preferences", selecting the "Search" tab, checking the "Disable AJAX suggestions" box, and saving. There may be a limit on the number of completions shown, based on settings at the server or client end. Thus, the list of completions shown for a partial string should not be treated as the entire list of completions. ===The go button=== The search default is that after you type a search query and press "Enter", the "Go" button is activated (the reason the "Go" button appears darker than the "Search" button is because it is the activated option). The "Go" button proceeds as follows: * First, it looks for an article with an exact title match. If an exact title match is not found, a very similar title match (that is the same, except for some uppercase-lowercase differences) is determined. If such a match is found, you are taken directly to the article. Note that the search is not completely case-insensitive, even though it may overcome slight differences in case. * If this fails, a regular search is performed, the way pressing the "Search" button would have achieved. ===The search button=== The search button invokes a regular MediaWiki search. The MediaWiki search results page has the following structure: * The top of the page lists alternative links to "all pages starting with" the search term, and to "all pages linking to" the search term. These alternative links are useful in some cases. * Below this, there is a section titled '''No page title matches'''. This states that there is no page with the exact title, and links to relevant resource pages within that wiki that offer other search features, as well as things to bear in mind while searching on the wiki. * If there are partial page title matches, these matches are listed under a section called '''Page title matches'''. Following this section is a section titled '''Page text matches''', that finds matches within the page text. ==Other options== ===Controlling the appearance of search results=== There are three parameters that individual logged-in users can change in the appearance of search results. These include: * Hits per page: This controls the number of hits that appear on one search page. * Lines per hit: * Context per line: ===Search namespaces=== Every wiki has a collection of default namespaces to search in. All subject wiki search in the main namespace, as well as in the category namespace, the help namespace, the project namespace, and some other namespaces. As a logged-in user, you can change the namespaces you want searched by default by modifying them in the "search" tab under "my preferences". If you want to alter the namespaces being searched for a given search, go to the [[Special:Search]] page of the wiki. ===Operators in search=== The only operator allowed in the version of MediaWiki currently running is quotation marks, which search for entire strings. Some operators have been introduced in the newer version of MediaWiki, that is currently running on Wikipedia. These include intitle search, prefix search, wildcard and fuzzy search, and corrections for mis-spellings. bbc7ed227e3dc3ff043e9a5080bba8b9c3cce3e2 231 230 2008-11-17T22:40:43Z Vipul 2 /* Other options */ wikitext text/x-wiki This page gives detailed information about using MediaWiki search on subject wikis. Note that MediaWiki search in a given subject wiki searches ''only that wiki''. ==The search bar== The search bar appears at the top of the left-hand sidebar in every page, right below the logo (or logo region). Below the bar are two options: the "Go" button and the "Search" button. ===Search suggestions=== As you type queries into the search bar, a drop-down menu appears with a list of suggested completions. These completions include all pages whose name begins with the string typed so far. These suggestions are called "AJAX suggestions". As a logged-in user, you can disable these suggestions by clicking through "My Preferences", selecting the "Search" tab, checking the "Disable AJAX suggestions" box, and saving. There may be a limit on the number of completions shown, based on settings at the server or client end. Thus, the list of completions shown for a partial string should not be treated as the entire list of completions. ===The go button=== The search default is that after you type a search query and press "Enter", the "Go" button is activated (the reason the "Go" button appears darker than the "Search" button is because it is the activated option). The "Go" button proceeds as follows: * First, it looks for an article with an exact title match. If an exact title match is not found, a very similar title match (that is the same, except for some uppercase-lowercase differences) is determined. If such a match is found, you are taken directly to the article. Note that the search is not completely case-insensitive, even though it may overcome slight differences in case. * If this fails, a regular search is performed, the way pressing the "Search" button would have achieved. ===The search button=== The search button invokes a regular MediaWiki search. The MediaWiki search results page has the following structure: * The top of the page lists alternative links to "all pages starting with" the search term, and to "all pages linking to" the search term. These alternative links are useful in some cases. * Below this, there is a section titled '''No page title matches'''. This states that there is no page with the exact title, and links to relevant resource pages within that wiki that offer other search features, as well as things to bear in mind while searching on the wiki. * If there are partial page title matches, these matches are listed under a section called '''Page title matches'''. Following this section is a section titled '''Page text matches''', that finds matches within the page text. ==Other options== ===Controlling the appearance of search results=== There are three parameters that individual logged-in users can change in the appearance of search results. These include: * Hits per page: This controls the number of hits that appear on one search page. * Lines per hit: * Context per line: ===Search namespaces=== Every wiki has a collection of default namespaces to search in. All subject wiki search in the main namespace, as well as in the category namespace, the help namespace, the project namespace, and some other namespaces. As a logged-in user, you can change the namespaces you want searched by default by modifying them in the "search" tab under "my preferences". If you want to alter the namespaces being searched for a given search, go to the [[Special:Search]] page of the wiki. ===Operators in search=== The only operator allowed in the version of MediaWiki currently running is quotation marks, which search for entire strings. Some operators have been introduced in the newer version of MediaWiki, that is currently running on Wikipedia. These include intitle search, prefix search, wildcard and fuzzy search, and corrections for mis-spellings. We hope to provide these features and many more in the near future, once the new version is officially released. {{further|[[Wikipedia:Wikipedia:Searching]]}} 5afdd980d9af3fc461032f25af6be169e9b1b06c 232 231 2008-11-17T22:42:01Z Vipul 2 wikitext text/x-wiki This page gives detailed information about using MediaWiki search on subject wikis. Note that MediaWiki search in a given subject wiki searches ''only that wiki''. For more information on ''general'' search help, refer the pages: [[wikipedia:Wikipedia:Searching]] and [[Wikipedia:Help:Searching]]. ==The search bar== The search bar appears at the top of the left-hand sidebar in every page, right below the logo (or logo region). Below the bar are two options: the "Go" button and the "Search" button. ===Search suggestions=== As you type queries into the search bar, a drop-down menu appears with a list of suggested completions. These completions include all pages whose name begins with the string typed so far. These suggestions are called "AJAX suggestions". As a logged-in user, you can disable these suggestions by clicking through "My Preferences", selecting the "Search" tab, checking the "Disable AJAX suggestions" box, and saving. There may be a limit on the number of completions shown, based on settings at the server or client end. Thus, the list of completions shown for a partial string should not be treated as the entire list of completions. ===The go button=== The search default is that after you type a search query and press "Enter", the "Go" button is activated (the reason the "Go" button appears darker than the "Search" button is because it is the activated option). The "Go" button proceeds as follows: * First, it looks for an article with an exact title match. If an exact title match is not found, a very similar title match (that is the same, except for some uppercase-lowercase differences) is determined. If such a match is found, you are taken directly to the article. Note that the search is not completely case-insensitive, even though it may overcome slight differences in case. * If this fails, a regular search is performed, the way pressing the "Search" button would have achieved. ===The search button=== The search button invokes a regular MediaWiki search. The MediaWiki search results page has the following structure: * The top of the page lists alternative links to "all pages starting with" the search term, and to "all pages linking to" the search term. These alternative links are useful in some cases. * Below this, there is a section titled '''No page title matches'''. This states that there is no page with the exact title, and links to relevant resource pages within that wiki that offer other search features, as well as things to bear in mind while searching on the wiki. * If there are partial page title matches, these matches are listed under a section called '''Page title matches'''. Following this section is a section titled '''Page text matches''', that finds matches within the page text. ==Other options== ===Controlling the appearance of search results=== There are three parameters that individual logged-in users can change in the appearance of search results. These include: * Hits per page: This controls the number of hits that appear on one search page. * Lines per hit: * Context per line: ===Search namespaces=== Every wiki has a collection of default namespaces to search in. All subject wiki search in the main namespace, as well as in the category namespace, the help namespace, the project namespace, and some other namespaces. As a logged-in user, you can change the namespaces you want searched by default by modifying them in the "search" tab under "my preferences". If you want to alter the namespaces being searched for a given search, go to the [[Special:Search]] page of the wiki. ===Operators in search=== The only operator allowed in the version of MediaWiki currently running is quotation marks, which search for entire strings. Some operators have been introduced in the newer version of MediaWiki, that is currently running on Wikipedia. These include intitle search, prefix search, wildcard and fuzzy search, and corrections for mis-spellings. We hope to provide these features and many more in the near future, once the new version is officially released. {{further|[[Wikipedia:Wikipedia:Searching]]}} ee53a8c0dcffad946a82ccaeedf9f10e4feb24f8 8 122 233 2008-11-19T22:12:09Z Vipul 2 New page: //<source lang="JavaScript"> /** Change Special:Search to use a drop-down menu ******************************************************* * * Description: Change Special:Search to use a... javascript text/javascript //<source lang="JavaScript"> /** Change Special:Search to use a drop-down menu ******************************************************* * * Description: Change Special:Search to use a drop-down menu, with the default being * the internal MediaWiki engine. This script is copied from a similar script at Wikipedia, located at * http://en.wikipedia.org/wiki/MediaWiki:Common.js/search.js, and is thereby released under the same license terms (GNU Free Documentation License, Version 1.2). * The original script was ceated and maintained by: [[Wikipedia:User:Gracenotes]] */ function SpecialSearchEnhanced() { var createOption = function(site, action, mainQ, addQ, addV) { var opt = document.createElement('option'); opt.appendChild(document.createTextNode(site)); searchEngines[searchEngines.length] = [action, mainQ, addQ, addV]; return opt; } if (document.forms['powersearch']) var searchForm = document.forms['powersearch']; if (document.forms['search']) var searchForm = document.forms['search']; if (searchForm.lsearchbox) { var searchBox = searchForm.lsearchbox; } else { var searchBox = searchForm.search; } var selectBox = document.createElement('select'); selectBox.id = 'searchEngine'; searchForm.onsubmit = function() { var optSelected = searchEngines[document.getElementById('searchEngine').selectedIndex]; searchForm.action = optSelected[0]; searchBox.name = optSelected[1]; searchForm.title.value = optSelected[3]; searchForm.title.name = optSelected[2]; } selectBox.appendChild(createOption('Search Ref', wgScriptPath + '/index.php', 'search', 'title', 'Special:Search')); selectBox.appendChild(createOption('Groupprops', 'http://groupprops.subwiki.org/w', 'search', 'title', 'Special:Search')); selectBox.appendChild(createOption('Topospaces', 'http://topospaces.subwiki.org/w', 'search', 'title', 'Special:Search')); selectBox.appendChild(createOption('Commalg', 'http://commalg.subwiki.org/w', 'search', 'title', 'Special:Search')); selectBox.appendChild(createOption('Diffgeom', 'http://diffgeom.subwiki.org/w', 'search', 'title', 'Special:Search')); selectBox.appendChild(createOption('Companal', 'http://companal.subwiki.org/w', 'search', 'title', 'Special:Search')); searchBox.style.marginLeft = '0px'; if (document.getElementById('loadStatus')) { var lStat = document.getElementById('loadStatus'); } else { var lStat = searchForm.fulltext; } lStat.parentNode.insertBefore(selectBox, lStat); } var searchEngines = []; addOnloadHook(SpecialSearchEnhanced); //</source> 078e76d88029f02b613333bbf25d47372e798642 8 123 234 2008-11-19T22:12:39Z Vipul 2 New page: /* Any JavaScript here will be loaded for all users on every page load. */ if (wgPageName == "Special:Search") //scripts specific to Special:Search { importScript("MediaWiki:Common.js... javascript text/javascript /* Any JavaScript here will be loaded for all users on every page load. */ if (wgPageName == "Special:Search") //scripts specific to Special:Search { importScript("MediaWiki:Common.js/search.js") } c2422b9dd68a6769ba3f82df604fce05e88d280e 0 124 235 2008-12-10T01:29:01Z Vipul 2 New page: The term ''complete'' in mathematics is used in the following broad senses: * An algebraic, topological or geometric structure that is closed under certain kinds of operations. * A struct... wikitext text/x-wiki The term ''complete'' in mathematics is used in the following broad senses: * An algebraic, topological or geometric structure that is closed under certain kinds of operations. * A structure such that any bigger structure containing it has it as a ''summand'' or ''split piece''. ===In group theory=== {{:Complete group}} ===In category theory=== {{:Complete category}} ===In ring theory=== {{:Complete intersection ring}} ===In topology=== {{:Complete metric space}} {{:Completely metrizable space}} ===In differential geometry=== {{:Complete Riemannian manifold}} cb5fa626af3241e53609133f1f102fa66a15f5a5 0 125 236 2008-12-10T01:30:45Z Vipul 2 New page: <noinclude>[[Status::Basic definition| ]][[Topic::Group theory| ]][[Primary wiki::Cattheory| ]]</noinclude> '''Complete category''': A category that contains all [[small limit]]s. Primar... wikitext text/x-wiki <noinclude>[[Status::Basic definition| ]][[Topic::Group theory| ]][[Primary wiki::Cattheory| ]]</noinclude> '''Complete category''': A category that contains all [[small limit]]s. Primary subject wiki entry: [[Cattheory:Complete category]] e44d5351b2bcea353090a70be40f8f17c66b4346 0 126 237 2008-12-10T01:32:16Z Vipul 2 New page: <noinclude>[[Status::Basic definition| ]][[Topic::Group theory| ]][[Primary wiki::Groupprops| ]]</noinclude> '''Complete group''': A group whose center is trivial and in which every autom... wikitext text/x-wiki <noinclude>[[Status::Basic definition| ]][[Topic::Group theory| ]][[Primary wiki::Groupprops| ]]</noinclude> '''Complete group''': A group whose center is trivial and in which every automorphism is inner. Equivalently, it is a direct factor inside any group in which it is a normal subgroup. Primary subject wiki entry: [[Groupprops:Complete group]] 410c6cb4fb2a3b7ba107cb883009e028d6c0ec97 106 127 239 2008-12-25T11:11:24Z Vipul 2 New page: Property definition articles are particular kinds of [[subwiki:definition article|definition articles]], where the definiendum (term being defined) is a property. This page discusses the o... wikitext text/x-wiki Property definition articles are particular kinds of [[subwiki:definition article|definition articles]], where the definiendum (term being defined) is a property. This page discusses the overall design and organization of property definition articles. The principles discussed here build on the [[subwiki:definition article|general principles for definition articles]]. 52c40fb6af6bda82d64b02d253e304d37f8cf9a1 240 239 2008-12-25T11:28:02Z Vipul 2 wikitext text/x-wiki Property definition articles are particular kinds of [[subwiki:definition article|definition articles]], where the definiendum (term being defined) is a property. This page discusses the overall design and organization of property definition articles. The principles discussed here build on the [[subwiki:definition article|general principles for definition articles]]. ==Goals of the property definition article== ===Content goals=== The property definition article should provide: * A clear definition of the property, sufficient to give a clear criterion of whether an object satisfies or does not satisfy the property. * A representative range of examples. * The relation with other properties over the same collection of objects: what properties are stronger, what properties are weaker, and what properties are otherwise closely related. * The metaproperties satisfied and not satisfied by the property. ===Link goals=== The definition article should provide brief information with further links to: * Articles detailing the history of the property: how it was first considered, and what it is used for today. * Survey articles explaining how to understand the definition of the property and how to test whether an object in the collection satisfies the property. * Explanations of property satisfactions and dissatisfactions: For objects satisfying the property, links to proofs/explanations of why the objects satisfy the property. * Explanations of property implications and non-implications: For stronger and weaker properties, links to proofs/explanations of the implication and also of the breakdown of the ''reverse'' implication. * Explanations of metaproperty satisfactions and dissatisfactions. ==Article-tagging templates== {{further|[[Subwiki:Property space-specification template]]}} A property definition article has a [[subwiki:property space-specification template|property space-specification template]]. This template may be of the following kinds: * An ''ordinary'' template, which simply states that the given property is a property over a certain collection of objects. This includes the page in a category listing all properties over that collection of objects, and also provides links to the category list as well as to relevant definitions. * A ''conjunction'' template, which states that the given property is the conjunction ('''AND''') of two other properties. * A ''pivotal property'' template, which states that the given property is one of the pivotal, or important, properties over a certain collection of objects. b2118e4d235e9a2aa57c6e7ec10b6743269056ee 0 128 242 2008-12-26T06:30:14Z Vipul 2 New page: '''Normal monomorphism''': A monomorphism in a preadditive category that arises as the kernel of an epimorphism. Primary subject wiki entry: [[Cattheory:Normal monomorphism]] wikitext text/x-wiki '''Normal monomorphism''': A monomorphism in a preadditive category that arises as the kernel of an epimorphism. Primary subject wiki entry: [[Cattheory:Normal monomorphism]] b89aa706bc3930179ff79930ff1ec59f3a813106 246 242 2008-12-26T07:05:09Z Vipul 2 wikitext text/x-wiki '''Normal monomorphism''': A monomorphism in a preadditive category, or more generally, in a category enriched over pointed sets, that arises as the kernel of an epimorphism. Primary subject wiki entry: [[Cattheory:Normal monomorphism]] 1f73a800e8509b75aec6168f6bcbca534625862f 0 129 243 2008-12-26T06:30:58Z Vipul 2 New page: '''Normal epimorphism''': An epimorphism in a preadditive category that arises as the cokernel of a monomorphism. Primary subject wiki entry: [[Cattheory:Normal epimorphism]] wikitext text/x-wiki '''Normal epimorphism''': An epimorphism in a preadditive category that arises as the cokernel of a monomorphism. Primary subject wiki entry: [[Cattheory:Normal epimorphism]] 53f3cf20da1a82f1fa9d6af429ec96ad9c6cd08a 247 243 2008-12-26T07:06:11Z Vipul 2 wikitext text/x-wiki '''Normal epimorphism''': An epimorphism in a preadditive category, or more generally a category enriched over pointed sets, that arises as the cokernel of a monomorphism. Primary subject wiki entry: [[Cattheory:Normal epimorphism]] f9767ee1c544ae5445c1ab91bf11e8793926d0a4 0 3 244 120 2008-12-26T07:02:55Z Vipul 2 /* In category theory */ wikitext text/x-wiki The term ''normal'' in mathematics is used in the following broad senses: * To denote something upright or perpendicular * To denote something that is as it ''should be''. In this sense, normal means ''good'' or ''desirable'' rather than ''typical'' It is typically used as a limiting adjective, and in a binary sense. In other words, an object in a certain class is ''normal'' if it satisfies certain conditions, and every object in that class is either normal or not normal. The noun form is ''normality''. Thus, normality can be viewed as a property in various contexts. ===In group theory=== {{:Normal subgroup}} ===In topology=== {{:Normal space}} ===In linear algebra=== {{:Normal form (linear algebra)}} {{:Normal matrix}} {{:Normal operator}} ===In field theory=== {{:Normal field extension}} ===In commutative algebra=== {{:Normal domain}} {{:Normal ring}} ===In differential geometry=== Normal to a curve in the plane: The line perpendicular to the tangent to the curve at a point. {{:Normal bundle}} {{:Normal coordinate system}} ===In category theory=== {{:Normal monomorphism}} {{:Normal epimorphism}} {{:Normal category}} ===In probability/statistics=== {{:Normal distribution}} ===In axiomatic set theory=== {{:Normal function}} {{:Normal measure}} ===Others=== {{:Normal number}} 331e0830c08aae7e0eac2964c64aa4704ddac7fd 0 130 245 2008-12-26T07:04:08Z Vipul 2 New page: '''Normal category''': A preadditive category, or more generally a category enriched over pointed sets, in which every monomorphism is a normal monomorphism. Primary subject wiki entry: [... wikitext text/x-wiki '''Normal category''': A preadditive category, or more generally a category enriched over pointed sets, in which every monomorphism is a normal monomorphism. Primary subject wiki entry: [[Cattheory:Normal category]] bb33f007bd379760fb97d3128ab24add9024d1f0 106 22 248 42 2008-12-26T10:25:47Z Vipul 2 wikitext text/x-wiki * [[Groupprops:Main Page|Groupprops]]: The Group Properties Wiki (currently, around 2000 pages). Main topic is group theory * [[Topospaces:Main Page|Topospaces]]: The Topology Wiki (currently, around 400 pages). Main topics are point-set topology and algebraic topology * [[Commalg:Main Page|Commalg]]: The Commutative Algebra Wiki (currently, around 300 pages). Main topic is commutative algebra, with some algebraic geometry. * [[Diffgeom:Main Page|Diffgeom]]: The Differential Geometry Wiki * [[Measure:Main Page|Measure]]: The Measure Theory Wiki (to be set up) * [[Noncommutative:Main Page|Noncommutative]]: The Noncommutative Algebra Wiki * [[Companal:Main Page|Companal]]: The Complex Analysis Wiki * [[Cattheory:Main Page|Cattheory]]: The Category Theory Wiki (just started) Some upcoming subject wikis: * Number: The Number Theory Wiki * Mech: The Classical Mechanics Wiki * Linear: The Theoretical and Computational Linear Algebra Wiki * Galois: The Field Theory and Galois Theory Wiki * Market: The Economic Theory of Markets, Choices and Prices Wiki e029dc1f91ed05c95dbae0713922d18be538a2b6 106 131 249 2008-12-26T11:08:23Z Vipul 2 New page: The subwiki.org website contains a number of subject wikis in separate subdomain. Each subject wiki functions as an independent and separately managed unit. This article describes some gen... wikitext text/x-wiki The subwiki.org website contains a number of subject wikis in separate subdomain. Each subject wiki functions as an independent and separately managed unit. This article describes some general attributes of subject wikis. ==What is a subject wiki?== ===A quick definition=== A subject wiki is a wiki whose focus in terms of content and perspective is a specific subject. The ''content'' focus: Most content on the wiki is directly related to the specific subject, and most material related to the specific subject is covered in the wiki. The ''perspective'' focus: The organization of content, both within articles and across articles, is done keeping the needs and goals of the subject in mind. Thus, an article on the same topic may appear very different in different subject wikis, and may be organized or categorized differently. ===What is a subject?=== Subject wikis are currently at an experimental, pre-alpha stage, so the definition of subject is still very much in flux. Here are some possible general guidelines: * A subject that deserves a wiki of its own should have a reasonable coherent mass of content, or a basic body of knowledge, along with its own distinct perspectives, needs and goals. * The subject should have a strong core (giving character and cohesion to the subject wiki's content) with a large periphery linking the material to other subjects. * Under the current setup, ''subjects'' as used in subject wikis refer to ''academic'' subjects with huge bodies of knowledge, whose ''core'' is not determined by topical news and events. For instance, Salman Rushdie and Harry Potter are not suitable for subject wikis, even though there may be huge boies of knowledge on both. Karl Marx is not a subject, but communist theory may be. The distinction can get muddled at times, specifically for subjects whose significance is historical or cultural. * The core of a subject wiki should be strong enough that ''most'' of the links are internal. In other words, most of the questions that arise naturally from reading one topic article in the subject wiki should be answerable by reading other topic articles. ===Can subject wikis have overlapping themes and content?=== Overlap of content between subject wikis is a good thing, as long as each subject wiki has a different overall perspective and goal. For instance, topics in human behavior are of interest in economics, psychology, anthropology, history and many other subjects. Commonly used ideas within a broad discipline may get different kinds of treatment in its subdisciplines. For instance: * The concept of a chemical bond is of importance in chemistry. The subdiscipline of chemical bonding, the subdisciplines of chemical reactions and their kinetics and equilibrium, the subdiscipline of energy changes in chemical reactions, the subdiscipline of analytic chemistry (testing of compounds), all look at bonds from somewhat different angles. * The notion of price is of importance in economics. 1319e2a5e4ffc355bb890a2ea72b1246769cd9c9 250 249 2008-12-26T12:56:49Z Vipul 2 wikitext text/x-wiki The subwiki.org website contains a number of subject wikis in separate subdomain. Each subject wiki functions as an independent and separately managed unit. This article describes some general attributes of subject wikis. ==What is a subject wiki?== ===A quick definition=== A subject wiki is a wiki whose focus in terms of content and perspective is a specific subject. The ''content'' focus: Most content on the wiki is directly related to the specific subject, and most material related to the specific subject is covered in the wiki. The ''perspective'' focus: The organization of content, both within articles and across articles, is done keeping the needs and goals of the subject in mind. Thus, an article on the same topic may appear very different in different subject wikis, and may be organized or categorized differently. ===What is a subject?=== Subject wikis are currently at an experimental, pre-alpha stage, so the definition of subject is still very much in flux. Here are some possible general guidelines: * A subject that deserves a wiki of its own should have a reasonable coherent mass of content, or a basic body of knowledge, along with its own distinct perspectives, needs and goals. * The subject should have a strong core (giving character and cohesion to the subject wiki's content) with a large periphery linking the material to other subjects. * Under the current setup, ''subjects'' as used in subject wikis refer to ''academic'' subjects with huge bodies of knowledge, whose ''core'' is not determined by topical news and events. For instance, Salman Rushdie and Harry Potter are not suitable for subject wikis, even though there may be huge boies of knowledge on both. Karl Marx is not a subject, but communist theory may be. The distinction can get muddled at times, specifically for subjects whose significance is historical or cultural. * The core of a subject wiki should be strong enough that ''most'' of the links are internal. In other words, most of the questions that arise naturally from reading one topic article in the subject wiki should be answerable by reading other topic articles. ===Can subject wikis have overlapping themes and content?=== Overlap of content between subject wikis is a good thing, as long as each subject wiki has a different overall perspective and goal. For instance, topics in human behavior are of interest in economics, psychology, anthropology, history and many other subjects. Commonly used ideas within a broad discipline may get different kinds of treatment in its subdisciplines. For instance: * The concept of a chemical bond is of importance in chemistry. The subdiscipline of chemical bonding, the subdisciplines of chemical reactions and their kinetics and equilibrium, the subdiscipline of energy changes in chemical reactions, the subdiscipline of analytic chemistry (testing of compounds), all look at bonds from somewhat different angles. * The notion of price is of importance in economics. The subdisciplines of microeconomic theory (that focuses on market interactions between individuals transacting), macroeconomic theory, behavioural economics, evolutionary economics, and decision analysis all view this notion somewhat differently. What should differ between different subject wikis is the kind of core and perspective offered. It may also be possible to have ''broader level'' subject wikis -- subject wikis that cover a broad level topic from a general view and use generic organizational principles. For instance, there ma be a subject wiki on elementary organic chemistry, and there may be separate subject wikis on separate subdisciplines of organic chemistry. ===How are subject wikis being created? Is there a broad "body of knowledge" principle?=== As of now, there is no broad "body of knowledge" or "hierarchy of knowledge" used for the creation of subject wikis. Rather, individual subject wikis are being created experimentally. Once the experiment reaches a somewhat more advanced stage, we may introduce elements of hierarchical planning. 781e3208d3ff5900dd04a241907afb45518f5f47 106 132 251 2008-12-28T13:36:27Z Vipul 2 New page: This article describes the multiple steps of the setup procedure for a subject wiki. ==Deciding a name and a subtitle== The first step is to decide on a one-word name for the subject wik... wikitext text/x-wiki This article describes the multiple steps of the setup procedure for a subject wiki. ==Deciding a name and a subtitle== The first step is to decide on a one-word name for the subject wiki. This one-word name comes up as: * The official wiki name. * The subdomain name used to store the wiki. * The variable part of the name of the specific database where the wiki's contents are stored. The subtitle is a phrase of the form ''The ... Wiki'', where ... indicates the subject in two to four words. ==Installation== ===Basic installation and configuration=== Reference: [[mediawikiwiki:manual:installation guide]] This has the following important steps: * Create the database: The name of the database is determined by the official wiki name. * Download and extract the latest version of MediaWiki from http://www.mediawiki.org * Use the on-screen configuration process and complete the moves. ===Further steps=== # Set up short URLs for /wiki/ to direct to /w/index.php?title= (reference: [[mediawikiwiki:Manual:Short URLs]]). Or, copy the short URL snippet code from another subject wiki. # Set up math mode: The OCaml software has already been installed on the server. Math mode can be set using $wgUseTex. # Install extensions: Straightforward extensions to install: CategoryTree, LabeledSectionTransclusion, MultiCategorySearch, NewestPages, ParserFunctions. # Install Google Analytics extension and get an Analytics UID. # Install ConfirmAccount and SemanticMediaWiki extensions. Both these extensions require some database operations. 84723ab952251198c5e93aa6f741ac0d23bd14ce 252 251 2008-12-28T14:07:23Z Vipul 2 wikitext text/x-wiki This article describes the multiple steps of the setup procedure for a subject wiki. ==Deciding a name and a subtitle== The first step is to decide on a one-word name for the subject wiki. This one-word name comes up as: * The official wiki name. * The subdomain name used to store the wiki. * The variable part of the name of the specific database where the wiki's contents are stored. The subtitle is a phrase of the form ''The ... Wiki'', where ... indicates the subject in two to four words. ==Installation== ===Basic installation and configuration=== Reference: [[mediawikiwiki:manual:installation guide]] This has the following important steps: * Create the database: The name of the database is determined by the official wiki name. * Download and extract the latest version of MediaWiki from http://www.mediawiki.org * Use the on-screen configuration process and complete the moves. ===Further installation steps=== # Set up short URLs for /wiki/ to direct to /w/index.php?title= (reference: [[mediawikiwiki:Manual:Short URLs]]). Or, copy the short URL snippet code from another subject wiki. # Set up math mode: The OCaml software has already been installed on the server. Math mode can be set using $wgUseTex. # Set #wgUseAjax = true. # Copy searchsettings.inc, editpermissions.inc, and namespaces.inc from Ref or any other subject wiki, make suitable modifications and include from LocalSettings.php. Include them in the order: namespaces.inc, editpermissions.inc, searchsettings.inc. # Install extensions: Straightforward extensions to install: CategoryTree, LabeledSectionTransclusion, MultiCategorySearch, NewestPages, ParserFunctions, Social Bookmarking, Random Page. Also install ConfirmAccount. Copy the extensionlist.inc file from Ref and alter it suitably. Finally, do the necessary database operations for ConfirmAccount. # Install the SemanticMediaWiki extension, and copy and include semanticsettings.inc from Ref or Groupprops, making necessary modification. Include it after extensionlist.inc but before the cache clearing at the end of the file. # Install Google Analytics extension and get an Analytics UID. Copy the analytics.inc generic file from Ref and alter the UID. Include in LocalSettings.php at the end of the file, after the cache clearing. ==On-wiki changes== ===Necessary and immediate=== # Create two administrative accounts, one impersonal and one personal. Edit their user groups appropriately. # Load the 4AM logo as Logo.jpg. # Edit the sitenotice: The first line of the sitenotice should give the name and subtitle, followed by a parenthetical comment indicating the status (just setup or pre-pre-alpha, to begin with). Include the 4AM logo as a thumbnail on the right. # Edit the sidebar: Move SEARCH to the top, remove unnecessary navigational links. If the wiki will be organized using a similar paradigm as Groupprops, copy from [[Groupprops:MediaWiki:Sidebar]]. # Import or copy generic templates: {{fillin}} # Import or copy the Copyright notice from Ref. # Import or copy [[MediaWiki:Noarticletext]], [[MediaWiki:Newarticletext]]: {{fillin}} # Upgrade the on-wiki search features: Edit the search display text. Also, edit Common.js to allow for searching through external search engines starting from the site. ===Not immediately necessary=== # Commission a logo for the left upper corner display. # Convert the logo to an ICO file and set that as the favicon for the wiki. # Create search plugins for the wiki, in the manner of the search plugin for Groupprops. ===From other wikis=== # Add interwiki links to it from Ref and the other subject wikis. # Add it to the Subwiki list on Ref and also to [[MediaWiki:Sidebar]] on Ref. 635ba698e5b8526bc2ca09d8ef76f3ed95d51be1 259 252 2008-12-29T08:15:18Z Vipul 2 /* Further installation steps */ wikitext text/x-wiki This article describes the multiple steps of the setup procedure for a subject wiki. ==Deciding a name and a subtitle== The first step is to decide on a one-word name for the subject wiki. This one-word name comes up as: * The official wiki name. * The subdomain name used to store the wiki. * The variable part of the name of the specific database where the wiki's contents are stored. The subtitle is a phrase of the form ''The ... Wiki'', where ... indicates the subject in two to four words. ==Installation== ===Basic installation and configuration=== Reference: [[mediawikiwiki:manual:installation guide]] This has the following important steps: * Create the database: The name of the database is determined by the official wiki name. * Download and extract the latest version of MediaWiki from http://www.mediawiki.org * Use the on-screen configuration process and complete the moves. ===Further installation steps=== # Set up short URLs for /wiki/ to direct to /w/index.php?title= (reference: [[mediawikiwiki:Manual:Short URLs]]). Or, copy the short URL snippet code from another subject wiki. # Set up math mode: The OCaml software has already been installed on the server. Math mode can be set using $wgUseTex. # Set $wgUseAjax = true. # Copy searchsettings.inc, editpermissions.inc, and namespaces.inc from Ref or any other subject wiki, make suitable modifications and include from LocalSettings.php. Include them in the order: namespaces.inc, editpermissions.inc, searchsettings.inc. # Install extensions: Straightforward extensions to install: CategoryTree, LabeledSectionTransclusion, MultiCategorySearch, NewestPages, ParserFunctions, Social Bookmarking, Random Page. Also install ConfirmAccount. Copy the extensionlist.inc file from Ref and alter it suitably. Finally, do the necessary database operations for ConfirmAccount. # Install the SemanticMediaWiki extension, and copy and include semanticsettings.inc from Ref or Groupprops, making necessary modification. Include it after extensionlist.inc but before the cache clearing at the end of the file. # Install Google Analytics extension and get an Analytics UID. Copy the analytics.inc generic file from Ref and alter the UID. Include in LocalSettings.php at the end of the file, after the cache clearing. ==On-wiki changes== ===Necessary and immediate=== # Create two administrative accounts, one impersonal and one personal. Edit their user groups appropriately. # Load the 4AM logo as Logo.jpg. # Edit the sitenotice: The first line of the sitenotice should give the name and subtitle, followed by a parenthetical comment indicating the status (just setup or pre-pre-alpha, to begin with). Include the 4AM logo as a thumbnail on the right. # Edit the sidebar: Move SEARCH to the top, remove unnecessary navigational links. If the wiki will be organized using a similar paradigm as Groupprops, copy from [[Groupprops:MediaWiki:Sidebar]]. # Import or copy generic templates: {{fillin}} # Import or copy the Copyright notice from Ref. # Import or copy [[MediaWiki:Noarticletext]], [[MediaWiki:Newarticletext]]: {{fillin}} # Upgrade the on-wiki search features: Edit the search display text. Also, edit Common.js to allow for searching through external search engines starting from the site. ===Not immediately necessary=== # Commission a logo for the left upper corner display. # Convert the logo to an ICO file and set that as the favicon for the wiki. # Create search plugins for the wiki, in the manner of the search plugin for Groupprops. ===From other wikis=== # Add interwiki links to it from Ref and the other subject wikis. # Add it to the Subwiki list on Ref and also to [[MediaWiki:Sidebar]] on Ref. 325d11cb4eef3a329e583b38b89f11c598005475 260 259 2008-12-29T08:17:52Z Vipul 2 /* Basic installation and configuration */ wikitext text/x-wiki This article describes the multiple steps of the setup procedure for a subject wiki. ==Deciding a name and a subtitle== The first step is to decide on a one-word name for the subject wiki. This one-word name comes up as: * The official wiki name. * The subdomain name used to store the wiki. * The variable part of the name of the specific database where the wiki's contents are stored. The subtitle is a phrase of the form ''The ... Wiki'', where ... indicates the subject in two to four words. ==Installation== ===Basic installation and configuration=== Reference: [[mediawikiwiki:manual:installation guide]] This has the following important steps: * Create the database: The name of the database is determined by the official wiki name. * Download and extract the latest version of MediaWiki from http://www.mediawiki.org * Use the on-screen configuration process and complete the moves. * Configure the backup scripts for the database and other content. ===Further installation steps=== # Set up short URLs for /wiki/ to direct to /w/index.php?title= (reference: [[mediawikiwiki:Manual:Short URLs]]). Or, copy the short URL snippet code from another subject wiki. # Set up math mode: The OCaml software has already been installed on the server. Math mode can be set using $wgUseTex. # Set $wgUseAjax = true. # Copy searchsettings.inc, editpermissions.inc, and namespaces.inc from Ref or any other subject wiki, make suitable modifications and include from LocalSettings.php. Include them in the order: namespaces.inc, editpermissions.inc, searchsettings.inc. # Install extensions: Straightforward extensions to install: CategoryTree, LabeledSectionTransclusion, MultiCategorySearch, NewestPages, ParserFunctions, Social Bookmarking, Random Page. Also install ConfirmAccount. Copy the extensionlist.inc file from Ref and alter it suitably. Finally, do the necessary database operations for ConfirmAccount. # Install the SemanticMediaWiki extension, and copy and include semanticsettings.inc from Ref or Groupprops, making necessary modification. Include it after extensionlist.inc but before the cache clearing at the end of the file. # Install Google Analytics extension and get an Analytics UID. Copy the analytics.inc generic file from Ref and alter the UID. Include in LocalSettings.php at the end of the file, after the cache clearing. ==On-wiki changes== ===Necessary and immediate=== # Create two administrative accounts, one impersonal and one personal. Edit their user groups appropriately. # Load the 4AM logo as Logo.jpg. # Edit the sitenotice: The first line of the sitenotice should give the name and subtitle, followed by a parenthetical comment indicating the status (just setup or pre-pre-alpha, to begin with). Include the 4AM logo as a thumbnail on the right. # Edit the sidebar: Move SEARCH to the top, remove unnecessary navigational links. If the wiki will be organized using a similar paradigm as Groupprops, copy from [[Groupprops:MediaWiki:Sidebar]]. # Import or copy generic templates: {{fillin}} # Import or copy the Copyright notice from Ref. # Import or copy [[MediaWiki:Noarticletext]], [[MediaWiki:Newarticletext]]: {{fillin}} # Upgrade the on-wiki search features: Edit the search display text. Also, edit Common.js to allow for searching through external search engines starting from the site. ===Not immediately necessary=== # Commission a logo for the left upper corner display. # Convert the logo to an ICO file and set that as the favicon for the wiki. # Create search plugins for the wiki, in the manner of the search plugin for Groupprops. ===From other wikis=== # Add interwiki links to it from Ref and the other subject wikis. # Add it to the Subwiki list on Ref and also to [[MediaWiki:Sidebar]] on Ref. 42bad97fb680c569b85ff5c2cb9f1461929ade3f 261 260 2008-12-29T12:00:33Z Vipul 2 /* Further installation steps */ wikitext text/x-wiki This article describes the multiple steps of the setup procedure for a subject wiki. ==Deciding a name and a subtitle== The first step is to decide on a one-word name for the subject wiki. This one-word name comes up as: * The official wiki name. * The subdomain name used to store the wiki. * The variable part of the name of the specific database where the wiki's contents are stored. The subtitle is a phrase of the form ''The ... Wiki'', where ... indicates the subject in two to four words. ==Installation== ===Basic installation and configuration=== Reference: [[mediawikiwiki:manual:installation guide]] This has the following important steps: * Create the database: The name of the database is determined by the official wiki name. * Download and extract the latest version of MediaWiki from http://www.mediawiki.org * Use the on-screen configuration process and complete the moves. * Configure the backup scripts for the database and other content. ===Further installation steps=== # Set up short URLs for /wiki/ to direct to /w/index.php?title= (reference: [[mediawikiwiki:Manual:Short URLs]]). Or, copy the short URL snippet code from another subject wiki. # Enable image uploads. # Set up math mode: The OCaml software has already been installed on the server. Math mode can be set using $wgUseTex. # Set $wgUseAjax = true and enable PDF uploads. # Make sure that cache clearing instructions are present at the end of LocalSettings.php. # Copy and include namespaces.inc. # Install the ConfirmAccount extension, and in the process, copy and include editpermissions.inc, which includes this extension. Do the necessary database operations for the extension. # Install extensions: Straightforward extensions to install: CategoryTree, LabeledSectionTransclusion, MultiCategorySearch, NewestPages, ParserFunctions, Social Bookmarking, Random Page. Also install ConfirmAccount. Copy the extensionlist.inc file from Ref and alter it suitably. Finally, do the necessary database operations for ConfirmAccount. # Install the SemanticMediaWiki extension, and copy and include semanticsettings.inc from Ref or Groupprops, making necessary modification. Include it after extensionlist.inc but before the cache clearing at the end of the file. # Copy and include searchsettings.inc. # Install Google Analytics extension and get an Analytics UID. Copy the analytics.inc generic file from Ref and alter the UID. Include in LocalSettings.php at the end of the file, after the cache clearing. ==On-wiki changes== ===Necessary and immediate=== # Create two administrative accounts, one impersonal and one personal. Edit their user groups appropriately. # Load the 4AM logo as Logo.jpg. # Edit the sitenotice: The first line of the sitenotice should give the name and subtitle, followed by a parenthetical comment indicating the status (just setup or pre-pre-alpha, to begin with). Include the 4AM logo as a thumbnail on the right. # Edit the sidebar: Move SEARCH to the top, remove unnecessary navigational links. If the wiki will be organized using a similar paradigm as Groupprops, copy from [[Groupprops:MediaWiki:Sidebar]]. # Import or copy generic templates: {{fillin}} # Import or copy the Copyright notice from Ref. # Import or copy [[MediaWiki:Noarticletext]], [[MediaWiki:Newarticletext]]: {{fillin}} # Upgrade the on-wiki search features: Edit the search display text. Also, edit Common.js to allow for searching through external search engines starting from the site. ===Not immediately necessary=== # Commission a logo for the left upper corner display. # Convert the logo to an ICO file and set that as the favicon for the wiki. # Create search plugins for the wiki, in the manner of the search plugin for Groupprops. ===From other wikis=== # Add interwiki links to it from Ref and the other subject wikis. # Add it to the Subwiki list on Ref and also to [[MediaWiki:Sidebar]] on Ref. bf9b075c63870b5da9c821bbaaab19bcf15bbba8 263 261 2009-01-16T02:09:51Z Vipul 2 /* Necessary and immediate */ wikitext text/x-wiki This article describes the multiple steps of the setup procedure for a subject wiki. ==Deciding a name and a subtitle== The first step is to decide on a one-word name for the subject wiki. This one-word name comes up as: * The official wiki name. * The subdomain name used to store the wiki. * The variable part of the name of the specific database where the wiki's contents are stored. The subtitle is a phrase of the form ''The ... Wiki'', where ... indicates the subject in two to four words. ==Installation== ===Basic installation and configuration=== Reference: [[mediawikiwiki:manual:installation guide]] This has the following important steps: * Create the database: The name of the database is determined by the official wiki name. * Download and extract the latest version of MediaWiki from http://www.mediawiki.org * Use the on-screen configuration process and complete the moves. * Configure the backup scripts for the database and other content. ===Further installation steps=== # Set up short URLs for /wiki/ to direct to /w/index.php?title= (reference: [[mediawikiwiki:Manual:Short URLs]]). Or, copy the short URL snippet code from another subject wiki. # Enable image uploads. # Set up math mode: The OCaml software has already been installed on the server. Math mode can be set using $wgUseTex. # Set $wgUseAjax = true and enable PDF uploads. # Make sure that cache clearing instructions are present at the end of LocalSettings.php. # Copy and include namespaces.inc. # Install the ConfirmAccount extension, and in the process, copy and include editpermissions.inc, which includes this extension. Do the necessary database operations for the extension. # Install extensions: Straightforward extensions to install: CategoryTree, LabeledSectionTransclusion, MultiCategorySearch, NewestPages, ParserFunctions, Social Bookmarking, Random Page. Also install ConfirmAccount. Copy the extensionlist.inc file from Ref and alter it suitably. Finally, do the necessary database operations for ConfirmAccount. # Install the SemanticMediaWiki extension, and copy and include semanticsettings.inc from Ref or Groupprops, making necessary modification. Include it after extensionlist.inc but before the cache clearing at the end of the file. # Copy and include searchsettings.inc. # Install Google Analytics extension and get an Analytics UID. Copy the analytics.inc generic file from Ref and alter the UID. Include in LocalSettings.php at the end of the file, after the cache clearing. ==On-wiki changes== ===Necessary and immediate=== # Create two administrative accounts, one impersonal and one personal. Edit their user groups appropriately. # Load the 4AM logo as Logo.jpg. # Edit the sitenotice: The first line of the sitenotice should give the name and subtitle, followed by a parenthetical comment indicating the status (just setup or pre-pre-alpha, to begin with). Include the 4AM logo as a thumbnail on the right. # Edit the sidebar: Move SEARCH to the top, remove unnecessary navigational links. If the wiki will be organized using a similar paradigm as Groupprops, copy from [[Groupprops:MediaWiki:Sidebar]]. # Import or copy generic templates: {{fillin}} # Import or copy the [[Ref:Copyrights|Copyright notice]] and [[Ref:Privacy policy|privacy policy notice]] from Ref. # Import or copy [[MediaWiki:Noarticletext]], [[MediaWiki:Newarticletext]]: {{fillin}} # Upgrade the on-wiki search features: Edit the search display text. Also, edit Common.js to allow for searching through external search engines starting from the site. ===Not immediately necessary=== # Commission a logo for the left upper corner display. # Convert the logo to an ICO file and set that as the favicon for the wiki. # Create search plugins for the wiki, in the manner of the search plugin for Groupprops. ===From other wikis=== # Add interwiki links to it from Ref and the other subject wikis. # Add it to the Subwiki list on Ref and also to [[MediaWiki:Sidebar]] on Ref. dfe86095338e9f15a6beff55bcbc1a9a71019e53 4 133 253 2008-12-29T07:24:42Z Vipul 2 New page: This is a common copyright notice to all subject wikis. ==General license information== All content is put up under the [http://creativecommons.org/licenses/by-sa/3.0/ Creative Commons A... wikitext text/x-wiki This is a common copyright notice to all subject wikis. ==General license information== All content is put up under the [http://creativecommons.org/licenses/by-sa/3.0/ Creative Commons Attribute Share-Alike 3.0 license (unported)] (shorthand: CC-by-SA). Read the [http://creativecommons.org/licenses/by-sa/3.0/legalcode full legal code here]. ===Exceptions=== The software engine that runs the wiki is MediaWiki, and this software is licensed under the GNU General Public License, which is distinct from and currently incompatible with the Creative Commons license. Similarly, some of the extensions to the software engine are licensed by their developers using the GPL, which is incompatible with Creative Commons licenses. More information on the version of MediaWiki and extensions: [[Special:Version]]. In addition, some templates may have been copied or adapted from Wikipedia or other wiki-based websites that use licenses other than Creative Commons licenses. Wikipedia uses the GNU Free Documentation License (GFDL) that, as of now, is incompatible with CC licenses. In such cases, it is clearly indicated on the page that it is licensed differently. ===Short description of the license=== The CC-by-SA license has the following features: # Content can be used and reused for any purposes, commercial or noncommercial. # Any public use or reuse of the material requires attribution of the original source (for attribution formats, see later in the document). # In case of adaptation or creation of derivative works, the newly created works must also be licensed under the same license (CC-by-SA) unless the creator of the derivative work takes ''explicit'' permission from the creator of the original work. ===Fair use and other rights=== The license cannot and does not limit fair use rights. Fair use may include quoting or copying select passages for the purposes of criticism and review. The license also cannot limit the use of data or information contained in the content on subject wikis. This means that anybody can use or reuse the facts or knowledge they gain from the subject wiki in any way they want. ==Applicability of license to the subject wiki== ===Examples of personal use that do not necessitate use of the license terms=== * Saving, making copies, or taking printouts of pages on the subject wiki, as long as these copies are meant for personal use and are not to be distributed to others. * Learning facts from the subject wikis and using those facts as part of instruction or research. * Providing links to pages on the subject wikis. * Quoting short passages from the subject wiki in a criticism or review (it is preferable, though not legally enforceable, that a link to the original content being reviewed be provided). ===Examples of personal use that necessitate use of the license terms=== * Creating a mirror website by exporting pages or content in bulk: In this case, the mirror website must release the mirrored content under the same license terms and ''also'' attribute the original source with a link. Attribution formats are discussed below. * Giving handouts of pages or collections of pages to a group of people, such as students in a course: In this case, the handouts should be accompanied by an attribution that makes clear both the source and the original license terms, as well as a link to the Uniform Resource Identifier (URI) of the subject wiki. Attribution formats: * For websites: ''This content is from [[fullurl:{{Main Page}}|{{SITENAME}}]]'' or ''This content is from [[fullurl:{{Main Page}}|{{fullsitetitle}}]]''. Preferably, also a link to the specific pages (if any) used. The language of the attribution may be modified somewhat to indicate whether the content has been used as such, or whether changes have been made. * For print materials: ''This content is from {{SITENAME}}. The site URL is http://{{SITENAME}}.subwiki.org'' 84ef7efb99933b7dcc3bc01377e23bcfef36b510 254 253 2008-12-29T07:25:59Z Vipul 2 /* Examples of personal use that necessitate use of the license terms */ wikitext text/x-wiki This is a common copyright notice to all subject wikis. ==General license information== All content is put up under the [http://creativecommons.org/licenses/by-sa/3.0/ Creative Commons Attribute Share-Alike 3.0 license (unported)] (shorthand: CC-by-SA). Read the [http://creativecommons.org/licenses/by-sa/3.0/legalcode full legal code here]. ===Exceptions=== The software engine that runs the wiki is MediaWiki, and this software is licensed under the GNU General Public License, which is distinct from and currently incompatible with the Creative Commons license. Similarly, some of the extensions to the software engine are licensed by their developers using the GPL, which is incompatible with Creative Commons licenses. More information on the version of MediaWiki and extensions: [[Special:Version]]. In addition, some templates may have been copied or adapted from Wikipedia or other wiki-based websites that use licenses other than Creative Commons licenses. Wikipedia uses the GNU Free Documentation License (GFDL) that, as of now, is incompatible with CC licenses. In such cases, it is clearly indicated on the page that it is licensed differently. ===Short description of the license=== The CC-by-SA license has the following features: # Content can be used and reused for any purposes, commercial or noncommercial. # Any public use or reuse of the material requires attribution of the original source (for attribution formats, see later in the document). # In case of adaptation or creation of derivative works, the newly created works must also be licensed under the same license (CC-by-SA) unless the creator of the derivative work takes ''explicit'' permission from the creator of the original work. ===Fair use and other rights=== The license cannot and does not limit fair use rights. Fair use may include quoting or copying select passages for the purposes of criticism and review. The license also cannot limit the use of data or information contained in the content on subject wikis. This means that anybody can use or reuse the facts or knowledge they gain from the subject wiki in any way they want. ==Applicability of license to the subject wiki== ===Examples of personal use that do not necessitate use of the license terms=== * Saving, making copies, or taking printouts of pages on the subject wiki, as long as these copies are meant for personal use and are not to be distributed to others. * Learning facts from the subject wikis and using those facts as part of instruction or research. * Providing links to pages on the subject wikis. * Quoting short passages from the subject wiki in a criticism or review (it is preferable, though not legally enforceable, that a link to the original content being reviewed be provided). ===Examples of personal use that necessitate use of the license terms=== * Creating a mirror website by exporting pages or content in bulk: In this case, the mirror website must release the mirrored content under the same license terms and ''also'' attribute the original source with a link. Attribution formats are discussed below. * Giving handouts of pages or collections of pages to a group of people, such as students in a course: In this case, the handouts should be accompanied by an attribution that makes clear both the source and the original license terms, as well as a link to the Uniform Resource Identifier (URI) of the subject wiki. Attribution formats: * For websites: ''This content is from [[fullurl:Main Page|{{SITENAME}}]]'' or ''This content is from [[fullurl:Main Page|{{fullsitetitle}}]]''. Preferably, also a link to the specific pages (if any) used. The language of the attribution may be modified somewhat to indicate whether the content has been used as such, or whether changes have been made. * For print materials: ''This content is from {{SITENAME}}. The site URL is http://{{SITENAME}}.subwiki.org'' c3f0140df25963b62a7183c2c7e585bda2b39f25 256 254 2008-12-29T07:39:22Z Vipul 2 wikitext text/x-wiki This is a common copyright notice to all subject wikis. ==General license information== All content is put up under the [http://creativecommons.org/licenses/by-sa/3.0/ Creative Commons Attribute Share-Alike 3.0 license (unported)] (shorthand: CC-by-SA). Read the [http://creativehttp://ref.subwiki.org/w/skins/common/images/button_media.pngcommons.org/licenses/by-sa/3.0/legalcode full legal code here]. ===Exceptions=== The software engine that runs the wiki is MediaWiki, and this software is licensed under the GNU General Public License, which is distinct from and currently incompatible with the Creative Commons license. Similarly, some of the extensions to the software engine are licensed by their developers using the GPL, which is incompatible with Creative Commons licenses. More information on the version of MediaWiki and extensions: [[Special:Version]]. In addition, some templates may have been copied or adapted from Wikipedia or other wiki-based websites that use licenses other than Creative Commons licenses. Wikipedia uses the GNU Free Documentation License (GFDL) that, as of now, is incompatible with CC licenses. In such cases, it is clearly indicated on the page that it is licensed differently. ===Short description of the license=== The CC-by-SA license has the following features: # Content can be used and reused for any purposes, commercial or noncommercial. # Any public use or reuse of the material requires attribution of the original source (for attribution formats, see later in the document). # In case of adaptation or creation of derivative works, the newly created works must also be licensed under the same license (CC-by-SA) unless the creator of the derivative work takes ''explicit'' permission from the creator of the original work. ===Fair use and other rights=== The license cannot and does not limit fair use rights. Fair use may include quoting or copying select passages for the purposes of criticism and review. The license also cannot limit the use of data or information contained in the content on subject wikis. This means that anybody can use or reuse the facts or knowledge they gain from the subject wiki in any way they want. ==Applicability of license to the subject wiki== ===Examples of personal use that do not necessitate use of the license terms=== * Saving, making copies, or taking printouts of pages on the subject wiki, as long as these copies are meant for personal use and are not to be distributed to others. * Learning facts from the subject wikis and using those facts as part of instruction or research. * Providing links to pages on the subject wikis. * Quoting short passages from the subject wiki in a criticism or review (it is preferable, though not legally enforceable, that a link to the original content being reviewed be provided). ===Examples of personal use that necessitate use of the license terms=== * Creating a mirror website by exporting pages or content in bulk: In this case, the mirror website must release the mirrored content under the same license terms and ''also'' attribute the original source with a link. Attribution formats are discussed below. * Giving handouts of pages or collections of pages to a group of people, such as students in a course: In this case, the handouts should be accompanied by an attribution that makes clear both the source and the original license terms, as well as a link to the Uniform Resource Identifier (URI) of the subject wiki. Attribution formats: * For websites: ''This content is from [[fullurl:Main Page|{{SITENAME}}]]'' or ''This content is from [[fullurl:Main Page|{{fullsitetitle}}]]''. Preferably, also a link to the specific pages (if any) used. The language of the attribution may be modified somewhat to indicate whether the content has been used as such, or whether changes have been made. * For print materials: ''This content is from {{SITENAME}}. The site URL is http://{{SITENAME}}.subwiki.org'' Attribution of individuals who have contributed content to the subject wiki entries is ''not'' necessary. (However, this list of individuals can be tracked down, as discussed in the next section). ==Uses not within the ambit of the license or fair use rights== Those who have contributed all the content to a particular subject wiki entry can republish the same content anywhere else ''without restriction'' and can also release the content elsewhere under a different license, since they own the full copyright to their content. To use the content on the wiki in a manner that is outside the terms of the license or beyond fair use rights, the individuals who have contributed that particular content need to be contacted. For any particular page, this information can be accessed using the history tab of the page. The users who have made edits to the page are the ones who need to be contacted. In case of difficulty doing this, please contact vipul.wikis@gmail.com or vipul@math.uchicago.edu for more information. ==Copyright violations== All users using subject wikis are requested not to violate copyright of other authors or publishers. This includes providing links that may aid and abet copyright violation. Examples of things that may be considered copyright violation: * Putting up links to personal copies of restricted-access journal articles acquired through a personal or library subscription: This usually goes against the Terms of Service of the subscription. The exception is when the content of the articles is out of copyright. * Giving links to personal copies, or copies on filesharing systems, of books that are under copyright and where either owning or sharing a copy in that manner is against the terms of copyright. In case copyright violations are detected, please email vipul.wikis@gmail.com and vipul@math.uchicago.edu to have the matter looked into immediately. 854dfec6b4ce0fc56003085bfad060ba88436c12 257 256 2008-12-29T07:40:14Z Vipul 2 /* Examples of personal use that necessitate use of the license terms */ wikitext text/x-wiki This is a common copyright notice to all subject wikis. ==General license information== All content is put up under the [http://creativecommons.org/licenses/by-sa/3.0/ Creative Commons Attribute Share-Alike 3.0 license (unported)] (shorthand: CC-by-SA). Read the [http://creativehttp://ref.subwiki.org/w/skins/common/images/button_media.pngcommons.org/licenses/by-sa/3.0/legalcode full legal code here]. ===Exceptions=== The software engine that runs the wiki is MediaWiki, and this software is licensed under the GNU General Public License, which is distinct from and currently incompatible with the Creative Commons license. Similarly, some of the extensions to the software engine are licensed by their developers using the GPL, which is incompatible with Creative Commons licenses. More information on the version of MediaWiki and extensions: [[Special:Version]]. In addition, some templates may have been copied or adapted from Wikipedia or other wiki-based websites that use licenses other than Creative Commons licenses. Wikipedia uses the GNU Free Documentation License (GFDL) that, as of now, is incompatible with CC licenses. In such cases, it is clearly indicated on the page that it is licensed differently. ===Short description of the license=== The CC-by-SA license has the following features: # Content can be used and reused for any purposes, commercial or noncommercial. # Any public use or reuse of the material requires attribution of the original source (for attribution formats, see later in the document). # In case of adaptation or creation of derivative works, the newly created works must also be licensed under the same license (CC-by-SA) unless the creator of the derivative work takes ''explicit'' permission from the creator of the original work. ===Fair use and other rights=== The license cannot and does not limit fair use rights. Fair use may include quoting or copying select passages for the purposes of criticism and review. The license also cannot limit the use of data or information contained in the content on subject wikis. This means that anybody can use or reuse the facts or knowledge they gain from the subject wiki in any way they want. ==Applicability of license to the subject wiki== ===Examples of personal use that do not necessitate use of the license terms=== * Saving, making copies, or taking printouts of pages on the subject wiki, as long as these copies are meant for personal use and are not to be distributed to others. * Learning facts from the subject wikis and using those facts as part of instruction or research. * Providing links to pages on the subject wikis. * Quoting short passages from the subject wiki in a criticism or review (it is preferable, though not legally enforceable, that a link to the original content being reviewed be provided). ===Examples of personal use that necessitate use of the license terms=== * Creating a mirror website by exporting pages or content in bulk: In this case, the mirror website must release the mirrored content under the same license terms and ''also'' attribute the original source with a link. Attribution formats are discussed below. * Giving handouts of pages or collections of pages to a group of people, such as students in a course: In this case, the handouts should be accompanied by an attribution that makes clear both the source and the original license terms, as well as a link to the Uniform Resource Identifier (URI) of the subject wiki. Attribution formats: * For websites: ''This content is from [{{fullurl:Main Page}} {{SITENAME}}]]'' or ''This content is from [{{fullurl:Main Page}} {{fullsitetitle}}]''. Preferably, also a link to the specific pages (if any) used. The language of the attribution may be modified somewhat to indicate whether the content has been used as such, or whether changes have been made. * For print materials: ''This content is from {{SITENAME}}. The site URL is http://{{SITENAME}}.subwiki.org'' Attribution of individuals who have contributed content to the subject wiki entries is ''not'' necessary. (However, this list of individuals can be tracked down, as discussed in the next section). ==Uses not within the ambit of the license or fair use rights== Those who have contributed all the content to a particular subject wiki entry can republish the same content anywhere else ''without restriction'' and can also release the content elsewhere under a different license, since they own the full copyright to their content. To use the content on the wiki in a manner that is outside the terms of the license or beyond fair use rights, the individuals who have contributed that particular content need to be contacted. For any particular page, this information can be accessed using the history tab of the page. The users who have made edits to the page are the ones who need to be contacted. In case of difficulty doing this, please contact vipul.wikis@gmail.com or vipul@math.uchicago.edu for more information. ==Copyright violations== All users using subject wikis are requested not to violate copyright of other authors or publishers. This includes providing links that may aid and abet copyright violation. Examples of things that may be considered copyright violation: * Putting up links to personal copies of restricted-access journal articles acquired through a personal or library subscription: This usually goes against the Terms of Service of the subscription. The exception is when the content of the articles is out of copyright. * Giving links to personal copies, or copies on filesharing systems, of books that are under copyright and where either owning or sharing a copy in that manner is against the terms of copyright. In case copyright violations are detected, please email vipul.wikis@gmail.com and vipul@math.uchicago.edu to have the matter looked into immediately. 4761071b2ccb772e5c513a6ffa65f9eabdf40d13 258 257 2008-12-29T07:40:43Z Vipul 2 /* Examples of personal use that necessitate use of the license terms */ wikitext text/x-wiki This is a common copyright notice to all subject wikis. ==General license information== All content is put up under the [http://creativecommons.org/licenses/by-sa/3.0/ Creative Commons Attribute Share-Alike 3.0 license (unported)] (shorthand: CC-by-SA). Read the [http://creativehttp://ref.subwiki.org/w/skins/common/images/button_media.pngcommons.org/licenses/by-sa/3.0/legalcode full legal code here]. ===Exceptions=== The software engine that runs the wiki is MediaWiki, and this software is licensed under the GNU General Public License, which is distinct from and currently incompatible with the Creative Commons license. Similarly, some of the extensions to the software engine are licensed by their developers using the GPL, which is incompatible with Creative Commons licenses. More information on the version of MediaWiki and extensions: [[Special:Version]]. In addition, some templates may have been copied or adapted from Wikipedia or other wiki-based websites that use licenses other than Creative Commons licenses. Wikipedia uses the GNU Free Documentation License (GFDL) that, as of now, is incompatible with CC licenses. In such cases, it is clearly indicated on the page that it is licensed differently. ===Short description of the license=== The CC-by-SA license has the following features: # Content can be used and reused for any purposes, commercial or noncommercial. # Any public use or reuse of the material requires attribution of the original source (for attribution formats, see later in the document). # In case of adaptation or creation of derivative works, the newly created works must also be licensed under the same license (CC-by-SA) unless the creator of the derivative work takes ''explicit'' permission from the creator of the original work. ===Fair use and other rights=== The license cannot and does not limit fair use rights. Fair use may include quoting or copying select passages for the purposes of criticism and review. The license also cannot limit the use of data or information contained in the content on subject wikis. This means that anybody can use or reuse the facts or knowledge they gain from the subject wiki in any way they want. ==Applicability of license to the subject wiki== ===Examples of personal use that do not necessitate use of the license terms=== * Saving, making copies, or taking printouts of pages on the subject wiki, as long as these copies are meant for personal use and are not to be distributed to others. * Learning facts from the subject wikis and using those facts as part of instruction or research. * Providing links to pages on the subject wikis. * Quoting short passages from the subject wiki in a criticism or review (it is preferable, though not legally enforceable, that a link to the original content being reviewed be provided). ===Examples of personal use that necessitate use of the license terms=== * Creating a mirror website by exporting pages or content in bulk: In this case, the mirror website must release the mirrored content under the same license terms and ''also'' attribute the original source with a link. Attribution formats are discussed below. * Giving handouts of pages or collections of pages to a group of people, such as students in a course: In this case, the handouts should be accompanied by an attribution that makes clear both the source and the original license terms, as well as a link to the Uniform Resource Identifier (URI) of the subject wiki. Attribution formats: * For websites: ''This content is from [{{fullurl:Main Page}} {{SITENAME}}]'' or ''This content is from [{{fullurl:Main Page}} {{fullsitetitle}}]''. Preferably, also a link to the specific pages (if any) used. The language of the attribution may be modified somewhat to indicate whether the content has been used as such, or whether changes have been made. * For print materials: ''This content is from {{SITENAME}}. The site URL is http://{{SITENAME}}.subwiki.org'' Attribution of individuals who have contributed content to the subject wiki entries is ''not'' necessary. (However, this list of individuals can be tracked down, as discussed in the next section). ==Uses not within the ambit of the license or fair use rights== Those who have contributed all the content to a particular subject wiki entry can republish the same content anywhere else ''without restriction'' and can also release the content elsewhere under a different license, since they own the full copyright to their content. To use the content on the wiki in a manner that is outside the terms of the license or beyond fair use rights, the individuals who have contributed that particular content need to be contacted. For any particular page, this information can be accessed using the history tab of the page. The users who have made edits to the page are the ones who need to be contacted. In case of difficulty doing this, please contact vipul.wikis@gmail.com or vipul@math.uchicago.edu for more information. ==Copyright violations== All users using subject wikis are requested not to violate copyright of other authors or publishers. This includes providing links that may aid and abet copyright violation. Examples of things that may be considered copyright violation: * Putting up links to personal copies of restricted-access journal articles acquired through a personal or library subscription: This usually goes against the Terms of Service of the subscription. The exception is when the content of the articles is out of copyright. * Giving links to personal copies, or copies on filesharing systems, of books that are under copyright and where either owning or sharing a copy in that manner is against the terms of copyright. In case copyright violations are detected, please email vipul.wikis@gmail.com and vipul@math.uchicago.edu to have the matter looked into immediately. e359227e0b0bf082eda51f28b1c10fce86669fec 10 134 255 2008-12-29T07:26:50Z Vipul 2 New page: Ref, The Subject Wikis Reference Guide wikitext text/x-wiki Ref, The Subject Wikis Reference Guide 69ef4fa64dbb83c2106dde97a038a65c76dd751a 4 135 262 2009-01-16T01:39:00Z Vipul 2 New page: ==Privacy for readers== If you are surfing this website, your actions are logged in our usage logs. These usage logs are accessible to: * The site's administrators and technical support ... wikitext text/x-wiki ==Privacy for readers== If you are surfing this website, your actions are logged in our usage logs. These usage logs are accessible to: * The site's administrators and technical support group. For a full list of administrators, contact [[User:Vipul|Vipul Naik]] by email: vipul@math.uchicago.edu or vipul.wikis@gmail.com. The technical support is [http://www.4am.co.in 4AM]. * The service that hosts the data and servers, which is currently Dreamhost (http://www.dreamhost.com). * Google Analytics, which has been integrated to collect site statistics. View Google's privacy policy here: http://www.google.com/intl/en_ALL/privacypolicy.html Your usage logs are not made available to other parties. Aggregated data from logs, such as general usage patterns, may be used by the MedaWiki software as well as by site administrators in decision making. For instance, MediaWiki keeps track of the number of times each page is viewed. ==Privacy for editors== Editing on subject wikis is generally permitted only for registered users. Registered users must, at the time of registration, provide their real name, and enter basic information about their reason for interest. ''No'' private information such as date of birth, social security or taxation number, or home address is sought. Regarding personal information: * The email IDs of registered users are visible to site administrators only. For information about site administrators, contact vipul.wikis@gmail.com or vipul@math.uchicago.edu with the particular subject wiki and the reason for request. * All editing activity by registered users is recorded on the site and is visible to all site users. However, this information is not indexed by search engines that follows robots.txt. * For edits made by registered users when logged in, the originating IP addresses for the edits can be accessed only by the site administrators. * Passwords chosen by registered users are not humanly accessible, even to site administrators. f68bc226ffbc9a8d1bbae00c1eeecfe98690d7dc 4 136 264 2009-01-16T15:37:53Z Vipul 2 New page: This is a general disclaimer common to all subject wikis. For specific hazards of using this particular subject wiki, refer [[{{SITENAME}}:Hazards]]. '''SUBJECT WIKIS MAKE NO GUARANTEE OF... wikitext text/x-wiki This is a general disclaimer common to all subject wikis. For specific hazards of using this particular subject wiki, refer [[{{SITENAME}}:Hazards]]. '''SUBJECT WIKIS MAKE NO GUARANTEE OF VALIDITY''' Content on individual subject wikis need ''not'', in general, be correct or useful. There are the following general hazards: * Specific content pages may have wrong or misleading information. * Content pages may use terminology that is not standard or generally accepted, or differs from terminology or notation in other sources. * The content is not designed specifically for a particular use, and any use that you put the content to is ''at your own risk''. Also note that: * Adding content to subject wikis and reusing content from subject wikis is subject to copyright laws. See [[{{SITENAME}}:Copyrights]] for more details. * Your activities as a user or editor are tracked and may be used by site administrators and declared third parties. See [[{{SITENAME}}:Privacy policy]] for more details. 432abbbc8bbf3dead365ef5550d114def49fcefb 106 107 265 203 2009-01-16T16:02:10Z Vipul 2 wikitext text/x-wiki In all subject wikis, articles form the fundamental ''unit of knowledge''. Each article is treated as a separate, independent unit of knowledge. Articles are strongly linked to each other but all inter-article dependencies are loose. ==Goal of an article== The goal of each article is to: * Answer certain specific questions clearly and comprehensively. * Provide a vantage point for exploration of related ideas and for answers to related questions. The two goals together mean that each article should be very clear about what questions it provides comprehensive answers to, and what questions it provides relevant links for. ==Types of articles== ===Definition articles=== {{further|[[Subwiki:Definition article]], [[Subwiki:Definition]]}} Most subject wikis have articles defining terms. These articles are called definition articles or terminology articles. A definition article must contain the definition, and, depending on the type of term and the organizational paradigms of the subject wiki, it may contain other sections giving further context to the term. Definition articles follow a brief and terse style. Their main purpose is to serve as vantage points for exploring a term and understanding related terms and facts. Definition articles do ''not'' contain proofs, explanatory arguments, exploration and motivation. Rather, they give short statements with clear explanations, linking to more complete proofs and explanation. ===Fact articles=== {{further|[[Subwiki:Fact article]], [[Subwiki:Fact]]}} Fact articles aim to state a fact clearly, and provide a proof or explanation of the fact. Where the proof is too long, involved, or unilluminating, it is outlined and separate components are linked to. Fact articles follow a brief and terse style. Their main purpose is to serve as vantage points for exploring the fact, its underlying ideas, and related terms and facts. Fact articles contain definitions of important terms used, and state facts used, but do ''not'' attempt to prove other facts being used in their proof. ===Survey articles=== {{further|[[Subwiki:Survey article]]}} Survey articles explore an idea. The idea could be based on a definition or fact or it could be based on the relation between multiple definitions and facts. Survey articles do not follow rigid guidelines like definition and fact articles, and the value they offer is considerably more variable. This is because a survey article is not usually intended to answer a ''specific question'', but rather, it is intended to address general curiosities. Unlike definition and fact articles, survey articles are not obliged to provide complete and comprehensive statements of definitions and facts used. They can provide short summaries of the definitions and facts, linking to the full article for further information. b817a89477d2c2858198470c5a92d053ce401353 106 137 266 2009-01-16T20:37:46Z Vipul 2 New page: Definition and fact articles in subject wikis often have a section called '''References'''. The References section lists ''external'' references, usually published books, articles, or othe... wikitext text/x-wiki Definition and fact articles in subject wikis often have a section called '''References'''. The References section lists ''external'' references, usually published books, articles, or other material, related to the topic of the article. This article describes the nature of references. ==Kinds of references== ===Textbook references=== {{further|[[Subwiki:Textbook references]]}} Textbook references are references to related material in textbooks. A textbook reference has the following features: * It contains a short ''clip'' containing the title and author of the book, the ISBN (10-digit and 13-digit) and possibly the publisher's name. * It provides a link to a separate page giving more details about the book. This page may, for instance, contain a review of the material in the book, as well as how the material in the book correlates with the subject wiki's content. * It indicates the page number and chapter/section/part of the book where the given material is referenced. In some cases, there may be multiple locations within the same book. For more information on book pages in subject wikis, refer [[Subwiki:Book pages]]. * It may contain a short description of the ''manner'' in which the book contains material relevant to the topic. For instance, for the definition of a term, it may indicate whether the book defines the term similarly, or in a slightly different way. For a theorem or fact, it may indicate whether the book states the fact in its main text, and whether the book supplies proof/explanation. ===General book references=== {{further|[[Subwiki:General book references]]}} General book references are references to material in general books, that are not written as or intended as textbooks or reference books. Examples are books on science subjects written for popular audiences. The format for general book references is the same as that for textbook references; however, these are generally given in a separate subsection from the textbook references subsection. ===Journal references=== {{further|[[Subwiki:Journal references]]}} These are references to articles published in academic or subject-related journals. Unlike textbook references, journal references ''usually'' do not pinpoint specific pages within the journal article related to the topic. The journal reference comprises a short ''clip'' with the name of the paper, the journal in which it was published, the names of the authors, and other academic information. There is a link to a page with more detailed information. The more detailed information may include an intra-subject classification code of the paper and links to gateway pages or general subject databases (such as Mathscinet for mathematical papers). c684c2264cd1f755fdc03ff560eb80b1bac1002d 106 138 267 2009-01-16T20:59:26Z Vipul 2 New page: Textbook references are provided in individual articles on subject wikis, listing material related to the article topic in textbooks. This page discusses how textbook references are design... wikitext text/x-wiki Textbook references are provided in individual articles on subject wikis, listing material related to the article topic in textbooks. This page discusses how textbook references are designed and can be used. ==Quick explanation on book pages== {{further|[[Subwiki:Book pages]]}} Book pages on individual subject wikis give information about books related to material covered in the subject wiki. Each book page is about a particular book, and usually deals with its latest edition. Separate editions of the same book are ''not'' given separate book pages. The book page has two parts: a short ''clip'' giving the title, authors and ISBN numbers, that is included in any textbook reference to that book, and the ''rest'' of the page, which may comprise information about the book's contents and their correlation with material in the subject wiki. ==Textbook references== ===Where textbook references can be found=== Textbook references are present only on a ''per article'' basis. There is no ''general'' list of textbook references. There is, however, a general list of ''books'' that can be accessed by looking at the list of all book pages. In any article, the '''Textbook references''', if present, exist as a subsection of a section titled '''References'''. This is usually at the bottom of the page -- above the '''External links''' and semantically generated factbox, but below all other article content. ===The format of textbook references=== Textbook references are presented as bullet points. Each bullet point gives one textbook reference. An individual textbook reference comprises the following data: * The ''book clip'' for the textbook, with a link to the book page for more information. * The page number and chapter/section/part where related material in the book is covered. These are presented in a comma-separated format. * Some information on the kind of information the book does provide (more on this below). The information of the book where the topic is referenced, as well as the page number and other reference information, is stored in a semantic database using [[Property:Referenced in]]. This can be queried as described later. ==Textbook references for definition articles== For definition articles, any textbook where the term is defined, is marked using the semantic relation [[Property:Defined in]]: the term of the page is marked as ''defined in'' that textbook, along with the page number and the other reference information. In these cases, it is customary to give the following additional information: * If the definition in the textbook uses a different convention than what is used in the subject wiki, or uses one of many possible conventions as indicated in the subject wiki, specific information about which convention is used should be provided. * If the definition in the textbook is different in language from those given in the subject wiki, or if the textbook gives only some of the many equivalent definitions given in the subject wiki, information about which of the definitions is given in the textbook should be provided. * Further useful information includes: whether the definition is given as a separate definition (with its own header or paragraph), or whether it is a ''definition in paragraph'', along with other content. * If the textbook defines the term along with other terms, this should be stated, with links to the subject wiki entries on these other terms. ==Textbook references for fact articles== For fact articles, any textbook where the fact is stated, is marked using the semantic relation [[Property:Stated in]], and any fact proved in the textbook is marked using the semantic relation [[Property:Proved in]]. * It should be clearly stated whether the fact is simply stated or also proved. * Some indication should be provided of the context in which the fact is stated (e.g., if it is a lemma for something else, a corollary of something else, if it is stated as an exercise). * An idea of the length and approach of the proof in the book should be provided, particularly if this proof differs from the one in the subject wiki either in actual content or in presentation. 260a875f6993eb2e3d5059bd1eceb5651f04337b 268 267 2009-01-16T22:36:08Z Vipul 2 wikitext text/x-wiki Textbook references are provided in individual articles on subject wikis, listing material related to the article topic in textbooks. This page discusses how textbook references are designed and can be used. ==Quick explanation on book pages== {{further|[[Subwiki:Book pages]]}} Book pages on individual subject wikis give information about books related to material covered in the subject wiki. Each book page is about a particular book, and usually deals with its latest edition. Separate editions of the same book are ''not'' given separate book pages. The book page has two parts: a short ''clip'' giving the title, authors and ISBN numbers, that is included in any textbook reference to that book, and the ''rest'' of the page, which may comprise information about the book's contents and their correlation with material in the subject wiki. ==Textbook references== ===Where textbook references can be found=== Textbook references are present only on a ''per article'' basis. There is no ''general'' list of textbook references. There is, however, a general list of ''books'' that can be accessed by looking at the list of all book pages. In any article, the '''Textbook references''', if present, exist as a subsection of a section titled '''References'''. This is usually at the bottom of the page -- above the '''External links''' and semantically generated factbox, but below all other article content. ===The format of textbook references=== Textbook references are presented as bullet points. Each bullet point gives one textbook reference. An individual textbook reference comprises the following data: * The ''book clip'' for the textbook, with a link to the book page for more information. * The page number and chapter/section/part where related material in the book is covered. These are presented in a comma-separated format. * Some information on the kind of information the book does provide (more on this below). The information of the book where the topic is referenced, as well as the page number and other reference information, is stored in a semantic database using [[Property:Referenced in]]. This can be queried as described later. ==Textbook references for definition articles== For definition articles, any textbook where the term is defined, is marked using the semantic relation [[Property:Defined in]]: the term of the page is marked as ''defined in'' that textbook, along with the page number and the other reference information. In these cases, it is customary to give the following additional information: * If the definition in the textbook uses a different convention than what is used in the subject wiki, or uses one of many possible conventions as indicated in the subject wiki, specific information about which convention is used should be provided. * If the definition in the textbook is different in language from those given in the subject wiki, or if the textbook gives only some of the many equivalent definitions given in the subject wiki, information about which of the definitions is given in the textbook should be provided. * Further useful information includes: whether the definition is given as a separate definition (with its own header or paragraph), or whether it is a ''definition in paragraph'', along with other content. * If the textbook defines the term along with other terms, this should be stated, with links to the subject wiki entries on these other terms. ==Textbook references for fact articles== For fact articles, any textbook where the fact is stated, is marked using the semantic relation [[Property:Stated in]], and any fact proved in the textbook is marked using the semantic relation [[Property:Proved in]]. * It should be clearly stated whether the fact is simply stated or also proved. * Some indication should be provided of the context in which the fact is stated (e.g., if it is a lemma for something else, a corollary of something else, if it is stated as an exercise). * An idea of the length and approach of the proof in the book should be provided, particularly if this proof differs from the one in the subject wiki either in actual content or in presentation. ==Using textbook references== ===Forward search: from wiki to book=== Suppose you are looking up a certain topic and want references on the topic. Steps: * Open the wiki page (article) on that topic. This could be a definition or fact article. Generally, the more ''specific'' the page that you go to, the more relevant the reference is. * Go to the textbook references section. * Find the names of books with the page numbers, section information and other contextual information. If you have a copy of the book accessible, this information should suffice. For books that you do not have or want more information on, you can follow the link for more information. ===Backward search: from book to wiki=== The steps are: * Search for the book on the wiki. This can be done by looking at the list of all books -- available at [[:Category:Books]], or alternatively, by searching for the ISBN number of the book, title or authors. You can restrict your search to the "Book" namespace. * The book page has general information about the content, structure and organization of the book, along with links to relevant pages on the wiki. In addition, there are links to lists of all facts stated, facts proved, and terms defined in the book. ===Semantic search=== Semantic search allows for searching for terms based on which books they are defined in. Semantic search can be done using the [[Special:Ask]] feature. Here are examples in the individual wikis. ===Groupprops=== Go to the [[Groupprops:Special:Ask|Special:Ask page of the Groupprops wiki]]. The text of each example needs to be typed under '''Query'''. * For a complete list of basic definitions in group theory that are referenced in the book by Dummit and Foote, try: <pre>[[Category:Basic definitions in group theory]][[Referenced in::Book:DummitFoote]]</pre> * For a list of terms and facts that are referenced ''both'' in the book by Dummit and Foote and in the book by Alperin and Bell, try: <pre>[[Referenced in::Book:DummitFoote]][[Referenced in::Book:AlperinBell]]</pre> * For a list of ''facts'' that are ''stated'' in the book by Alperin and Bell, try: <pre>[[Stated in::Book:AlperinBell]]</pre> * For a list of ''facts'' that are ''proved'' in the book by Dummit and Foote, try: <pre>[[Proved in::Book:DummitFoote]]</pre> * For a list of subgroup properties that are stronger than the property of being a [[normal subgroup]], and are defined in the book by Dummit and Foote, try: <pre>[[Stronger than::Normal subgroup]][[Defined in::Book:DummitFoote]]</pre> ===Topopspaces=== Go to the [[Topospaces:Special:Ask|Special:Ask page of the Topospaces wiki]]. The text of each example needs to be typed under '''Query'''. * For a complete list of properties of topological spaces that are defined in the book on topology by Munkres, try: <pre>[[Category:Properties of topological spaces]][[Defined in::Book:Munkres]]</pre> * For a list of all terms and facts that are referenced in the book by Munkres as well as the topology book by Singer and Thorpe, try: <pre>[[Referenced in::Book:SingerThorpe]][[Referenced in::Book:Munkres]]</pre> * For a complete list of property implications of topological spaces that are stated in the book on topology by Munkres, try: <pre>[[Stated in::Book:Munkres]][[Category:Topological space property implications]]</pre> 1caec35ccffbf03e64b6e2fa18da3f5f1300b72c 106 109 269 206 2009-01-16T23:07:38Z Vipul 2 wikitext text/x-wiki [[Subwiki:Definition article|Definition articles]], or article defining new terms, have ''relation'' sections. These relation sections describe how the term being defined (henceforth, called the ''definiendum'') is related to other terms and concepts. ==Relation with similarly typed terms== The ''type'' of a term is the specific role that term plays. For instance, some terms may be of the ''property'' type: ''prime number'' is a property of natural numbers. Some terms may be of the ''species'' type, so ''mangifera indica'' is a species of plant. (Learn more about typing at [[Subwiki:Type-based organization]]). The relation with similarly typed terms describes how the definiendum is related to terms of the same ''type''. ===Stronger and weaker=== For properties over some context space, one can talk of ''stronger properties'' and ''weaker properties'', to describe the relation with other properties over the same context space. Thus, for instance, the property of natural numbers of being a perfect fourth power is ''stronger than'' the property of being a perfect square, while the property of being a perfect power is ''weaker than'' the property of being a perfect square. Something similar is true for extra structures placed. A ''stronger structure'' in this case is a structure that carries more information (or more constraints) than a weaker structure, and from which the weaker structure can be recovered. For instance, the structure of a Riemannian manifold is ''stronger than'' the structure of a topological manifold, and the structure of a group is ''stronger than'' the structure of a magma (a set with a binary operation). We typically say that the stronger property (resp., structure) ''implies'' (resp., gives rise to) the weaker property (resp., structure). Here's how this is presented in the case of properties: * In a section titled '''Relation with other properties''', there are subsections titled '''Stronger properties''' and '''Weaker properties'''. * In the section titled '''Stronger properties''', there are bullet points listing the stronger properties. When available or desired, each property is accompanied by a short description either of the property or of why it is stronger; there is usually also a link to a more complete proof, both of the implication, and of why the converse implication fails in general. There may also be a link to a survey article comparing the properties. * In the section titled '''Weaker properties''', there are bullet points listing the weaker properties. When available or desired, each property is accompanied by a short description either of the property or of why it is weaker; there is usually also a link to a more complete proof, both of the implication, and of why the converse implication fails in general. There may also be a link to a survey article comparing the properties. * There may also be a section titled '''Related properties''' or '''Incomparable properties''' that lists properties that are neither stronger nor weaker but are still closely related. A similar format is followed for stronger/weaker structure and for stronger notions and weaker notions in other senses. ===Conjunctions and disjunctions=== Sometimes new terms are obtained by taking the definiendum and '''AND'''ing it with something else. For instance, we can take the '''AND''' of the property of being a prime number and the property of being an odd number, and get the property of being an odd prime. We can take the '''OR''' of being a number that is 1 mod 3 and a number tha tis 2 mod 3, to get the property of being a number that is not a multiple of 3. For property articles, conjunctions and disjunctions are typically handled in separate subsections under the '''Relation with other properties''' section. The section with conjunctions is titled '''Conjunction with other properties''' while the section on disjunctions is titled '''Disjunction with other properties'''. ===Semantic information for stronger and weaker=== If <math>B</math> is stronger than <math>A</math>, and is mentioned in the article on <math>A</math>, the semantic property [[Property:Weaker than]] is used. This stores the information that <math>A</math> is weaker than <math>B</math>. Conversely, if <math>A</math> is stronger than <math>B</math>, the property [[Property:Stronger than]] is used. These can be used in semantic search to locate particular properties or structures that are stronger than or weaker than given ones. Note that conjunctions are ''stronger than'' the components, and disjunctions are ''weaker than'' the components, and this information is stored, ''even though'' conjunctions and disjunctions are usually listed in separate subsections. ==Relation with terms of related but different types== Apart from being related with terms of the same type, the definiendum may also be related with terms of different types. These relations may be somewhat more complicated. ===A broad banner of related properties=== Suppose a subgroup property has some related group properties. Then, these group properties are listed in a subsection titled '''Related group properties''' under the section '''Relation with other properties'''. ==Examples== ===Groupprops=== Here are some examples of pages on properties with a '''Relation with other properties''' section: * [[Groupprops:Normal subgroup]]: Here is its [[Groupprops:Normal subgroup#Relation with other properties|relation with other properties section]]. This has many subsections, including stronger properties, weaker properties, conjunction with other properties. For some of them, there are links to proofs of the implications and of the reverse implications being false, and for some, there are links to survey articles explaining the differences between the properties. * [[Groupprops:Pronormal subgroup]]: Here is its [[Groupprops:Pronormal subgroup#Relation with other properties|relation with other properties section]]. This has a list of stronger properties, a list of weaker properties, and a separate subsection titled '''Conjunction with other properties'''. * [[Groupprops:Complete group]]: Here is its [[Groupprops:Complete group#Relation with other properties|relation with other properties section]]. This has a list of stronger properties and a list of weaker properties. For some of these, there are links to proofs of the implications. * [[Groupprops:Monoid]]: Here is its [[Groupprops:Monoid#Relation with other structures|relation with other structures section]]. This has a list of stronger structures and a list of weaker structures. * [[Groupprops:Conjugate subgroups]]: Here is its [[Groupprops:Conjugate subgroups#Relation with other equivalence relations|relation with other equivalence relations section]]. This has a list of stronger equivalence relations and weaker equivalence relations. * [[Groupprops:Transitive subgroup property]]: Here is its [[Groupprops:Transitive subgroup property#Relation with other metaproperties|relation with other metaproperties section]]. This has a list of stronger metaproperties and a list of weaker metaproperties. These relations between properties, structures and equivalence relations of various sorts can be used to locate properties. Here are some examples of semantic queries (that can be executed by typing the query term in [[Groupprops:Special:Ask|the Special:Ask page]]: * For a list of subgroup properties that are ''stronger than'' subnormality and ''weaker than'' normality: <pre>[[Stronger than::Subnormal subgroup]][[Weaker than::Normal subgroup]]</pre> b007dfa122500d061db57f6a1f0c2dbe89723aa5 270 269 2009-01-17T01:22:23Z Vipul 2 wikitext text/x-wiki [[Subwiki:Definition article|Definition articles]], or article defining new terms, have ''relation'' sections. These relation sections describe how the term being defined (henceforth, called the ''definiendum'') is related to other terms and concepts. ==Relation with similarly typed terms== The ''type'' of a term is the specific role that term plays. For instance, some terms may be of the ''property'' type: ''prime number'' is a property of natural numbers. Some terms may be of the ''species'' type, so ''mangifera indica'' is a species of plant. (Learn more about typing at [[Subwiki:Type-based organization]]). The relation with similarly typed terms describes how the definiendum is related to terms of the same ''type''. ===Stronger and weaker=== For properties over some context space, one can talk of ''stronger properties'' and ''weaker properties'', to describe the relation with other properties over the same context space. Thus, for instance, the property of natural numbers of being a perfect fourth power is ''stronger than'' the property of being a perfect square, while the property of being a perfect power is ''weaker than'' the property of being a perfect square. Something similar is true for extra structures placed. A ''stronger structure'' in this case is a structure that carries more information (or more constraints) than a weaker structure, and from which the weaker structure can be recovered. For instance, the structure of a Riemannian manifold is ''stronger than'' the structure of a topological manifold, and the structure of a group is ''stronger than'' the structure of a magma (a set with a binary operation). We typically say that the stronger property (resp., structure) ''implies'' (resp., gives rise to) the weaker property (resp., structure). Here's how this is presented in the case of properties: * In a section titled '''Relation with other properties''', there are subsections titled '''Stronger properties''' and '''Weaker properties'''. * In the section titled '''Stronger properties''', there are bullet points listing the stronger properties. When available or desired, each property is accompanied by a short description either of the property or of why it is stronger; there is usually also a link to a more complete proof, both of the implication, and of why the converse implication fails in general. There may also be a link to a survey article comparing the properties. * In the section titled '''Weaker properties''', there are bullet points listing the weaker properties. When available or desired, each property is accompanied by a short description either of the property or of why it is weaker; there is usually also a link to a more complete proof, both of the implication, and of why the converse implication fails in general. There may also be a link to a survey article comparing the properties. * There may also be a section titled '''Related properties''' or '''Incomparable properties''' that lists properties that are neither stronger nor weaker but are still closely related. A similar format is followed for stronger/weaker structure and for stronger notions and weaker notions in other senses. ===Conjunctions and disjunctions=== Sometimes new terms are obtained by taking the definiendum and '''AND'''ing it with something else. For instance, we can take the '''AND''' of the property of being a prime number and the property of being an odd number, and get the property of being an odd prime. We can take the '''OR''' of being a number that is 1 mod 3 and a number tha tis 2 mod 3, to get the property of being a number that is not a multiple of 3. For property articles, conjunctions and disjunctions are typically handled in separate subsections under the '''Relation with other properties''' section. The section with conjunctions is titled '''Conjunction with other properties''' while the section on disjunctions is titled '''Disjunction with other properties'''. ===Semantic information for stronger and weaker=== If <math>B</math> is stronger than <math>A</math>, and is mentioned in the article on <math>A</math>, the semantic property [[Property:Weaker than]] is used. This stores the information that <math>A</math> is weaker than <math>B</math>. Conversely, if <math>A</math> is stronger than <math>B</math>, the property [[Property:Stronger than]] is used. These can be used in semantic search to locate particular properties or structures that are stronger than or weaker than given ones. Note that conjunctions are ''stronger than'' the components, and disjunctions are ''weaker than'' the components, and this information is stored, ''even though'' conjunctions and disjunctions are usually listed in separate subsections. ==Relation with terms of related but different types== Apart from being related with terms of the same type, the definiendum may also be related with terms of different types. These relations may be somewhat more complicated. ===A broad banner of related properties=== Suppose a subgroup property has some related group properties. Then, these group properties are listed in a subsection titled '''Related group properties''' under the section '''Relation with other properties'''. ==Examples== ===Groupprops=== Here are some examples of pages with relations sections: * [[Groupprops:Normal subgroup]]: Here is its [[Groupprops:Normal subgroup#Relation with other properties|relation with other properties section]]. This has many subsections, including stronger properties, weaker properties, conjunction with other properties. For some of them, there are links to proofs of the implications and of the reverse implications being false, and for some, there are links to survey articles explaining the differences between the properties. * [[Groupprops:Pronormal subgroup]]: Here is its [[Groupprops:Pronormal subgroup#Relation with other properties|relation with other properties section]]. This has a list of stronger properties, a list of weaker properties, and a separate subsection titled '''Conjunction with other properties'''. * [[Groupprops:Complete group]]: Here is its [[Groupprops:Complete group#Relation with other properties|relation with other properties section]]. This has a list of stronger properties and a list of weaker properties. For some of these, there are links to proofs of the implications. * [[Groupprops:Monoid]]: Here is its [[Groupprops:Monoid#Relation with other structures|relation with other structures section]]. This has a list of stronger structures and a list of weaker structures. * [[Groupprops:Conjugate subgroups]]: Here is its [[Groupprops:Conjugate subgroups#Relation with other equivalence relations|relation with other equivalence relations section]]. This has a list of stronger equivalence relations and weaker equivalence relations. * [[Groupprops:Transitive subgroup property]]: Here is its [[Groupprops:Transitive subgroup property#Relation with other metaproperties|relation with other metaproperties section]]. This has a list of stronger metaproperties and a list of weaker metaproperties. These relations between properties, structures and equivalence relations of various sorts can be used to locate properties. Here are some examples of semantic queries (that can be executed by typing the query term in [[Groupprops:Special:Ask|the Special:Ask page]]: * For a list of subgroup properties that are ''stronger than'' subnormality and ''weaker than'' normality: <pre>[[Stronger than::Subnormal subgroup]][[Weaker than::Normal subgroup]]</pre> * For a list of group properties that are ''stronger than'' solvable group and ''weaker than'' Abelian group: <pre>[[Stronger than::Solvable group]][[Weaker than::Abelian group]]</pre> * We can restrict attention to ''pivotal'' subgroup properties that are stronger than normality: <pre>[[Stronger than::Normal subgroup]][[Category:Pivotal subgroup properties]]</pre> * To look at all the subgroup properties that are formed as '''AND''' of the property of normality with something else: <pre>[[Conjunction involving::Normal subgroup]][[Category:Subgroup properties]]</pre> ===Topospaces=== Here are some examples of pages with relations sections: * [[Topospaces:Normal space]]: Here is its [[Topospaces:Normal space#Relation with other properties|relation with other properties section]]. This has two subsections: stronger properties and weaker properties. For some of these, links to proof of the implication are given. These relations between properties, structures and equivalence relations of various sorts can be used to locate properties. Here are some examples of semantic queries (that can be executed by typing the query term in [[Groupprops:Special:Ask|the Special:Ask page]]: * For a list of properties that are stronger than Hausdorffness but weaker than normality: <pre>[[Weaker than::Normal space]][[Stronger than::Hausdorff space]]</pre> faa635bbafee604b9eec901c8366f63b4b26b437 271 270 2009-01-17T17:06:53Z Vipul 2 /* Examples */ wikitext text/x-wiki [[Subwiki:Definition article|Definition articles]], or article defining new terms, have ''relation'' sections. These relation sections describe how the term being defined (henceforth, called the ''definiendum'') is related to other terms and concepts. ==Relation with similarly typed terms== The ''type'' of a term is the specific role that term plays. For instance, some terms may be of the ''property'' type: ''prime number'' is a property of natural numbers. Some terms may be of the ''species'' type, so ''mangifera indica'' is a species of plant. (Learn more about typing at [[Subwiki:Type-based organization]]). The relation with similarly typed terms describes how the definiendum is related to terms of the same ''type''. ===Stronger and weaker=== For properties over some context space, one can talk of ''stronger properties'' and ''weaker properties'', to describe the relation with other properties over the same context space. Thus, for instance, the property of natural numbers of being a perfect fourth power is ''stronger than'' the property of being a perfect square, while the property of being a perfect power is ''weaker than'' the property of being a perfect square. Something similar is true for extra structures placed. A ''stronger structure'' in this case is a structure that carries more information (or more constraints) than a weaker structure, and from which the weaker structure can be recovered. For instance, the structure of a Riemannian manifold is ''stronger than'' the structure of a topological manifold, and the structure of a group is ''stronger than'' the structure of a magma (a set with a binary operation). We typically say that the stronger property (resp., structure) ''implies'' (resp., gives rise to) the weaker property (resp., structure). Here's how this is presented in the case of properties: * In a section titled '''Relation with other properties''', there are subsections titled '''Stronger properties''' and '''Weaker properties'''. * In the section titled '''Stronger properties''', there are bullet points listing the stronger properties. When available or desired, each property is accompanied by a short description either of the property or of why it is stronger; there is usually also a link to a more complete proof, both of the implication, and of why the converse implication fails in general. There may also be a link to a survey article comparing the properties. * In the section titled '''Weaker properties''', there are bullet points listing the weaker properties. When available or desired, each property is accompanied by a short description either of the property or of why it is weaker; there is usually also a link to a more complete proof, both of the implication, and of why the converse implication fails in general. There may also be a link to a survey article comparing the properties. * There may also be a section titled '''Related properties''' or '''Incomparable properties''' that lists properties that are neither stronger nor weaker but are still closely related. A similar format is followed for stronger/weaker structure and for stronger notions and weaker notions in other senses. ===Conjunctions and disjunctions=== Sometimes new terms are obtained by taking the definiendum and '''AND'''ing it with something else. For instance, we can take the '''AND''' of the property of being a prime number and the property of being an odd number, and get the property of being an odd prime. We can take the '''OR''' of being a number that is 1 mod 3 and a number tha tis 2 mod 3, to get the property of being a number that is not a multiple of 3. For property articles, conjunctions and disjunctions are typically handled in separate subsections under the '''Relation with other properties''' section. The section with conjunctions is titled '''Conjunction with other properties''' while the section on disjunctions is titled '''Disjunction with other properties'''. ===Semantic information for stronger and weaker=== If <math>B</math> is stronger than <math>A</math>, and is mentioned in the article on <math>A</math>, the semantic property [[Property:Weaker than]] is used. This stores the information that <math>A</math> is weaker than <math>B</math>. Conversely, if <math>A</math> is stronger than <math>B</math>, the property [[Property:Stronger than]] is used. These can be used in semantic search to locate particular properties or structures that are stronger than or weaker than given ones. Note that conjunctions are ''stronger than'' the components, and disjunctions are ''weaker than'' the components, and this information is stored, ''even though'' conjunctions and disjunctions are usually listed in separate subsections. ==Relation with terms of related but different types== Apart from being related with terms of the same type, the definiendum may also be related with terms of different types. These relations may be somewhat more complicated. ===A broad banner of related properties=== Suppose a subgroup property has some related group properties. Then, these group properties are listed in a subsection titled '''Related group properties''' under the section '''Relation with other properties'''. ==Examples== ===Groupprops=== Here are some examples of pages with relations sections: * [[Groupprops:Normal subgroup]]: Here is its [[Groupprops:Normal subgroup#Relation with other properties|relation with other properties section]]. This has many subsections, including stronger properties, weaker properties, conjunction with other properties. For some of them, there are links to proofs of the implications and of the reverse implications being false, and for some, there are links to survey articles explaining the differences between the properties. * [[Groupprops:Pronormal subgroup]]: Here is its [[Groupprops:Pronormal subgroup#Relation with other properties|relation with other properties section]]. This has a list of stronger properties, a list of weaker properties, and a separate subsection titled '''Conjunction with other properties'''. * [[Groupprops:Complete group]]: Here is its [[Groupprops:Complete group#Relation with other properties|relation with other properties section]]. This has a list of stronger properties and a list of weaker properties. For some of these, there are links to proofs of the implications. * [[Groupprops:Monoid]]: Here is its [[Groupprops:Monoid#Relation with other structures|relation with other structures section]]. This has a list of stronger structures and a list of weaker structures. * [[Groupprops:Conjugate subgroups]]: Here is its [[Groupprops:Conjugate subgroups#Relation with other equivalence relations|relation with other equivalence relations section]]. This has a list of stronger equivalence relations and weaker equivalence relations. * [[Groupprops:Transitive subgroup property]]: Here is its [[Groupprops:Transitive subgroup property#Relation with other metaproperties|relation with other metaproperties section]]. This has a list of stronger metaproperties and a list of weaker metaproperties. These relations between properties, structures and equivalence relations of various sorts can be used to locate properties. Here are some examples of semantic queries (that can be executed by typing the query term in [[Groupprops:Special:Ask|the Special:Ask page]]: * For a list of subgroup properties that are ''stronger than'' subnormality and ''weaker than'' normality: <pre>[[Stronger than::Subnormal subgroup]][[Weaker than::Normal subgroup]]</pre> * For a list of group properties that are ''stronger than'' solvable group and ''weaker than'' Abelian group: <pre>[[Stronger than::Solvable group]][[Weaker than::Abelian group]]</pre> * We can restrict attention to ''pivotal'' subgroup properties that are stronger than normality: <pre>[[Stronger than::Normal subgroup]][[Category:Pivotal subgroup properties]]</pre> * For a list of the subgroup properties that are formed as '''AND''' of the property of normality with something else: <pre>[[Conjunction involving::Normal subgroup]][[Category:Subgroup properties]]</pre> * For a list of the ''transitive'' subgroup properties that are stronger than the property of normality: <pre>[[Stronger than::Normal subgroup]][[Category:Transitive subgroup properties]]</pre> * For a list of the subgroup properties that are variations of normality, and are stronger than the property of being a subnormal subgroup: <pre>[[Stronger than::Subnormal subgroup]][[Category:Variations of normality]]</pre> ===Topospaces=== Here are some examples of pages with relations sections: * [[Topospaces:Normal space]]: Here is its [[Topospaces:Normal space#Relation with other properties|relation with other properties section]]. This has two subsections: stronger properties and weaker properties. For some of these, links to proof of the implication are given. * [[Topospaces:Closed subset]]: Here is its [[Topospaces:Closed subset#Relation with other properties}relation with other properties section]]. This has two subsections: stronger properties and weaker properties. These relations between properties, structures and equivalence relations of various sorts can be used to locate properties. Here are some examples of semantic queries (that can be executed by typing the query term in [[Groupprops:Special:Ask|the Special:Ask page]]: * For a list of properties that are stronger than Hausdorffness but weaker than normality: <pre>[[Weaker than::Normal space]][[Stronger than::Hausdorff space]]</pre> * For a list of properties that are stronger than Hausdorffness and are ''variations'' of Hausdorffness: <pre>[[Stronger than::Hausdorff space]][[Category:Variations of Hausdorffness]]</pre> * For a list of retract-hereditary properties of topological spaces that are weaker than contractibility: <pre>[[Weaker than::Contractible space]][[Category:Retract-hereditary properties of topological spaces]]</pre> ===Commalg=== d4724c9c54c0dc1f8e188e266704fc179b4ef9b5 272 271 2009-01-17T20:29:54Z Vipul 2 /* Examples */ wikitext text/x-wiki [[Subwiki:Definition article|Definition articles]], or article defining new terms, have ''relation'' sections. These relation sections describe how the term being defined (henceforth, called the ''definiendum'') is related to other terms and concepts. ==Relation with similarly typed terms== The ''type'' of a term is the specific role that term plays. For instance, some terms may be of the ''property'' type: ''prime number'' is a property of natural numbers. Some terms may be of the ''species'' type, so ''mangifera indica'' is a species of plant. (Learn more about typing at [[Subwiki:Type-based organization]]). The relation with similarly typed terms describes how the definiendum is related to terms of the same ''type''. ===Stronger and weaker=== For properties over some context space, one can talk of ''stronger properties'' and ''weaker properties'', to describe the relation with other properties over the same context space. Thus, for instance, the property of natural numbers of being a perfect fourth power is ''stronger than'' the property of being a perfect square, while the property of being a perfect power is ''weaker than'' the property of being a perfect square. Something similar is true for extra structures placed. A ''stronger structure'' in this case is a structure that carries more information (or more constraints) than a weaker structure, and from which the weaker structure can be recovered. For instance, the structure of a Riemannian manifold is ''stronger than'' the structure of a topological manifold, and the structure of a group is ''stronger than'' the structure of a magma (a set with a binary operation). We typically say that the stronger property (resp., structure) ''implies'' (resp., gives rise to) the weaker property (resp., structure). Here's how this is presented in the case of properties: * In a section titled '''Relation with other properties''', there are subsections titled '''Stronger properties''' and '''Weaker properties'''. * In the section titled '''Stronger properties''', there are bullet points listing the stronger properties. When available or desired, each property is accompanied by a short description either of the property or of why it is stronger; there is usually also a link to a more complete proof, both of the implication, and of why the converse implication fails in general. There may also be a link to a survey article comparing the properties. * In the section titled '''Weaker properties''', there are bullet points listing the weaker properties. When available or desired, each property is accompanied by a short description either of the property or of why it is weaker; there is usually also a link to a more complete proof, both of the implication, and of why the converse implication fails in general. There may also be a link to a survey article comparing the properties. * There may also be a section titled '''Related properties''' or '''Incomparable properties''' that lists properties that are neither stronger nor weaker but are still closely related. A similar format is followed for stronger/weaker structure and for stronger notions and weaker notions in other senses. ===Conjunctions and disjunctions=== Sometimes new terms are obtained by taking the definiendum and '''AND'''ing it with something else. For instance, we can take the '''AND''' of the property of being a prime number and the property of being an odd number, and get the property of being an odd prime. We can take the '''OR''' of being a number that is 1 mod 3 and a number tha tis 2 mod 3, to get the property of being a number that is not a multiple of 3. For property articles, conjunctions and disjunctions are typically handled in separate subsections under the '''Relation with other properties''' section. The section with conjunctions is titled '''Conjunction with other properties''' while the section on disjunctions is titled '''Disjunction with other properties'''. ===Semantic information for stronger and weaker=== If <math>B</math> is stronger than <math>A</math>, and is mentioned in the article on <math>A</math>, the semantic property [[Property:Weaker than]] is used. This stores the information that <math>A</math> is weaker than <math>B</math>. Conversely, if <math>A</math> is stronger than <math>B</math>, the property [[Property:Stronger than]] is used. These can be used in semantic search to locate particular properties or structures that are stronger than or weaker than given ones. Note that conjunctions are ''stronger than'' the components, and disjunctions are ''weaker than'' the components, and this information is stored, ''even though'' conjunctions and disjunctions are usually listed in separate subsections. ==Relation with terms of related but different types== Apart from being related with terms of the same type, the definiendum may also be related with terms of different types. These relations may be somewhat more complicated. ===A broad banner of related properties=== Suppose a subgroup property has some related group properties. Then, these group properties are listed in a subsection titled '''Related group properties''' under the section '''Relation with other properties'''. ==Examples== ===Groupprops=== Here are some examples of pages with relations sections: * [[Groupprops:Normal subgroup]]: Here is its [[Groupprops:Normal subgroup#Relation with other properties|relation with other properties section]]. This has many subsections, including stronger properties, weaker properties, conjunction with other properties. For some of them, there are links to proofs of the implications and of the reverse implications being false, and for some, there are links to survey articles explaining the differences between the properties. * [[Groupprops:Pronormal subgroup]]: Here is its [[Groupprops:Pronormal subgroup#Relation with other properties|relation with other properties section]]. This has a list of stronger properties, a list of weaker properties, and a separate subsection titled '''Conjunction with other properties'''. * [[Groupprops:Complete group]]: Here is its [[Groupprops:Complete group#Relation with other properties|relation with other properties section]]. This has a list of stronger properties and a list of weaker properties. For some of these, there are links to proofs of the implications. * [[Groupprops:Monoid]]: Here is its [[Groupprops:Monoid#Relation with other structures|relation with other structures section]]. This has a list of stronger structures and a list of weaker structures. * [[Groupprops:Conjugate subgroups]]: Here is its [[Groupprops:Conjugate subgroups#Relation with other equivalence relations|relation with other equivalence relations section]]. This has a list of stronger equivalence relations and weaker equivalence relations. * [[Groupprops:Transitive subgroup property]]: Here is its [[Groupprops:Transitive subgroup property#Relation with other metaproperties|relation with other metaproperties section]]. This has a list of stronger metaproperties and a list of weaker metaproperties. These relations between properties, structures and equivalence relations of various sorts can be used to locate properties. Here are some examples of semantic queries (that can be executed by typing the query term in [[Groupprops:Special:Ask|the Special:Ask page]]: * For a list of subgroup properties that are ''stronger than'' subnormality and ''weaker than'' normality: <pre>[[Stronger than::Subnormal subgroup]][[Weaker than::Normal subgroup]]</pre> * For a list of group properties that are ''stronger than'' solvable group and ''weaker than'' Abelian group: <pre>[[Stronger than::Solvable group]][[Weaker than::Abelian group]]</pre> * We can restrict attention to ''pivotal'' subgroup properties that are stronger than normality: <pre>[[Stronger than::Normal subgroup]][[Category:Pivotal subgroup properties]]</pre> * For a list of the subgroup properties that are formed as '''AND''' of the property of normality with something else: <pre>[[Conjunction involving::Normal subgroup]][[Category:Subgroup properties]]</pre> * For a list of the ''transitive'' subgroup properties that are stronger than the property of normality: <pre>[[Stronger than::Normal subgroup]][[Category:Transitive subgroup properties]]</pre> * For a list of the subgroup properties that are variations of normality, and are stronger than the property of being a subnormal subgroup: <pre>[[Stronger than::Subnormal subgroup]][[Category:Variations of normality]]</pre> ===Topospaces=== Here are some examples of pages with relations sections: * [[Topospaces:Normal space]]: Here is its [[Topospaces:Normal space#Relation with other properties|relation with other properties section]]. This has two subsections: stronger properties and weaker properties. For some of these, links to proof of the implication are given. * [[Topospaces:Closed subset]]: Here is its [[Topospaces:Closed subset#Relation with other properties}relation with other properties section]]. This has two subsections: stronger properties and weaker properties. These relations between properties, structures and equivalence relations of various sorts can be used to locate properties. Here are some examples of semantic queries (that can be executed by typing the query term in [[Groupprops:Special:Ask|the Special:Ask page]]: * For a list of properties that are stronger than Hausdorffness but weaker than normality: <pre>[[Weaker than::Normal space]][[Stronger than::Hausdorff space]]</pre> * For a list of properties that are stronger than Hausdorffness and are ''variations'' of Hausdorffness: <pre>[[Stronger than::Hausdorff space]][[Category:Variations of Hausdorffness]]</pre> * For a list of retract-hereditary properties of topological spaces that are weaker than contractibility: <pre>[[Weaker than::Contractible space]][[Category:Retract-hereditary properties of topological spaces]]</pre> ===Commalg=== Here are some examples of pages with a relations section: * [[Commalg:Principal ideal domain]]: This has a [[Commalg:Principal ideal domain#Relation with other properties|relation with other properties section]]. The section includes subsections titled '''stronger properties''', '''weaker properties''', as well as a subsection giving pairs of properties whose '''AND''' gives the property of being a principal ideal domain. * [[Commalg:Noetherian ring]]: This has a [[Commalg:Noetherian ring#Relation with other properties|relation with other properties section]]. This section includes subsections titled '''stronger properties''', '''weaker properties'''', as well as '''conjunction with other properties'''. * [[Commalg:Intersection of maximal ideals]]: This has a [[Commalg:Intersection of maximal ideals#Relation with other properties|relation with other properties section]], listing stronger properties as well as weaker properties. These relations between properties, structures and equivalence relations of various sorts can be used to locate properties. Here are some examples of semantic queries (that can be executed by typing the query term in [[Commalg:Special:Ask|the Special:Ask page]]: * For a list of properties of commutative unital rings that are stronger than being a Noetherian domain but weaker than being a Euclidean domain, try: <pre>[[Weaker than::Euclidean domain]][[Stronger than::Noetherian domain]]</pre> a3b7370be2456ad3c0ff582a688bee7973600b16 106 94 273 182 2009-01-17T20:40:38Z Vipul 2 wikitext text/x-wiki ''Formalisms'' are formal expression systems, usually involving symbols, that may be used to define a term, state a fact, or describe a relationship between existing ideas. Formalisms range from mathematically and technically rigorous specifications to loose analogical formalisms. Each subject wiki has its own collection of formalisms. This article discusses, with examples and illustrations, the general principles behind the use of formalisms. ==How formalisms arise== ===Formalisms as a common pattern=== The idea behind formalisms for defining terms is that, often, a lot of closely related definitions have a similar structure, with some inputs to the structure changed. For instance, many definitions of group properties have the form: ''a group where every subgroup has property <math>p</math>''. Many definitions of topological space properties, closely related to compactness, have the form: ''a topological space where every open cover satisfying property <math>p</math> has a refinement having property <math>q</math>''. A formalism for definitions of this structure helps ''unite'' all these definitions and provides insight into how to manipulate the common features across all the terms. The advantages are: * It allows for grouping together of terms expressible using the same formalism * It helps in reverse search * It helps provide a better understanding of the relation between different terms * It sometimes suggests general tools for reasoning that can be applied whenever something is expressible in the given formalism, allowing for a better understanding of individual proofs. ===Formalisms as a language at the right level of generality=== While some formalisms are simply a way of recognizing and acknowledging existing patterns, other formalisms involve a more proactive approach in ''creating'' formalisms. These may involve certain special language tools within the subject that specialize in expressing a diverse range of ideas, while at the same time cannot be used to express ''everything''. For instance, the languages of model theory, category theory, and universal algebra provide some general frameworks in which a large number (though far from all) of definitions, facts and relationships can be stated. ==Presenting formalisms== ===Formalisms in definition articles=== In articles about terminology (called ''terminology articles'' or ''definition articles''), there may be a separate section title '''Formalisms'''. This section is usually after the '''History''', '''Definition''', and '''Examples''' sections. One subsection is devoted to each formalism. The heading of the subsection usually states the name of the formalism, or the ''operator'' used to obtain that formalism. Just below the heading is a box stating the general name of the formalism or operator, along with a link to more information about the formalism and to other terms expressible using the formalism. This is followed by a precise explanation of how the given term is expressed using the formalism. ===Formalisms in proof articles=== Proofs involving terms expressible using a certain formalism can occasionally be simplified using general techniques for manipulating expressions in that formalism. Usually, a standard proof is given in any case, and an ''alternative'' proof in terms of the formalism is provided in a separate section or subsection. This proof clearly translates the statement in terms of the formalism and explains the manipulation rules being used. ==Examples== ===Groupprops=== Here are some examples of definition articles on Groupprops that have formalisms sections: * [[Groupprops:Normal subgroup#Formalisms]]: This gives several different formal expressions for the property of normality. It begins with a ''first-order description'' of normality, then proceeds to give function restriction expressions for normality, then proceeds to describe normality in terms of relations with other subgroups, and finally describes how normal subgroups can be defined in the language of universal algebra. * [[Groupprops:Directly indecomposable group#Formalisms]]: This gives only one formalism for the notion of directly indecomposable group: viewing it as a consequence of applying the simple group operator to the subgroup property of being a direct factor. * [[Groupprops:Dedekind group#Formalisms]]: This gives two formalisms for the property of being a Dedekind group. ===Topospaces=== Here are some examples of definition articles on Topospaces that have formalisms sections: * [[Topospaces:Compact space#Formalisms]]: This gives the '''refinement formal expression''' for the property of compactness. * [[Topospaces:Orthocompact space#Formalisms]] bcc59139d7ea49fc0dfa27c18f26e105a0ca486a 274 273 2009-01-17T20:41:21Z Vipul 2 wikitext text/x-wiki ''Formalisms'' are formal expression systems, usually involving symbols, that may be used to define a term, state a fact, or describe a relationship between existing ideas. Formalisms range from mathematically and technically rigorous specifications to loose analogical formalisms. Each subject wiki has its own collection of formalisms. This article discusses, with examples and illustrations, the general principles behind the use of formalisms. ==How formalisms arise== ===Formalisms as a common pattern=== The idea behind formalisms for defining terms is that, often, a lot of closely related definitions have a similar structure, with some inputs to the structure changed. For instance, many definitions of group properties have the form: ''a group where every subgroup has property <math>p</math>''. Many definitions of topological space properties, closely related to compactness, have the form: ''a topological space where every open cover satisfying property <math>p</math> has a refinement having property <math>q</math>''. A formalism for definitions of this structure helps ''unite'' all these definitions and provides insight into how to manipulate the common features across all the terms. The advantages are: * It allows for grouping together of terms expressible using the same formalism * It helps in reverse search * It helps provide a better understanding of the relation between different terms * It sometimes suggests general tools for reasoning that can be applied whenever something is expressible in the given formalism, allowing for a better understanding of individual proofs. ===Formalisms as a language at the right level of generality=== While some formalisms are simply a way of recognizing and acknowledging existing patterns, other formalisms involve a more proactive approach in ''creating'' formalisms. These may involve certain special language tools within the subject that specialize in expressing a diverse range of ideas, while at the same time cannot be used to express ''everything''. For instance, the languages of model theory, category theory, and universal algebra provide some general frameworks in which a large number (though far from all) of definitions, facts and relationships can be stated. ==Presenting formalisms== ===Formalisms in definition articles=== In articles about terminology (called ''terminology articles'' or [[Subwiki:Definition article|definition articles]]), there may be a separate section title '''Formalisms'''. This section is usually after the '''History''', '''Definition''', and '''Examples''' sections. One subsection is devoted to each formalism. The heading of the subsection usually states the name of the formalism, or the ''operator'' used to obtain that formalism. Just below the heading is a box stating the general name of the formalism or operator, along with a link to more information about the formalism and to other terms expressible using the formalism. This is followed by a precise explanation of how the given term is expressed using the formalism. ===Formalisms in proof articles=== Proofs involving terms expressible using a certain formalism can occasionally be simplified using general techniques for manipulating expressions in that formalism. Usually, a standard proof is given in any case, and an ''alternative'' proof in terms of the formalism is provided in a separate section or subsection. This proof clearly translates the statement in terms of the formalism and explains the manipulation rules being used. ==Examples== ===Groupprops=== Here are some examples of definition articles on Groupprops that have formalisms sections: * [[Groupprops:Normal subgroup#Formalisms]]: This gives several different formal expressions for the property of normality. It begins with a ''first-order description'' of normality, then proceeds to give function restriction expressions for normality, then proceeds to describe normality in terms of relations with other subgroups, and finally describes how normal subgroups can be defined in the language of universal algebra. * [[Groupprops:Directly indecomposable group#Formalisms]]: This gives only one formalism for the notion of directly indecomposable group: viewing it as a consequence of applying the simple group operator to the subgroup property of being a direct factor. * [[Groupprops:Dedekind group#Formalisms]]: This gives two formalisms for the property of being a Dedekind group. ===Topospaces=== Here are some examples of definition articles on Topospaces that have formalisms sections: * [[Topospaces:Compact space#Formalisms]]: This gives the '''refinement formal expression''' for the property of compactness. * [[Topospaces:Orthocompact space#Formalisms]] c2649b8ec5ce0f4fcb7734535108709a599f4303 0 139 275 2009-01-20T16:43:05Z Vipul 2 New page: ===In group theory=== {{:Simple group}} ===In noncommutative ring theory=== {{:Simple ring}} ===In topology=== {{:Simple space}} wikitext text/x-wiki ===In group theory=== {{:Simple group}} ===In noncommutative ring theory=== {{:Simple ring}} ===In topology=== {{:Simple space}} f31586b6105db969413fc9143977d8c929474c38 285 275 2009-01-20T17:26:18Z Vipul 2 wikitext text/x-wiki ===In group theory=== {{:Simple group}} ===In noncommutative ring theory=== {{:Simple ring}} ===In topology=== {{:Simple space}} ===In measure theory=== {{:Simple function}} 4c5431f3f830ad688b6c2daebb22fe4ccbfe6db4 0 140 276 2009-01-20T16:45:46Z Vipul 2 New page: <noinclude>[[Status::Basic definition| ]][[Topic::Group theory| ]][[Primary wiki::Groupprops| ]]</noinclude> '''Simple group''': A nontrivial group that has only two normal subgroups: the ... wikitext text/x-wiki <noinclude>[[Status::Basic definition| ]][[Topic::Group theory| ]][[Primary wiki::Groupprops| ]]</noinclude> '''Simple group''': A nontrivial group that has only two normal subgroups: the whole group and the trivial subgroup. Related terms: [[Groupprops:Almost simple group|Almost simple group]], [[Groupprops:Quasisimple group|quasisimple group]], [[Groupprops:Simple algebraic group|simple algebraic group]] Primary subject wiki entry: [[Groupprops:Simple group]] Also located at: [[Wikipedia:Simple group]], [[Planetmath:SimpleGroup]], [[Mathworld:SimpleGroup]] 2d048b2ef02ba6542c39e8269c0b14474caead36 277 276 2009-01-20T16:47:35Z Vipul 2 wikitext text/x-wiki <noinclude>[[Status::Basic definition| ]][[Topic::Group theory| ]][[Primary wiki::Groupprops| ]]</noinclude> '''Simple group''': A nontrivial group that has only two normal subgroups: the whole group and the trivial subgroup. Related terms: [[Groupprops:Almost simple group|Almost simple group]], [[Groupprops:Quasisimple group|quasisimple group]], [[Groupprops:Simple algebraic group|simple algebraic group]] Primary subject wiki entry: [[Groupprops:Simple group]] Also located at: [[Wikipedia:Simple group]], [[Mathworld:SimpleGroup]], [[sor:S/s085220|Springer Online Reference Works]] 71f87dbd925e074c656dc9c36c20890efaf639b0 278 277 2009-01-20T16:50:00Z Vipul 2 wikitext text/x-wiki <noinclude>[[Status::Basic definition| ]][[Topic::Group theory| ]][[Primary wiki::Groupprops| ]]</noinclude> '''Simple group''': A nontrivial group that has only two normal subgroups: the whole group and the trivial subgroup. Related terms: [[Groupprops:Almost simple group|Almost simple group]], [[Groupprops:Quasisimple group|quasisimple group]], [[Groupprops:Characteristically simple group|characteristically simple group]], [[Groupprops:Simple algebraic group|simple algebraic group]] Term variations: [[Groupprops:Category:Variations of simplicity]] Primary subject wiki entry: [[Groupprops:Simple group]] Also located at: [[Wikipedia:Simple group]], [[Mathworld:SimpleGroup]], [[sor:S/s085220|Springer Online Reference Works]] f2054b79aa5605137bef3c8172d0ae8b446b2e0e 0 141 279 2009-01-20T16:52:13Z Vipul 2 New page: <noinclude>[[Status::Basic definition| ]][[Topic::Noncommutative algebra| ]][[Primary wiki::Noncommutative| ]]</noinclude> '''Simple ring''': A nonzero unital ring in which the only two-si... wikitext text/x-wiki <noinclude>[[Status::Basic definition| ]][[Topic::Noncommutative algebra| ]][[Primary wiki::Noncommutative| ]]</noinclude> '''Simple ring''': A nonzero unital ring in which the only two-sided ideals are the whole ring and the zero ideal. Primary subject wiki entry: [[Noncommutative:Simple ring]] 80e3258cd1ff96727277fd5dfc0a48d42f55b188 0 142 280 2009-01-20T16:53:58Z Vipul 2 New page: <noinclude>[[Status::Basic definition| ]][[Topic::Algebraic topology| ]][[Primary wiki::Topospaces| ]]</noinclude> '''Simple space''': A path-connected space with Abelian fundamental group... wikitext text/x-wiki <noinclude>[[Status::Basic definition| ]][[Topic::Algebraic topology| ]][[Primary wiki::Topospaces| ]]</noinclude> '''Simple space''': A path-connected space with Abelian fundamental group whose induced action on all higher homotopy groups is trivial. Primary subject wiki entry: [[Topospaces:Simple space]] 61188bda92e001bc4f2ecb8cd3086cccc5d95ce1 0 143 281 2009-01-20T17:15:13Z Vipul 2 New page: Main form: ''subnormal'', [[word type::adjective]]. Related forms: ''subnormality'' Typical use: Used in common parlance to indicate something below normal, inferior, or less than usual... wikitext text/x-wiki Main form: ''subnormal'', [[word type::adjective]]. Related forms: ''subnormality'' Typical use: Used in common parlance to indicate something below normal, inferior, or less than usual. Has different specific meanings in scientific contexts. ==Mathematics== {{:subnormal (mathematics)}} da844cf373d2346c8b9e1b4fb39cd120da581ff3 0 144 282 2009-01-20T17:15:36Z Vipul 2 New page: ===In group theory=== {{:subnormal subgroup}} wikitext text/x-wiki ===In group theory=== {{:subnormal subgroup}} 47639159e51a9dc043205d69eea50bc42024b44f 0 6 283 190 2009-01-20T17:25:09Z Vipul 2 wikitext text/x-wiki Main form: ''normal'', [[word type::adjective]]. Used both descriptively (adding detail) and restrictively (restricting the subject). In sciences, typically used in a binary sense (something is either normal or is not normal), whereas in the social sciences and daily parlance, typically used to describe position on a wider scale. Related forms: ''normality'' (how normal something is), ''normalcy'' (the condition of being normal), ''normalize''/''normalise'' (make normal) Same roots: [[Root related::norm]] Typical use: * Something customary, typical, routine, expected, or standard. Similar words: [[Similar::ordinary]], [[Similar::typical]], [[Similar::standard]], [[Similar::average]], [[Similar::expected]], [[Similar::routine]] * Something good or desirable, or something one optimistically hopes for. Similar words: [[Similar::conventional]], [[Similar::standard]], [[Similar::canonical]] * Upright or perpendicular. [[Similar::orthogonal]], [[Similar::perpendicular]] Opposite words: [[Opposite::abnormal]], [[Opposite::paranormal]], [[Opposite::strange]] Derived words: [[Derived::conormal]], [[Derived::abnormal]], [[Derived::subnormal]], [[Derived::supernormal]], [[Derived::paranormal]], [[Derived::contranormal]], [[Derived::malnormal]], [[Derived::pseudonormal]], [[Derived::orthonormal]], [[Derived::seminormal]], [[Derived::quasinormal]] ==Mathematics== {{:Normal (mathematics)}} ==Computer science== {{:Normal (computer science)}} ==Physics== {{:Normal (physics)}} ==Chemistry== {{:Normal (chemistry)}} ==Economics== {{:Normal (economics)}} 05b8b40ad2e974988d35f30d6a97f15ebc8e87be 0 40 284 124 2009-01-20T17:25:17Z Vipul 2 wikitext text/x-wiki Main form: ''simple'', [[word type::adjective]]. Used both descriptively (adding detail) and restrictively (restricting the subject). In sciences, typically used in a binary sense (something is either simple or is not simple), whereas in the social sciences and daily parlance, typically used to describe position on a wider scale. Related forms: ''simplicity'' (whether or not something is simple, extent to which something is simple) Typical use: * Easy to handle, basic, lacking in complexity. Similar words: [[Similar::easy]], [[Similar::basic]] * Something that cannot be decomposed, reduced or broken up. Similar words: [[Similar::indecomposable]], [[Similar::irreducible]]. * Honest, open and straightforward, not deceitful, designing, or guileful. Similar words: [[Similar::straightforward]] Opposite words: [[Opposite::compound]], [[Opposite::complex]], [[Opposite::convoluted]], [[Opposite:decomposable]], [[Opposite::reducible]]. Derived words: [[Derived::semisimple]], [[Derived::quasisimple]], [[Derived::pseudosimple]]. ==Mathematics== {{:Simple (mathematics)}} 1c5c982fefc7074e1f9d8663351a83965a9474c3 0 145 286 2009-01-20T17:28:42Z Vipul 2 New page: <noinclude>[[Status::Basic definition| ]][[Topic::Measure theory| ]][[Primary wiki::Measure| ]]</noinclude> '''Simple function''': A real-valued or complex-valued function on a measure spa... wikitext text/x-wiki <noinclude>[[Status::Basic definition| ]][[Topic::Measure theory| ]][[Primary wiki::Measure| ]]</noinclude> '''Simple function''': A real-valued or complex-valued function on a measure space that is expressible as a finite linear combination of indicator functions of measurable subsets. Primary subject wiki entry: [[Measure:Simple function]] Also located at: [[Wikipedia:Simple function]] 7e10183fe8c9bc4fb272de30f3b2eaa7dc5b5c8a 0 146 287 2009-01-20T17:37:56Z Vipul 2 New page: In mathematics, ''local'' generally means that a property or behaviour at a point depends only on that point, or one points close to it. Often, it means ''locally determined'' -- some glob... wikitext text/x-wiki In mathematics, ''local'' generally means that a property or behaviour at a point depends only on that point, or one points close to it. Often, it means ''locally determined'' -- some global property is determined completely by the way things behave locally. In addition to ''local'' being used as an adjective, ''locally'' is also used as an adverb, typically as a modifier to existing properties to indicate that these properties are only satisfied locally. ===In ring theory=== In both commutative and noncommutative algebra: {{:Local ring}} {{:Local field}} ===In topology=== {{:Locally compact space}} {{:Locally connected space}} {{:Locally path-connected space}} ===In group theory=== {{:Local subgroup}} a556ef9289758ad313b234766b56982cf5471179 0 147 288 2009-01-20T17:39:01Z Vipul 2 New page: <noinclude>[[Status::Standard non-basic definition| ]][[Topic::Group theory| ]][[Primary wiki::Groupprops| ]]</noinclude> '''Local subgroup''': A subgroup of a group that occurs as the nor... wikitext text/x-wiki <noinclude>[[Status::Standard non-basic definition| ]][[Topic::Group theory| ]][[Primary wiki::Groupprops| ]]</noinclude> '''Local subgroup''': A subgroup of a group that occurs as the normalizer of a nontrivial solvable subgroup. '''Primary subject wiki entry''': [[Groupprops:Local subgroup]] 671fe7c2adf8e8c4b831adf819e7af274c029290 0 148 289 2009-01-20T17:39:51Z Vipul 2 Redirecting to [[Center]] wikitext text/x-wiki #redirect [[center]] 0ec456e1323644d1844ef0f3cf529c9df127d658 0 149 290 2009-01-20T17:46:49Z Vipul 2 New page: Main form: ''centre'' or ''centre'', [[word type::noun]]. Other forms: ''central'' (adjective), center''/''centre'' (verb) Typical usage: * Something right in the middle, such as the ce... wikitext text/x-wiki Main form: ''centre'' or ''centre'', [[word type::noun]]. Other forms: ''central'' (adjective), center''/''centre'' (verb) Typical usage: * Something right in the middle, such as the center of a circle. * Something pivotal, crucial, a key ingredient. * A basin of attraction, something to which things get drawn. * The average. * In the middle, a compromise position, typically use for ideologies or viewpoints. * A source or point of origin, typically for something transmitted in all directions, such as light. * A facility providing a service. Opposite words: [[Opposite::periphery]], [[Opposite::boundary]], [[Opposite::endpoint]]. Derived words: [[Derived::barycenter]], [[Derived::epicenter]], [[Derived::incenter]], [[Derived::curcumcenter]], [[Derived::excenter]]. ==Mathematics== {{:Center (mathematics)}} 8c6d9763264681f23a383ed76772f519bdbedde1 0 150 291 2009-01-20T17:48:16Z Vipul 2 New page: Typically used for an element or a subset in a set with additional structure, satisfying certain special ''core'' properties. ===In group theory=== {{:Center of a group}} ===In ring the... wikitext text/x-wiki Typically used for an element or a subset in a set with additional structure, satisfying certain special ''core'' properties. ===In group theory=== {{:Center of a group}} ===In ring theory=== {{:Center of a ring}} ===In graph theory=== {{:Center of a graph}} 0980dac1d866fb7f69a1f2f62c8c47c48843882b 0 151 292 2009-01-20T17:50:14Z Vipul 2 New page: <noinclude>[[Status::Basic definition| ]][[Topic::Group theory| ]][[Primary wiki::Groupprops| ]]</noinclude> '''Center of a group''': The set of those elements of the group that commute wi... wikitext text/x-wiki <noinclude>[[Status::Basic definition| ]][[Topic::Group theory| ]][[Primary wiki::Groupprops| ]]</noinclude> '''Center of a group''': The set of those elements of the group that commute with every element of the group. It forms a subgroup, in fact, a characteristic subgroup. '''Primary subject wiki entry''': [[Groupprops:Center]] '''Also located at''': [[Wikipedia:Center (group theory)]] 3c1b032b93a9408d9b96574d3c20459383922799 0 152 293 2009-01-20T17:53:31Z Vipul 2 New page: <noinclude>[[Status::Basic definition| ]][[Topic::Noncommutative algebra| ]][[Primary wiki::Noncommutative| ]]</noinclude> '''Center of a ring''': The set of those elements of a (usually u... wikitext text/x-wiki <noinclude>[[Status::Basic definition| ]][[Topic::Noncommutative algebra| ]][[Primary wiki::Noncommutative| ]]</noinclude> '''Center of a ring''': The set of those elements of a (usually unital) ring that commute with every element of the ring. The center of a ring is a subring; for a unital ring, it is a subring containing the multiplicative identity. '''Primary subject wiki entry''': [[Noncommutative:Center of a ring]] 99791c79285e120d31ba8d8aa831944a317e3559 0 153 294 2009-01-20T17:55:20Z Vipul 2 New page: ===In group theory=== {{:Paranormal subgroup}} ===In topology=== {{:Paranormal space}} wikitext text/x-wiki ===In group theory=== {{:Paranormal subgroup}} ===In topology=== {{:Paranormal space}} 26d07e6f647bee74e3ab04ffc89aee6cc36e4da5 296 294 2009-01-20T17:59:14Z Vipul 2 wikitext text/x-wiki ===In group theory=== {{:Paranormal subgroup}} 527bd74243c1678ca0ba0415317ea27eb8a5fa77 0 154 295 2009-01-20T17:58:21Z Vipul 2 New page: <noinclude>[[Status::Standard non-basic definition| ]][[Topic::Group theory| ]][[Primary wiki::Groupprops| ]]</noinclude> '''Paranormal subgroup''': A subgroup whose normal closure in the ... wikitext text/x-wiki <noinclude>[[Status::Standard non-basic definition| ]][[Topic::Group theory| ]][[Primary wiki::Groupprops| ]]</noinclude> '''Paranormal subgroup''': A subgroup whose normal closure in the subgroup generated with any conjugate is the whole subgroup generated. '''Primary subject wiki entry''': [[Groupprops:Paranormal subgroup]] 45149be297cf16aa7ce3fda72dc36b51899dd0c3 0 155 297 2009-01-20T18:02:06Z Vipul 2 New page: Main form: ''paranormal'', adjective. ==Mathematics== {{:paranormal (mathematics)}} wikitext text/x-wiki Main form: ''paranormal'', adjective. ==Mathematics== {{:paranormal (mathematics)}} fc1ecfb8a25af894b6bca64fe11a75af9d993c7b 0 156 298 2009-01-20T18:05:29Z Vipul 2 New page: <noinclude>[[Status::Semi-basic definition| ]][[Topic::Point-set topology| ]][[Primary wiki::Topospaces| ]]</noinclude> '''Paracompact space''': A topological space for which every open co... wikitext text/x-wiki <noinclude>[[Status::Semi-basic definition| ]][[Topic::Point-set topology| ]][[Primary wiki::Topospaces| ]]</noinclude> '''Paracompact space''': A topological space for which every open cover has a locally finite open refinement. Some definitions also require the topological space to be Hausdorff, which gives a different and stronger notion. '''Primary subject wiki entry''': [[Topospaces:Paracompact space]]. 9b092e0045b044e5e0d3ce6c736571fc9dfbe859 106 22 299 248 2009-01-29T16:58:12Z Vipul 2 wikitext text/x-wiki * [[Groupprops:Main Page|Groupprops]]: The Group Properties Wiki (currently, around 2000 pages). Main topic is group theory * [[Topospaces:Main Page|Topospaces]]: The Topology Wiki (currently, around 400 pages). Main topics are point-set topology and algebraic topology * [[Commalg:Main Page|Commalg]]: The Commutative Algebra Wiki (currently, around 300 pages). Main topic is commutative algebra, with some algebraic geometry. * [[Diffgeom:Main Page|Diffgeom]]: The Differential Geometry Wiki * [[Measure:Main Page|Measure]]: The Measure Theory Wiki (to be set up) * [[Noncommutative:Main Page|Noncommutative]]: The Noncommutative Algebra Wiki * [[Companal:Main Page|Companal]]: The Complex Analysis Wiki * [[Cattheory:Main Page|Cattheory]]: The Category Theory Wiki (just started) * [[Market:Main Page|Market]]: The Economic Theory of Markets, Choices and Prices Wiki Some upcoming subject wikis: * Number: The Number Theory Wiki * Mech: The Classical Mechanics Wiki * Linear: The Theoretical and Computational Linear Algebra Wiki * Galois: The Field Theory and Galois Theory Wiki * Graph: The Graph Theory Wiki * Complexity: The Complexity Theory Wiki Also see the [http://blog.subwiki.org subject wikis blog]. d5102bafbaa584898543422f5a9793355d33f755 0 79 300 157 2009-01-29T17:41:20Z Vipul 2 wikitext text/x-wiki <noinclude>[[Topic::Economics| ]][[Status::Basic definition| ]][[Primary wiki::Market| ]]</noinclude> '''Normal good''': A good for which demand increases with an increase in (real) income (other things remaining the same). Primary subject wiki entry: [[Market:Normal good]] Also located at [[Wikipedia:Normal good]] abd795b7bc861c2470344065429c2643b2508e38 0 80 301 159 2009-01-29T17:42:21Z Vipul 2 wikitext text/x-wiki <noinclude>[[Status::Basic definition| ]][[Topic::Economics| ]][[Primary wiki::Market| ]] '''Ordinary good''': A good for which demand increases with a drop in price (other things remaining the same). Opposite: [[Giffen good]] Primary subject wiki entry: [[Market:Ordinary good]] Also located at: [[Wikipedia:Ordinary good]] 6509eb266f0f0e7ea4fda1d73b29d3dac1a6adc7 106 131 302 250 2009-02-03T01:25:28Z Vipul 2 wikitext text/x-wiki The subwiki.org website contains a number of subject wikis in separate subdomain. Each subject wiki functions as an independent and separately managed unit. This article describes some general attributes of subject wikis. ==What is a subject wiki?== ===A quick definition=== A subject wiki is a wiki whose focus in terms of content and perspective is a specific subject. The ''content'' focus: Most content on the wiki is directly related to the specific subject, and most material related to the specific subject is covered in the wiki. The ''perspective'' focus: The organization of content, both within articles and across articles, is done keeping the needs and goals of the subject in mind. Thus, an article on the same topic may appear very different in different subject wikis, and may be organized or categorized differently. ===What is a subject?=== Subject wikis are currently at an experimental, pre-alpha stage, so the definition of subject is still very much in flux. Here are some possible general guidelines: * A subject that deserves a wiki of its own should have a reasonable coherent mass of content, or a basic body of knowledge, along with its own distinct perspectives, needs and goals. * The subject should have a strong core (giving character and cohesion to the subject wiki's content) with a large periphery linking the material to other subjects. * Under the current setup, ''subjects'' as used in subject wikis refer to ''academic'' subjects with huge bodies of knowledge, whose ''core'' is not determined by topical news and events. For instance, Salman Rushdie and Harry Potter are not suitable for subject wikis, even though there may be huge boies of knowledge on both. Karl Marx is not a subject, but communist theory may be. The distinction can get muddled at times, specifically for subjects whose significance is historical or cultural. * The core of a subject wiki should be strong enough that ''most'' of the links are internal. In other words, most of the questions that arise naturally from reading one topic article in the subject wiki should be answerable by reading other topic articles. ===Can subject wikis have overlapping themes and content?=== Overlap of content between subject wikis is a good thing, as long as each subject wiki has a different overall perspective and goal. For instance, topics in human behavior are of interest in economics, psychology, anthropology, history and many other subjects. Commonly used ideas within a broad discipline may get different kinds of treatment in its subdisciplines. For instance: * The concept of a chemical bond is of importance in chemistry. The subdiscipline of chemical bonding, the subdisciplines of chemical reactions and their kinetics and equilibrium, the subdiscipline of energy changes in chemical reactions, the subdiscipline of analytic chemistry (testing of compounds), all look at bonds from somewhat different angles. * The notion of price is of importance in economics. The subdisciplines of microeconomic theory (that focuses on market interactions between individuals transacting), macroeconomic theory, behavioural economics, evolutionary economics, and decision analysis all view this notion somewhat differently. What should differ between different subject wikis is the kind of core and perspective offered. It may also be possible to have ''broader level'' subject wikis -- subject wikis that cover a broad level topic from a general view and use generic organizational principles. For instance, there ma be a subject wiki on elementary organic chemistry, and there may be separate subject wikis on separate subdisciplines of organic chemistry. ===How are subject wikis being created? Is there a broad "body of knowledge" principle?=== As of now, there is no broad "body of knowledge" or "hierarchy of knowledge" used for the creation of subject wikis. Rather, individual subject wikis are being created experimentally. Once the experiment reaches a somewhat more advanced stage, we may introduce elements of hierarchical planning. ===Examples=== {{:Ref:subwiki list}} Groupprops is currently the most developed prototype of a subject wiki. ==How is an individual subject wiki designed?== There are several elements of choice in the design of individual subject wikis. ===Granularity: what makes an article?=== {{further|[[Subwiki:Article]]}} Each subject wiki must have a reasonable answer to the question: ''what deserves an article?'' In other words, there need to be reasonable guidelines to determine what kind of topics deserve separate articles. These guidelines are generally ''logical'' or ''subject-based'' guidelines, as opposed to guidelines based on timing, size and ease of maintenance. ===Content goals and link goals=== The subject wikis are best seen as highly networked collections of individual articles. Every article, and every part of an article, has two kinds of goals: ''content goals'' and ''link goals''. Content goals describe what content should appear in that part. Link goals describe what other material should be linked to. Subject wikis may evolve, over time, detailed guidelines about what sections an article of a certain kind should have, and what the content goals and link goals of each section should be. Sometimes, certain paradigms of organization suggest certain content goals and link goals; see the next section for more on this. ===Paradigms of organization=== Once the overall focus has been determined, the organization of content needs to be determined. There are several organizational models: * [[Subwiki:Type-based organization]]: Type-based organization -- articles featuring the same ''type'' of entity are placed in the same category. For instance, all ''countries'' may be in one category, all ''cities'' may be in another category, all ''lakes'' may be in another category. Type-based organization must usually handle issues such as the issue of subtypes (does something belonging to a subtype automatically belong to the type), the issue of recursive types (do types themselves have types?), the question of how to capture relations, and many others. * [[Subwiki:Property-theoretic organization]]: Property-theoretic organization builds on, and extends, the basic idea of type-based organization. A ''property'' over a collection is something that any particular member of the collection either satisfies or does not satisfy. Property-theoretic organization focuses on properties as the key things being defined and uses methods to organize these properties and the relations between them. * [[Subwiki:Relational organization]]: An array of methods used to capture different relations between definitions, facts and ideas. Each of these forms of organization brings its own content goals and link goals to individual articles and sections of articles. For instance, [[Subwiki:Relations in definition article]] gives a description of what the relations section in a definition article should generally say. The specifics of how this is done depends on the specific organizational paradigm. ===Editing models=== The default editing model for subject wikis is a model of editing only by confirmed registered users, who need to fill a short form to become registered users (we do this using the [[mediawikiwiki:Extension:ConfirmAccount|ConfirmAccount extension to MediaWiki]]). However, this model may be altered for individual subject wikis to models allowing editing by unregistered users to certain parts, in order to invite input and feedback from casual users. ===A window into a subject wiki's organizational model=== The organizational model of a subject wiki can become clear quickly by looking at the left-hand sidebar on the wiki, under the ''lookup'' header. This links to supercategories from which it is possible to browse downward to specific articles. The many items in the lookup header link to different organizational paradigms. Each of the paradigms may cover a certain subset of the content on the wiki. ==How do subject wikis interact?== Each subject wiki has a different database of content, a different emphasis, and a different organization paradigm. Interaction between subject wikis is more at an informal stage. ===Consistency or lack thereof=== As more people get involved with subject wikis, it is possible that different subject wikis will be managed by people with no interaction with each other. In general, there is no need for consistency of conventions across wikis. rather, different subect wikis can offer different conventions, and thus different perspectives. Different subject wikis may have articles on the ''same'' topic with different takes (see also [[{{FULLPAGENAME}}:Can subject wikis have overlapping themes and content|the earlier section on overlapping themes and content]]). ===Interwiki linking=== Different subject wikis can provide links to each other to cover ideas ''not'' covered in their own subject wiki, or for alternative, more in-depth perspective on ideas covered scantily in their own subject wiki. There is no reciprocity policy that governs or regulates such linking. ===Common pool of templates=== Templates developed on one subject wiki may be used on another. Examples are [[Subwiki:article-tagging template|article-tagging templates]], that stand on top of articles and describe the kind of article it is, with links to similar lists of articles. For instance, the [[Groupprops:Property implication (generic)|generic property implication template on Groupprops]] has been copied to other subject wikis. We may move in future to a central repository of generic use templates associated with a particular paradigm of organization, so that these templates can be imported ''en masse'' to a subject wiki following that paradigm. ==Interaction with the outside world== {{further|[[Subwiki:External linking]]}} There is no uniform policy on external linking. The main goal of a wiki is to have a dense network with strong internal linking, so external links take a backseat. In general, such links appear at the end of articles. External links that accompany references (by pointing to the URL of the reference) are strongly encouraged. Independent external links to other online resources are also encouraged, but should be clearly demarcated from the internal links. 4d11b1ff5ffa6b9d2ff966f187b7ea2c523cee3a 106 157 303 2009-02-03T01:37:54Z Vipul 2 New page: This wiki (the subject wikis reference guide) is intended to be a guide to the other subject wikis, as well as a more general guide through the sea of information. The goal of this article... wikitext text/x-wiki This wiki (the subject wikis reference guide) is intended to be a guide to the other subject wikis, as well as a more general guide through the sea of information. The goal of this article is to explain how it works. ==Entry point== ===Quick lookup=== The subject wikis reference guide serves as an entry point for ''generic'' queries, for a person whose query may not be related to a particular subject, or who is unaware of what subject wiki will answer his or her question. It does not attempt to answer very specific questions. For instance, suppose you read a sentence in a mathematical proof saying ''this shows that <math>R</math> is normal.'' You aren't sure what ''normal'' here means, though you know it has something to do with some mathematical set with additional structure. If you type "normal" into the search bar, you're taken to the reference guide page on [[normal]]. This in turn includes information on [[normal (mathematics)]], [[normal (physics)]], [[normal (chemistry)]], and [[normal (economics)]], and possibly more. (If you're sure at the beginning that your term is a mathematics term, you can select "normal (mathematics)" when the drop-down suggestions bar appears as you type in "normal"). You notice that the mathematics section is rather long, with several possible uses of the word "normal" popping up. Based on the context in which you read the word, you can probably narrow down which of these meanings is applicable. Next, if the short definition given doesn't satisfy you, you can follow the link to the subject wiki entry with more information. By clicking on the link, you leave the reference guide and enter a subject wiki. For instance, if you're interested in [[normal subgroup]], you can click the link to the [[Groupprops:Normal subgroup|Groupprops entry]], while if you're interested in [[normal space]], you can click the link to the [[Topospaces:Normal space|Topospaces entry]]. In some cases, there are also links to other related terms. In general, these links point to within the reference guide itself; however, some links may point directly to the subject wiki. Links of the latter kind are typically used if the related term does not yet have an entry on the reference guide. In addition to links to related terms, there may also be links to categories and list pages on the subject wiki that list related terms, facts, and other articles. ===Fun exploration=== The reference guide entry on [[normal]], [[perfect]], [[simple]], and many other terms, not only contains a list of the specific uses, but also contains general information on how the word is used, both in general and in particular subject. This includes a dictionary-like meaning, synonyms and antonyms, common word variations, and words with similar roots. Sometimes, a section on a particular subject begins by explaining how generic usage within that subject differs from standard usage in daily parlance. This background allows exploring meanings to be more of a fun activity. ffba98e7cbeae07d93a3906f796dcb5a554409cb 4 158 304 2009-02-03T01:40:59Z Vipul 2 New page: Do visit the [http://blog.subwiki.org subject wikis blog]! wikitext text/x-wiki Do visit the [http://blog.subwiki.org subject wikis blog]! 087424cffc246925ab72afd6d76ace53704b733d 106 22 305 299 2009-02-03T01:41:20Z Vipul 2 wikitext text/x-wiki * [[Groupprops:Main Page|Groupprops]]: The Group Properties Wiki (currently, around 2000 pages). Main topic is group theory * [[Topospaces:Main Page|Topospaces]]: The Topology Wiki (currently, around 400 pages). Main topics are point-set topology and algebraic topology * [[Commalg:Main Page|Commalg]]: The Commutative Algebra Wiki (currently, around 300 pages). Main topic is commutative algebra, with some algebraic geometry. * [[Diffgeom:Main Page|Diffgeom]]: The Differential Geometry Wiki * [[Measure:Main Page|Measure]]: The Measure Theory Wiki (to be set up) * [[Noncommutative:Main Page|Noncommutative]]: The Noncommutative Algebra Wiki * [[Companal:Main Page|Companal]]: The Complex Analysis Wiki * [[Cattheory:Main Page|Cattheory]]: The Category Theory Wiki (just started) * [[Market:Main Page|Market]]: The Economic Theory of Markets, Choices and Prices Wiki Some upcoming subject wikis: * Number: The Number Theory Wiki * Mech: The Classical Mechanics Wiki * Linear: The Theoretical and Computational Linear Algebra Wiki * Galois: The Field Theory and Galois Theory Wiki * Graph: The Graph Theory Wiki * Complexity: The Complexity Theory Wiki 1562a135ae8ccb801c287471361e748401e83656 0 1 306 54 2009-02-03T01:43:03Z Vipul 2 wikitext text/x-wiki {{top notice}} {{blog ad}} Here is our list of subject wikis: {{:Ref:Subwiki list}} 96189731d05aa93d358899d9ac7231a213d91de5 309 306 2009-02-03T01:47:04Z Vipul 2 wikitext text/x-wiki {{top notice}} Here is our list of subject wikis: {{:Ref:Subwiki list}} 95ce252de371efedc9cd74a7514d7b5112c7fc53 312 309 2009-02-03T01:54:00Z Vipul 2 wikitext text/x-wiki {{top notice}} {{quick word}} Here is our list of subject wikis: {{:Ref:Subwiki list}} 4bae0c39e7acb564186a2d4bd8896d356191d017 10 159 307 2009-02-03T01:43:36Z Vipul 2 New page: {{quotation|Do visit the [http://blog.subwiki.org subject wikis blog]!}} wikitext text/x-wiki {{quotation|Do visit the [http://blog.subwiki.org subject wikis blog]!}} 62bd25365bef975a14d05254c24641f9499ddba8 10 160 308 2009-02-03T01:46:35Z Vipul 2 New page: {{quotation|Welcome to the '''Subject Wikis Reference Guide'''. The subject wikis are run by [[User:Vipul|Vipul Naik]], and include wikis in many parts of mathematics (group theory, topolo... wikitext text/x-wiki {{quotation|Welcome to the '''Subject Wikis Reference Guide'''. The subject wikis are run by [[User:Vipul|Vipul Naik]], and include wikis in many parts of mathematics (group theory, topology, commutative algebra, and many others). Quick links: * [[Subwiki:Reference guide|A quick explanation for how the reference guide works]] -- more detailed help will be put up later. * [[Subwiki:Subject wiki|An explanation of what a subject wiki is]] * [http://blog.subwiki.org The Subject Wikis Blog]: A blog chronicling the development of subject wikis.}} dedd0cae4de1f38d066ff748518eb323658a0fd1 10 161 310 2009-02-03T01:52:51Z Vipul 2 New page: {{quotation|'''QUICK WORD''': <random>[[Normal]]: This much overused word has several meanings in [[normal (mathematics)|mathematics]], and a few in [[normal (physics)|physics]], [[normal... wikitext text/x-wiki {{quotation|'''QUICK WORD''': <random>[[Normal]]: This much overused word has several meanings in [[normal (mathematics)|mathematics]], and a few in [[normal (physics)|physics]], [[normal (economics)|economics]], [[normal (chemistry)|chemistry]] and [[normal (psychology)|psychology]]. Too many think they're normal!@@@[[Simple]]: A much overused word for the straightforward, ''simple'' is also sometimes used for that which is too hard and difficult to break down.@@@[[Perfect]]: Too good to be true! You bet, except that the perfection sought by its definers in [[perfect (mathematics)|mathematics]], [[perfect (physics)]], and [[perfect (economics)|economics]] often had different and extremely controversial ideas of what it means to be perfect.</random>}} e54b86cc41711e8e951ea05a143331b1b5c144f7 311 310 2009-02-03T01:53:23Z Vipul 2 wikitext text/x-wiki {{quotation|'''QUICK WORD''': <random>[[Normal]]: This much overused word has several meanings in [[normal (mathematics)|mathematics]], and a few in [[normal (physics)|physics]], [[normal (economics)|economics]], [[normal (chemistry)|chemistry]] and [[normal (psychology)|psychology]]. Too many think they're normal!@@@[[Simple]]: A much overused word for the straightforward, ''simple'' is also sometimes used for that which is too hard and difficult to break down.@@@[[Perfect]]: Too good to be true! You bet, except that the perfection sought by its definers in [[perfect (mathematics)|mathematics]], [[perfect (physics)|physics]], and [[perfect (economics)|economics]] often had different and extremely controversial ideas of what it means to be perfect.</random>}} a7f55a18064cbbaf6da9c1320020ddebb77c0856 0 40 313 284 2009-02-03T01:54:23Z Vipul 2 wikitext text/x-wiki Main form: ''simple'', [[word type::adjective]]. Used both descriptively (adding detail) and restrictively (restricting the subject). In sciences, typically used in a binary sense (something is either simple or is not simple), whereas in the social sciences and daily parlance, typically used to describe position on a wider scale. Related forms: ''simplicity'' (whether or not something is simple, extent to which something is simple) Typical use: * Easy to handle, basic, lacking in complexity. Similar words: [[Similar::easy]], [[Similar::basic]] * Something that cannot be decomposed, reduced or broken up. Similar words: [[Similar::indecomposable]], [[Similar::irreducible]]. * Honest, open and straightforward, not deceitful, designing, or guileful. Similar words: [[Similar::straightforward]] Opposite words: [[Opposite::compound]], [[Opposite::complex]], [[Opposite::convoluted]], [[Opposite::decomposable]], [[Opposite::reducible]]. Derived words: [[Derived::semisimple]], [[Derived::quasisimple]], [[Derived::pseudosimple]]. ==Mathematics== {{:Simple (mathematics)}} 63ab15260a32052479f7a072811b047639c6f3be 315 313 2009-02-03T02:09:37Z Vipul 2 wikitext text/x-wiki Main form: ''simple'', [[word type::adjective]]. Used both descriptively (adding detail) and restrictively (restricting the subject). In sciences, typically used in a binary sense (something is either simple or is not simple), whereas in the social sciences and daily parlance, typically used to describe position on a wider scale. Related forms: ''simplicity'' (whether or not something is simple, extent to which something is simple) Typical use: * Easy to handle, basic, lacking in complexity. Similar words: [[Similar::easy]], [[Similar::basic]] * Something that cannot be decomposed, reduced or broken up. Similar words: [[Similar::indecomposable]], [[Similar::irreducible]]. * Honest, open and straightforward, not deceitful, designing, or guileful. Similar words: [[Similar::straightforward]] Opposite words: [[Opposite::compound]], [[Opposite::complex]], [[Opposite::convoluted]], [[Opposite::decomposable]], [[Opposite::reducible]]. Derived words: [[Derived::semisimple]], [[Derived::quasisimple]], [[Derived::pseudosimple]]. {{wordbox| word = simple| oednumber = 00327034}} ==Mathematics== {{:Simple (mathematics)}} efec032149ac2967ae83f63ee1df46e83e6c1600 317 315 2009-02-03T02:10:54Z Vipul 2 wikitext text/x-wiki Main form: ''simple'', [[word type::adjective]]. Used both descriptively (adding detail) and restrictively (restricting the subject). In sciences, typically used in a binary sense (something is either simple or is not simple), whereas in the social sciences and daily parlance, typically used to describe position on a wider scale. Related forms: ''simplicity'' (whether or not something is simple, extent to which something is simple) Typical use: * Easy to handle, basic, lacking in complexity. Similar words: [[Similar::easy]], [[Similar::basic]] * Something that cannot be decomposed, reduced or broken up. Similar words: [[Similar::indecomposable]], [[Similar::irreducible]]. * Honest, open and straightforward, not deceitful, designing, or guileful. Similar words: [[Similar::straightforward]] Opposite words: [[Opposite::compound]], [[Opposite::complex]], [[Opposite::convoluted]], [[Opposite::decomposable]], [[Opposite::reducible]]. Derived words: [[Derived::semisimple]], [[Derived::quasisimple]], [[Derived::pseudosimple]]. {{wordbox| word = simple| oednumber = 50225127}} ==Mathematics== {{:Simple (mathematics)}} 935c49b30c96833ea1918ad684c0217a2d564d47 10 162 314 2009-02-03T02:08:06Z Vipul 2 New page: {{quotation|Learn more about {{{word}}}<br> Dictionary definitions: [[oed:{{{oednumber}}}|Oxford English Dictionary]], [[merriamwebster:{{{word}}}|Merriam Webster]], [[Wiktionary:{{{word}... wikitext text/x-wiki {{quotation|Learn more about {{{word}}}<br> Dictionary definitions: [[oed:{{{oednumber}}}|Oxford English Dictionary]], [[merriamwebster:{{{word}}}|Merriam Webster]], [[Wiktionary:{{{word}}}|Wiktionary]], [[thefreedictionary:{{{word}}}|The Free Dictionary]]<br> Visual Thesaurus: [[visualthesaurus:{{{word}}}|Visual Thesaurus]]}} ad0f02fbd74f433341b165d03f3b38def7b6fc12 0 6 316 283 2009-02-03T02:10:23Z Vipul 2 wikitext text/x-wiki Main form: ''normal'', [[word type::adjective]]. Used both descriptively (adding detail) and restrictively (restricting the subject). In sciences, typically used in a binary sense (something is either normal or is not normal), whereas in the social sciences and daily parlance, typically used to describe position on a wider scale. Related forms: ''normality'' (how normal something is), ''normalcy'' (the condition of being normal), ''normalize''/''normalise'' (make normal) Same roots: [[Root related::norm]] Typical use: * Something customary, typical, routine, expected, or standard. Similar words: [[Similar::ordinary]], [[Similar::typical]], [[Similar::standard]], [[Similar::average]], [[Similar::expected]], [[Similar::routine]] * Something good or desirable, or something one optimistically hopes for. Similar words: [[Similar::conventional]], [[Similar::standard]], [[Similar::canonical]] * Upright or perpendicular. [[Similar::orthogonal]], [[Similar::perpendicular]] Opposite words: [[Opposite::abnormal]], [[Opposite::paranormal]], [[Opposite::strange]] Derived words: [[Derived::conormal]], [[Derived::abnormal]], [[Derived::subnormal]], [[Derived::supernormal]], [[Derived::paranormal]], [[Derived::contranormal]], [[Derived::malnormal]], [[Derived::pseudonormal]], [[Derived::orthonormal]], [[Derived::seminormal]], [[Derived::quasinormal]] {{wordbox| word = normal| oednumber = 00327034}} ==Mathematics== {{:Normal (mathematics)}} ==Computer science== {{:Normal (computer science)}} ==Physics== {{:Normal (physics)}} ==Chemistry== {{:Normal (chemistry)}} ==Economics== {{:Normal (economics)}} 4e052c18f0a70af9b19fc63fbda243ce3a92202b 2 163 318 2009-02-25T20:08:03Z Vipul 2 Creating user page with biography of new user. wikitext text/x-wiki See [http://www.mywikibiz.com/Directory:Jon_Awbrey Web Vita] 17a98b7bf2c88d4846d716e20a1e0a3b6011c0f4 3 164 319 2009-02-25T20:08:03Z Vipul 2 Welcome! wikitext text/x-wiki '''Welcome to ''Ref''!''' We hope you will contribute much and well. You will probably want to read the [[Help:Contents|help pages]]. Again, welcome and have fun! [[User:Vipul|Vipul]] 20:08, 25 February 2009 (UTC) 353cc09e5ce6f9f13ca0ae49d5df56da256c3675 107 165 320 2009-02-25T20:20:01Z Vipul 2 New page: This discussion page is intended for a discussion on using property theory as a means of organizing material in subject wikis, as well as the underlying issues of mathematics, logic, compu... wikitext text/x-wiki This discussion page is intended for a discussion on using property theory as a means of organizing material in subject wikis, as well as the underlying issues of mathematics, logic, computation and philosophy. [[User:Vipul|Vipul]] 20:20, 25 February 2009 (UTC) e602b1ee620d1610fc6b28fe69725f4b46af95d4 321 320 2009-02-25T21:14:25Z Jon Awbrey 3 previously ... wikitext text/x-wiki This discussion page is intended for a discussion on using property theory as a means of organizing material in subject wikis, as well as the underlying issues of mathematics, logic, computation and philosophy. [[User:Vipul|Vipul]] 20:20, 25 February 2009 (UTC) ==Prospects for a Logic Wiki== Previously on Buffer &hellip; * [http://blog.subwiki.org/?p=13 The History of Subject Wikis] * [http://blog.subwiki.org/?p=13&cpage=1#comment-2 Comments] <blockquote> <p>Vipul,</p> <p>From the initial sample of your writing distribution that I’ve read so far, your approach to mathematical objects by way of their properties is essentially a logical angle.</p> <p>If one elects to think in functional, computational terms from the very beginning, then it is convenient to think of a property <math>p</math> as being a mapping from a space <math>X,</math> the universe of discourse, to a space of 2 elements, say, <math>\mathbb{B} = \{ 0, 1 \}.</math> So a property is something of the form <math>p : X \to \mathbb{B}.</math> That level of consideration amounts to propositional calculus. Trivial as it may seem, it’s been my experience that getting efficient computational support at this level is key to many of the things we’d like to do at higher levels in the way of mathematical knowledge management.</p> <p>Well, enough for now, as I’m not even sure the above formatting will work here. Let’s look for a wiki discussion page where it will be easier to talk. Either at MyWikiBiz, one of your Subject Wikis, or if you like email archiving there is the Inquiry List that an e-friend set up for me.</p> <p>[http://blog.subwiki.org/?p=13&cpage=1#comment-6 Jon Awbrey, 24 Feb 2009, 20:42 UTC]</p> </blockquote> 03dd9421609fc6177bf5767000e67fbc8a19a4e3 322 321 2009-02-25T21:15:21Z Jon Awbrey 3 /* Prospects for a Logic Wiki */ wikitext text/x-wiki This discussion page is intended for a discussion on using property theory as a means of organizing material in subject wikis, as well as the underlying issues of mathematics, logic, computation and philosophy. [[User:Vipul|Vipul]] 20:20, 25 February 2009 (UTC) ==Prospects for a Logic Wiki== Previously on Buffer &hellip; * [http://blog.subwiki.org/?p=13 The History of Subject Wikis] * [http://blog.subwiki.org/?p=13&cpage=1#comment-2 Comments] <blockquote> <p>Vipul,</p> <p>From the initial sample of your writing distribution that I’ve read so far, your approach to mathematical objects by way of their properties is essentially a logical angle.</p> <p>If one elects to think in functional, computational terms from the very beginning, then it is convenient to think of a property <math>p\!</math> as being a mapping from a space <math>X,\!</math> the universe of discourse, to a space of 2 elements, say, <math>\mathbb{B} = \{ 0, 1 \}.</math> So a property is something of the form <math>p : X \to \mathbb{B}.</math> That level of consideration amounts to propositional calculus. Trivial as it may seem, it’s been my experience that getting efficient computational support at this level is key to many of the things we’d like to do at higher levels in the way of mathematical knowledge management.</p> <p>Well, enough for now, as I’m not even sure the above formatting will work here. Let’s look for a wiki discussion page where it will be easier to talk. Either at MyWikiBiz, one of your Subject Wikis, or if you like email archiving there is the Inquiry List that an e-friend set up for me.</p> <p>[http://blog.subwiki.org/?p=13&cpage=1#comment-6 Jon Awbrey, 24 Feb 2009, 20:42 UTC]</p> </blockquote> f73a47ce4ce5827575415f7ab2d8f13adcac5204 323 322 2009-02-25T22:13:58Z Vipul 2 wikitext text/x-wiki This discussion page is intended for a discussion on using property theory as a means of organizing material in subject wikis, as well as the underlying issues of mathematics, logic, computation and philosophy. [[User:Vipul|Vipul]] 20:20, 25 February 2009 (UTC) ==Prospects for a Logic Wiki== Previously on Buffer &hellip; * [http://blog.subwiki.org/?p=13 The History of Subject Wikis] * [http://blog.subwiki.org/?p=13&cpage=1#comment-2 Comments] <blockquote> <p>Vipul,</p> <p>From the initial sample of your writing distribution that I’ve read so far, your approach to mathematical objects by way of their properties is essentially a logical angle.</p> <p>If one elects to think in functional, computational terms from the very beginning, then it is convenient to think of a property <math>p\!</math> as being a mapping from a space <math>X,\!</math> the universe of discourse, to a space of 2 elements, say, <math>\mathbb{B} = \{ 0, 1 \}.</math> So a property is something of the form <math>p : X \to \mathbb{B}.</math> That level of consideration amounts to propositional calculus. Trivial as it may seem, it’s been my experience that getting efficient computational support at this level is key to many of the things we’d like to do at higher levels in the way of mathematical knowledge management.</p> <p>Well, enough for now, as I’m not even sure the above formatting will work here. Let’s look for a wiki discussion page where it will be easier to talk. Either at MyWikiBiz, one of your Subject Wikis, or if you like email archiving there is the Inquiry List that an e-friend set up for me.</p> <p>[http://blog.subwiki.org/?p=13&cpage=1#comment-6 Jon Awbrey, 24 Feb 2009, 20:42 UTC]</p> </blockquote> VN: Jon, I agree with your basic ideas. There's of course the issue that the space <math>X</math> under consideration is often ''not'' a ''set'' (it's too large to be one -- for instance, a [[Groupprops:group property|group property]] is defined as a ''function'' on the collection of all groups), so we cnanot simply treat it as a set map. Still, it behaves largely like a set map. And as you point out, properties behave a lot like propositions with the ''parameter'' being from the space <math>X</math>. At some point in time, I was interested in the mathematical aspects of properties. Properties form a Boolean algebra, inheriting conjunction and disjunction etc. (in set-theoretic jargon, this would just be the power set of <math>X</math>, except that <math>X</math> isn't a set). Then, we can use various partial binary operations on <math>X</math> to define corresponding binary operations on the property space. I believe this, too, has been explored by some people, who came up with structures such as quantales and gaggles. The broad idea, as far as I understand, is that the binary operation must distribute over disjunctions (ORs). Many of the operators that I discuss in the group theory wiki, such as the [[groupprops:composition operator for subgroup properties|composition operator for subgroup properties]] and the [[groupprops:join operator|join operator]], are quantalic operators. Some of the terminology that I use for some property-theoretic notation is borrowed from terminology used for such operators, for instance, the [[groupprops:left residual operator for composition|left residual]] and [[groupprops:right residual operator for composition|right residual]]. Some of the basic structural theorems and approaches are also taken from corresponding ideas that already existed in logic. (I arrived at some of them before becoming aware of the corresponding structures explored in logic, and later found that much of this had been done before). As far as I know, though, there has been no systematic application of the ideas developed in different parts of logic to this ''real world'' (?) application to understanding properties in different mathematical domains. This puzzled me since I think that using the ideas of logic and logical structure in this way can help achieve a better understanding of many disciplines, particularly those in mathematics and computer science. Perhaps it is because there are no deep theorems here? My own use of property theory as an organizational principle seems to rely very minimally on the logical ideas beneath and rests more on the practical results that are important within the specific subject. But perhaps there are deeper theoretical logic issues that should be explored. I'd be glad to hear about your perspective, logical, philosophical, or any other. I'd also like to hear more about your views on what you mean by supporting such a model computationally. On the group theory wiki, I'm using MediaWiki's basic tools as well as Semantic MediaWiki to help in property-theoretic exploration. This isn't ''smart'' in any sense on the machine's part, but it meets some of the practical needs. Nonetheless, if you have ideas for something more sophisticated, I'd be glad to hear. [[User:Vipul|Vipul]] 22:13, 25 February 2009 (UTC) 37128ff9534423eb9b4b82f0d6f55160ab8295a4 324 323 2009-02-25T23:08:26Z Jon Awbrey 3 begin reply wikitext text/x-wiki This discussion page is intended for a discussion on using property theory as a means of organizing material in subject wikis, as well as the underlying issues of mathematics, logic, computation and philosophy. [[User:Vipul|Vipul]] 20:20, 25 February 2009 (UTC) ==Prospects for a Logic Wiki== Previously on Buffer &hellip; * [http://blog.subwiki.org/?p=13 The History of Subject Wikis] * [http://blog.subwiki.org/?p=13&cpage=1#comment-2 Comments] <blockquote> <p>Vipul,</p> <p>From the initial sample of your writing distribution that I’ve read so far, your approach to mathematical objects by way of their properties is essentially a logical angle.</p> <p>If one elects to think in functional, computational terms from the very beginning, then it is convenient to think of a property <math>p\!</math> as being a mapping from a space <math>X,\!</math> the universe of discourse, to a space of 2 elements, say, <math>\mathbb{B} = \{ 0, 1 \}.</math> So a property is something of the form <math>p : X \to \mathbb{B}.</math> That level of consideration amounts to propositional calculus. Trivial as it may seem, it’s been my experience that getting efficient computational support at this level is key to many of the things we’d like to do at higher levels in the way of mathematical knowledge management.</p> <p>Well, enough for now, as I’m not even sure the above formatting will work here. Let’s look for a wiki discussion page where it will be easier to talk. Either at MyWikiBiz, one of your Subject Wikis, or if you like email archiving there is the Inquiry List that an e-friend set up for me.</p> <p>[http://blog.subwiki.org/?p=13&cpage=1#comment-6 Jon Awbrey, 24 Feb 2009, 20:42 UTC]</p> </blockquote> VN: Jon, I agree with your basic ideas. There's of course the issue that the space <math>X</math> under consideration is often ''not'' a ''set'' (it's too large to be one -- for instance, a [[Groupprops:group property|group property]] is defined as a ''function'' on the collection of all groups), so we cnanot simply treat it as a set map. Still, it behaves largely like a set map. And as you point out, properties behave a lot like propositions with the ''parameter'' being from the space <math>X</math>. At some point in time, I was interested in the mathematical aspects of properties. Properties form a Boolean algebra, inheriting conjunction and disjunction etc. (in set-theoretic jargon, this would just be the power set of <math>X</math>, except that <math>X</math> isn't a set). Then, we can use various partial binary operations on <math>X</math> to define corresponding binary operations on the property space. I believe this, too, has been explored by some people, who came up with structures such as quantales and gaggles. The broad idea, as far as I understand, is that the binary operation must distribute over disjunctions (ORs). Many of the operators that I discuss in the group theory wiki, such as the [[groupprops:composition operator for subgroup properties|composition operator for subgroup properties]] and the [[groupprops:join operator|join operator]], are quantalic operators. Some of the terminology that I use for some property-theoretic notation is borrowed from terminology used for such operators, for instance, the [[groupprops:left residual operator for composition|left residual]] and [[groupprops:right residual operator for composition|right residual]]. Some of the basic structural theorems and approaches are also taken from corresponding ideas that already existed in logic. (I arrived at some of them before becoming aware of the corresponding structures explored in logic, and later found that much of this had been done before). As far as I know, though, there has been no systematic application of the ideas developed in different parts of logic to this ''real world'' (?) application to understanding properties in different mathematical domains. This puzzled me since I think that using the ideas of logic and logical structure in this way can help achieve a better understanding of many disciplines, particularly those in mathematics and computer science. Perhaps it is because there are no deep theorems here? My own use of property theory as an organizational principle seems to rely very minimally on the logical ideas beneath and rests more on the practical results that are important within the specific subject. But perhaps there are deeper theoretical logic issues that should be explored. I'd be glad to hear about your perspective, logical, philosophical, or any other. I'd also like to hear more about your views on what you mean by supporting such a model computationally. On the group theory wiki, I'm using MediaWiki's basic tools as well as Semantic MediaWiki to help in property-theoretic exploration. This isn't ''smart'' in any sense on the machine's part, but it meets some of the practical needs. Nonetheless, if you have ideas for something more sophisticated, I'd be glad to hear. [[User:Vipul|Vipul]] 22:13, 25 February 2009 (UTC) JA: Sure, there is all that. My main interest at present is focused on using logic as a tool for practical knowledge management in mathematical domains. That means starting small, breaking off this or that chunk of suitably interesting or useful domains and representing them in such a way that you can not only get computers to help you reason about them but in such a way that human beings can grok what's going on. JA: Uh-oh, I'm being called to dinner, I will have to continue later tonight. [[User:Jon Awbrey|Jon Awbrey]] 23:08, 25 February 2009 (UTC) 5011e94e78a0b9a54c5ef8d32ae718c27966a22f 325 324 2009-02-26T00:00:00Z Vipul 2 /* Prospects for a Logic Wiki */ wikitext text/x-wiki This discussion page is intended for a discussion on using property theory as a means of organizing material in subject wikis, as well as the underlying issues of mathematics, logic, computation and philosophy. [[User:Vipul|Vipul]] 20:20, 25 February 2009 (UTC) ==Prospects for a Logic Wiki== Previously on Buffer &hellip; * [http://blog.subwiki.org/?p=13 The History of Subject Wikis] * [http://blog.subwiki.org/?p=13&cpage=1#comment-2 Comments] <blockquote> <p>Vipul,</p> <p>From the initial sample of your writing distribution that I’ve read so far, your approach to mathematical objects by way of their properties is essentially a logical angle.</p> <p>If one elects to think in functional, computational terms from the very beginning, then it is convenient to think of a property <math>p\!</math> as being a mapping from a space <math>X,\!</math> the universe of discourse, to a space of 2 elements, say, <math>\mathbb{B} = \{ 0, 1 \}.</math> So a property is something of the form <math>p : X \to \mathbb{B}.</math> That level of consideration amounts to propositional calculus. Trivial as it may seem, it’s been my experience that getting efficient computational support at this level is key to many of the things we’d like to do at higher levels in the way of mathematical knowledge management.</p> <p>Well, enough for now, as I’m not even sure the above formatting will work here. Let’s look for a wiki discussion page where it will be easier to talk. Either at MyWikiBiz, one of your Subject Wikis, or if you like email archiving there is the Inquiry List that an e-friend set up for me.</p> <p>[http://blog.subwiki.org/?p=13&cpage=1#comment-6 Jon Awbrey, 24 Feb 2009, 20:42 UTC]</p> </blockquote> VN: Jon, I agree with your basic ideas. There's of course the issue that the space <math>X</math> under consideration is often ''not'' a ''set'' (it's too large to be one -- for instance, a [[Groupprops:group property|group property]] is defined as a ''function'' on the collection of all groups), so we cnanot simply treat it as a set map. Still, it behaves largely like a set map. And as you point out, properties behave a lot like propositions with the ''parameter'' being from the space <math>X</math>. At some point in time, I was interested in the mathematical aspects of properties. Properties form a Boolean algebra, inheriting conjunction and disjunction etc. (in set-theoretic jargon, this would just be the power set of <math>X</math>, except that <math>X</math> isn't a set). Then, we can use various partial binary operations on <math>X</math> to define corresponding binary operations on the property space. I believe this, too, has been explored by some people, who came up with structures such as quantales and gaggles. The broad idea, as far as I understand, is that the binary operation must distribute over disjunctions (ORs). Many of the operators that I discuss in the group theory wiki, such as the [[groupprops:composition operator for subgroup properties|composition operator for subgroup properties]] and the [[groupprops:join operator|join operator]], are quantalic operators. Some of the terminology that I use for some property-theoretic notation is borrowed from terminology used for such operators, for instance, the [[groupprops:left residual operator for composition|left residual]] and [[groupprops:right residual operator for composition|right residual]]. Some of the basic structural theorems and approaches are also taken from corresponding ideas that already existed in logic. (I arrived at some of them before becoming aware of the corresponding structures explored in logic, and later found that much of this had been done before). As far as I know, though, there has been no systematic application of the ideas developed in different parts of logic to this ''real world'' (?) application to understanding properties in different mathematical domains. This puzzled me since I think that using the ideas of logic and logical structure in this way can help achieve a better understanding of many disciplines, particularly those in mathematics and computer science. Perhaps it is because there are no deep theorems here? My own use of property theory as an organizational principle seems to rely very minimally on the logical ideas beneath and rests more on the practical results that are important within the specific subject. But perhaps there are deeper theoretical logic issues that should be explored. I'd be glad to hear about your perspective, logical, philosophical, or any other. I'd also like to hear more about your views on what you mean by supporting such a model computationally. On the group theory wiki, I'm using MediaWiki's basic tools as well as Semantic MediaWiki to help in property-theoretic exploration. This isn't ''smart'' in any sense on the machine's part, but it meets some of the practical needs. Nonetheless, if you have ideas for something more sophisticated, I'd be glad to hear. [[User:Vipul|Vipul]] 22:13, 25 February 2009 (UTC) JA: Sure, there is all that. My main interest at present is focused on using logic as a tool for practical knowledge management in mathematical domains. That means starting small, breaking off this or that chunk of suitably interesting or useful domains and representing them in such a way that you can not only get computers to help you reason about them but in such a way that human beings can grok what's going on. JA: Uh-oh, I'm being called to dinner, I will have to continue later tonight. [[User:Jon Awbrey|Jon Awbrey]] 23:08, 25 February 2009 (UTC) VN: Well, that's interesting. I'm myself interested in the practical aspects, too, though perhaps from a different angle. I'd like to hear more. I'll also go through some of your past writings to understand your perspective. If you have comments on the way things have been done on the subject wikis specifically, I'd like to hear those as well. [[User:Vipul|Vipul]] 00:00, 26 February 2009 (UTC) b21340296589f395faa7b1687c09a4489eb28439 326 325 2009-02-26T03:26:41Z Jon Awbrey 3 /* Prospects for a Logic Wiki */ + some links wikitext text/x-wiki This discussion page is intended for a discussion on using property theory as a means of organizing material in subject wikis, as well as the underlying issues of mathematics, logic, computation and philosophy. [[User:Vipul|Vipul]] 20:20, 25 February 2009 (UTC) ==Prospects for a Logic Wiki== Previously on Buffer &hellip; * [http://blog.subwiki.org/?p=13 The History of Subject Wikis] * [http://blog.subwiki.org/?p=13&cpage=1#comment-2 Comments] <blockquote> <p>Vipul,</p> <p>From the initial sample of your writing distribution that I’ve read so far, your approach to mathematical objects by way of their properties is essentially a logical angle.</p> <p>If one elects to think in functional, computational terms from the very beginning, then it is convenient to think of a property <math>p\!</math> as being a mapping from a space <math>X,\!</math> the universe of discourse, to a space of 2 elements, say, <math>\mathbb{B} = \{ 0, 1 \}.</math> So a property is something of the form <math>p : X \to \mathbb{B}.</math> That level of consideration amounts to propositional calculus. Trivial as it may seem, it’s been my experience that getting efficient computational support at this level is key to many of the things we’d like to do at higher levels in the way of mathematical knowledge management.</p> <p>Well, enough for now, as I’m not even sure the above formatting will work here. Let’s look for a wiki discussion page where it will be easier to talk. Either at MyWikiBiz, one of your Subject Wikis, or if you like email archiving there is the Inquiry List that an e-friend set up for me.</p> <p>[http://blog.subwiki.org/?p=13&cpage=1#comment-6 Jon Awbrey, 24 Feb 2009, 20:42 UTC]</p> </blockquote> VN: Jon, I agree with your basic ideas. There's of course the issue that the space <math>X</math> under consideration is often ''not'' a ''set'' (it's too large to be one -- for instance, a [[Groupprops:group property|group property]] is defined as a ''function'' on the collection of all groups), so we cnanot simply treat it as a set map. Still, it behaves largely like a set map. And as you point out, properties behave a lot like propositions with the ''parameter'' being from the space <math>X</math>. At some point in time, I was interested in the mathematical aspects of properties. Properties form a Boolean algebra, inheriting conjunction and disjunction etc. (in set-theoretic jargon, this would just be the power set of <math>X</math>, except that <math>X</math> isn't a set). Then, we can use various partial binary operations on <math>X</math> to define corresponding binary operations on the property space. I believe this, too, has been explored by some people, who came up with structures such as quantales and gaggles. The broad idea, as far as I understand, is that the binary operation must distribute over disjunctions (ORs). Many of the operators that I discuss in the group theory wiki, such as the [[groupprops:composition operator for subgroup properties|composition operator for subgroup properties]] and the [[groupprops:join operator|join operator]], are quantalic operators. Some of the terminology that I use for some property-theoretic notation is borrowed from terminology used for such operators, for instance, the [[groupprops:left residual operator for composition|left residual]] and [[groupprops:right residual operator for composition|right residual]]. Some of the basic structural theorems and approaches are also taken from corresponding ideas that already existed in logic. (I arrived at some of them before becoming aware of the corresponding structures explored in logic, and later found that much of this had been done before). As far as I know, though, there has been no systematic application of the ideas developed in different parts of logic to this ''real world'' (?) application to understanding properties in different mathematical domains. This puzzled me since I think that using the ideas of logic and logical structure in this way can help achieve a better understanding of many disciplines, particularly those in mathematics and computer science. Perhaps it is because there are no deep theorems here? My own use of property theory as an organizational principle seems to rely very minimally on the logical ideas beneath and rests more on the practical results that are important within the specific subject. But perhaps there are deeper theoretical logic issues that should be explored. I'd be glad to hear about your perspective, logical, philosophical, or any other. I'd also like to hear more about your views on what you mean by supporting such a model computationally. On the group theory wiki, I'm using MediaWiki's basic tools as well as Semantic MediaWiki to help in property-theoretic exploration. This isn't ''smart'' in any sense on the machine's part, but it meets some of the practical needs. Nonetheless, if you have ideas for something more sophisticated, I'd be glad to hear. [[User:Vipul|Vipul]] 22:13, 25 February 2009 (UTC) JA: Sure, there is all that. My main interest at present is focused on using logic as a tool for practical knowledge management in mathematical domains. That means starting small, breaking off this or that chunk of suitably interesting or useful domains and representing them in such a way that you can not only get computers to help you reason about them but in such a way that human beings can grok what's going on. JA: Uh-oh, I'm being called to dinner, I will have to continue later tonight. [[User:Jon Awbrey|Jon Awbrey]] 23:08, 25 February 2009 (UTC) VN: Well, that's interesting. I'm myself interested in the practical aspects, too, though perhaps from a different angle. I'd like to hear more. I'll also go through some of your past writings to understand your perspective. If you have comments on the way things have been done on the subject wikis specifically, I'd like to hear those as well. [[User:Vipul|Vipul]] 00:00, 26 February 2009 (UTC) JA: Maybe it will be quickest just to list a few links. Most of the longer pieces at MyWikBiz are unpublished papers and project workups of mine that I'm still in the process TeXing and Wikifying and redoing the graphics that got mushed over moving between different platforms over the years, so most of them have a point where I left off reformatting or rewriting. At any rate, here are a few serving suggestions on the basic logic side of things: * [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] * [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems Differential Logic and Dynamic Systems] * [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Functional_Logic_:_Quantification_Theory Functional Logic : Quantification Theory] JA: [[User:Jon Awbrey|Jon Awbrey]] 03:26, 26 February 2009 (UTC) 7826a4e65e9a8eeb43013650b0f647d113a9606c 327 326 2009-02-26T03:32:27Z Jon Awbrey 3 typo wikitext text/x-wiki This discussion page is intended for a discussion on using property theory as a means of organizing material in subject wikis, as well as the underlying issues of mathematics, logic, computation and philosophy. [[User:Vipul|Vipul]] 20:20, 25 February 2009 (UTC) ==Prospects for a Logic Wiki== Previously on Buffer &hellip; * [http://blog.subwiki.org/?p=13 The History of Subject Wikis] * [http://blog.subwiki.org/?p=13&cpage=1#comment-2 Comments] <blockquote> <p>Vipul,</p> <p>From the initial sample of your writing distribution that I’ve read so far, your approach to mathematical objects by way of their properties is essentially a logical angle.</p> <p>If one elects to think in functional, computational terms from the very beginning, then it is convenient to think of a property <math>p\!</math> as being a mapping from a space <math>X,\!</math> the universe of discourse, to a space of 2 elements, say, <math>\mathbb{B} = \{ 0, 1 \}.</math> So a property is something of the form <math>p : X \to \mathbb{B}.</math> That level of consideration amounts to propositional calculus. Trivial as it may seem, it’s been my experience that getting efficient computational support at this level is key to many of the things we’d like to do at higher levels in the way of mathematical knowledge management.</p> <p>Well, enough for now, as I’m not even sure the above formatting will work here. Let’s look for a wiki discussion page where it will be easier to talk. Either at MyWikiBiz, one of your Subject Wikis, or if you like email archiving there is the Inquiry List that an e-friend set up for me.</p> <p>[http://blog.subwiki.org/?p=13&cpage=1#comment-6 Jon Awbrey, 24 Feb 2009, 20:42 UTC]</p> </blockquote> VN: Jon, I agree with your basic ideas. There's of course the issue that the space <math>X</math> under consideration is often ''not'' a ''set'' (it's too large to be one -- for instance, a [[Groupprops:group property|group property]] is defined as a ''function'' on the collection of all groups), so we cnanot simply treat it as a set map. Still, it behaves largely like a set map. And as you point out, properties behave a lot like propositions with the ''parameter'' being from the space <math>X</math>. At some point in time, I was interested in the mathematical aspects of properties. Properties form a Boolean algebra, inheriting conjunction and disjunction etc. (in set-theoretic jargon, this would just be the power set of <math>X</math>, except that <math>X</math> isn't a set). Then, we can use various partial binary operations on <math>X</math> to define corresponding binary operations on the property space. I believe this, too, has been explored by some people, who came up with structures such as quantales and gaggles. The broad idea, as far as I understand, is that the binary operation must distribute over disjunctions (ORs). Many of the operators that I discuss in the group theory wiki, such as the [[groupprops:composition operator for subgroup properties|composition operator for subgroup properties]] and the [[groupprops:join operator|join operator]], are quantalic operators. Some of the terminology that I use for some property-theoretic notation is borrowed from terminology used for such operators, for instance, the [[groupprops:left residual operator for composition|left residual]] and [[groupprops:right residual operator for composition|right residual]]. Some of the basic structural theorems and approaches are also taken from corresponding ideas that already existed in logic. (I arrived at some of them before becoming aware of the corresponding structures explored in logic, and later found that much of this had been done before). As far as I know, though, there has been no systematic application of the ideas developed in different parts of logic to this ''real world'' (?) application to understanding properties in different mathematical domains. This puzzled me since I think that using the ideas of logic and logical structure in this way can help achieve a better understanding of many disciplines, particularly those in mathematics and computer science. Perhaps it is because there are no deep theorems here? My own use of property theory as an organizational principle seems to rely very minimally on the logical ideas beneath and rests more on the practical results that are important within the specific subject. But perhaps there are deeper theoretical logic issues that should be explored. I'd be glad to hear about your perspective, logical, philosophical, or any other. I'd also like to hear more about your views on what you mean by supporting such a model computationally. On the group theory wiki, I'm using MediaWiki's basic tools as well as Semantic MediaWiki to help in property-theoretic exploration. This isn't ''smart'' in any sense on the machine's part, but it meets some of the practical needs. Nonetheless, if you have ideas for something more sophisticated, I'd be glad to hear. [[User:Vipul|Vipul]] 22:13, 25 February 2009 (UTC) JA: Sure, there is all that. My main interest at present is focused on using logic as a tool for practical knowledge management in mathematical domains. That means starting small, breaking off this or that chunk of suitably interesting or useful domains and representing them in such a way that you can not only get computers to help you reason about them but in such a way that human beings can grok what's going on. JA: Uh-oh, I'm being called to dinner, I will have to continue later tonight. [[User:Jon Awbrey|Jon Awbrey]] 23:08, 25 February 2009 (UTC) VN: Well, that's interesting. I'm myself interested in the practical aspects, too, though perhaps from a different angle. I'd like to hear more. I'll also go through some of your past writings to understand your perspective. If you have comments on the way things have been done on the subject wikis specifically, I'd like to hear those as well. [[User:Vipul|Vipul]] 00:00, 26 February 2009 (UTC) JA: Maybe it will be quickest just to list a few links. Most of the longer pieces at MyWikBiz are unpublished papers and project workups of mine that I'm still in the process TeXing and Wikifying and redoing the graphics that got mushed over moving between different platforms over the years, so most of them have a point where I left off reformatting or rewriting. At any rate, here are a few serving suggestions on the basic logic side of things: * [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] * [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] * [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Functional_Logic_:_Quantification_Theory Functional Logic : Quantification Theory] JA: [[User:Jon Awbrey|Jon Awbrey]] 03:26, 26 February 2009 (UTC) d7cbcaf28a96c69ad479d338249027583362468d 328 327 2009-02-26T03:38:21Z Jon Awbrey 3 /* Prospects for a Logic Wiki */ typos wikitext text/x-wiki This discussion page is intended for a discussion on using property theory as a means of organizing material in subject wikis, as well as the underlying issues of mathematics, logic, computation and philosophy. [[User:Vipul|Vipul]] 20:20, 25 February 2009 (UTC) ==Prospects for a Logic Wiki== Previously on Buffer &hellip; * [http://blog.subwiki.org/?p=13 The History of Subject Wikis] * [http://blog.subwiki.org/?p=13&cpage=1#comment-2 Comments] <blockquote> <p>Vipul,</p> <p>From the initial sample of your writing distribution that I’ve read so far, your approach to mathematical objects by way of their properties is essentially a logical angle.</p> <p>If one elects to think in functional, computational terms from the very beginning, then it is convenient to think of a property <math>p\!</math> as being a mapping from a space <math>X,\!</math> the universe of discourse, to a space of 2 elements, say, <math>\mathbb{B} = \{ 0, 1 \}.</math> So a property is something of the form <math>p : X \to \mathbb{B}.</math> That level of consideration amounts to propositional calculus. Trivial as it may seem, it’s been my experience that getting efficient computational support at this level is key to many of the things we’d like to do at higher levels in the way of mathematical knowledge management.</p> <p>Well, enough for now, as I’m not even sure the above formatting will work here. Let’s look for a wiki discussion page where it will be easier to talk. Either at MyWikiBiz, one of your Subject Wikis, or if you like email archiving there is the Inquiry List that an e-friend set up for me.</p> <p>[http://blog.subwiki.org/?p=13&cpage=1#comment-6 Jon Awbrey, 24 Feb 2009, 20:42 UTC]</p> </blockquote> VN: Jon, I agree with your basic ideas. There's of course the issue that the space <math>X</math> under consideration is often ''not'' a ''set'' (it's too large to be one -- for instance, a [[Groupprops:group property|group property]] is defined as a ''function'' on the collection of all groups), so we cnanot simply treat it as a set map. Still, it behaves largely like a set map. And as you point out, properties behave a lot like propositions with the ''parameter'' being from the space <math>X</math>. At some point in time, I was interested in the mathematical aspects of properties. Properties form a Boolean algebra, inheriting conjunction and disjunction etc. (in set-theoretic jargon, this would just be the power set of <math>X</math>, except that <math>X</math> isn't a set). Then, we can use various partial binary operations on <math>X</math> to define corresponding binary operations on the property space. I believe this, too, has been explored by some people, who came up with structures such as quantales and gaggles. The broad idea, as far as I understand, is that the binary operation must distribute over disjunctions (ORs). Many of the operators that I discuss in the group theory wiki, such as the [[groupprops:composition operator for subgroup properties|composition operator for subgroup properties]] and the [[groupprops:join operator|join operator]], are quantalic operators. Some of the terminology that I use for some property-theoretic notation is borrowed from terminology used for such operators, for instance, the [[groupprops:left residual operator for composition|left residual]] and [[groupprops:right residual operator for composition|right residual]]. Some of the basic structural theorems and approaches are also taken from corresponding ideas that already existed in logic. (I arrived at some of them before becoming aware of the corresponding structures explored in logic, and later found that much of this had been done before). As far as I know, though, there has been no systematic application of the ideas developed in different parts of logic to this ''real world'' (?) application to understanding properties in different mathematical domains. This puzzled me since I think that using the ideas of logic and logical structure in this way can help achieve a better understanding of many disciplines, particularly those in mathematics and computer science. Perhaps it is because there are no deep theorems here? My own use of property theory as an organizational principle seems to rely very minimally on the logical ideas beneath and rests more on the practical results that are important within the specific subject. But perhaps there are deeper theoretical logic issues that should be explored. I'd be glad to hear about your perspective, logical, philosophical, or any other. I'd also like to hear more about your views on what you mean by supporting such a model computationally. On the group theory wiki, I'm using MediaWiki's basic tools as well as Semantic MediaWiki to help in property-theoretic exploration. This isn't ''smart'' in any sense on the machine's part, but it meets some of the practical needs. Nonetheless, if you have ideas for something more sophisticated, I'd be glad to hear. [[User:Vipul|Vipul]] 22:13, 25 February 2009 (UTC) JA: Sure, there is all that. My main interest at present is focused on using logic as a tool for practical knowledge management in mathematical domains. That means starting small, breaking off this or that chunk of suitably interesting or useful domains and representing them in such a way that you can not only get computers to help you reason about them but in such a way that human beings can grok what's going on. JA: Uh-oh, I'm being called to dinner, I will have to continue later tonight. [[User:Jon Awbrey|Jon Awbrey]] 23:08, 25 February 2009 (UTC) VN: Well, that's interesting. I'm myself interested in the practical aspects, too, though perhaps from a different angle. I'd like to hear more. I'll also go through some of your past writings to understand your perspective. If you have comments on the way things have been done on the subject wikis specifically, I'd like to hear those as well. [[User:Vipul|Vipul]] 00:00, 26 February 2009 (UTC) JA: Maybe it will be quickest just to list a few links. Most of the longer pieces at MyWikiBiz are unpublished papers and project workups of mine that I'm still in the process TeXing and Wikifying and redoing the graphics that got mushed up moving between different platforms over the years, so most of them have a point where I last left off reformatting or rewriting. At any rate, here are a few serving suggestions on the basic logic side of things: * [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] * [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] * [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Functional_Logic_:_Quantification_Theory Functional Logic : Quantification Theory] JA: [[User:Jon Awbrey|Jon Awbrey]] 03:26, 26 February 2009 (UTC) 990fa695874e0a5162d76aa699d033d7604a17d1 329 328 2009-02-27T18:08:34Z Jon Awbrey 3 /* Prospects for a Logic Wiki */ + [Scope and Purpose of a Logic Wiki] wikitext text/x-wiki This discussion page is intended for a discussion on using property theory as a means of organizing material in subject wikis, as well as the underlying issues of mathematics, logic, computation and philosophy. [[User:Vipul|Vipul]] 20:20, 25 February 2009 (UTC) ==Prospects for a Logic Wiki== ===Initial Discussion=== Previously on Buffer &hellip; * [http://blog.subwiki.org/?p=13 The History of Subject Wikis] * [http://blog.subwiki.org/?p=13&cpage=1#comment-2 Comments] <blockquote> <p>Vipul,</p> <p>From the initial sample of your writing distribution that I’ve read so far, your approach to mathematical objects by way of their properties is essentially a logical angle.</p> <p>If one elects to think in functional, computational terms from the very beginning, then it is convenient to think of a property <math>p\!</math> as being a mapping from a space <math>X,\!</math> the universe of discourse, to a space of 2 elements, say, <math>\mathbb{B} = \{ 0, 1 \}.</math> So a property is something of the form <math>p : X \to \mathbb{B}.</math> That level of consideration amounts to propositional calculus. Trivial as it may seem, it’s been my experience that getting efficient computational support at this level is key to many of the things we’d like to do at higher levels in the way of mathematical knowledge management.</p> <p>Well, enough for now, as I’m not even sure the above formatting will work here. Let’s look for a wiki discussion page where it will be easier to talk. Either at MyWikiBiz, one of your Subject Wikis, or if you like email archiving there is the Inquiry List that an e-friend set up for me.</p> <p>[http://blog.subwiki.org/?p=13&cpage=1#comment-6 Jon Awbrey, 24 Feb 2009, 20:42 UTC]</p> </blockquote> VN: Jon, I agree with your basic ideas. There's of course the issue that the space <math>X</math> under consideration is often ''not'' a ''set'' (it's too large to be one -- for instance, a [[Groupprops:group property|group property]] is defined as a ''function'' on the collection of all groups), so we cnanot simply treat it as a set map. Still, it behaves largely like a set map. And as you point out, properties behave a lot like propositions with the ''parameter'' being from the space <math>X</math>. At some point in time, I was interested in the mathematical aspects of properties. Properties form a Boolean algebra, inheriting conjunction and disjunction etc. (in set-theoretic jargon, this would just be the power set of <math>X</math>, except that <math>X</math> isn't a set). Then, we can use various partial binary operations on <math>X</math> to define corresponding binary operations on the property space. I believe this, too, has been explored by some people, who came up with structures such as quantales and gaggles. The broad idea, as far as I understand, is that the binary operation must distribute over disjunctions (ORs). Many of the operators that I discuss in the group theory wiki, such as the [[groupprops:composition operator for subgroup properties|composition operator for subgroup properties]] and the [[groupprops:join operator|join operator]], are quantalic operators. Some of the terminology that I use for some property-theoretic notation is borrowed from terminology used for such operators, for instance, the [[groupprops:left residual operator for composition|left residual]] and [[groupprops:right residual operator for composition|right residual]]. Some of the basic structural theorems and approaches are also taken from corresponding ideas that already existed in logic. (I arrived at some of them before becoming aware of the corresponding structures explored in logic, and later found that much of this had been done before). As far as I know, though, there has been no systematic application of the ideas developed in different parts of logic to this ''real world'' (?) application to understanding properties in different mathematical domains. This puzzled me since I think that using the ideas of logic and logical structure in this way can help achieve a better understanding of many disciplines, particularly those in mathematics and computer science. Perhaps it is because there are no deep theorems here? My own use of property theory as an organizational principle seems to rely very minimally on the logical ideas beneath and rests more on the practical results that are important within the specific subject. But perhaps there are deeper theoretical logic issues that should be explored. I'd be glad to hear about your perspective, logical, philosophical, or any other. I'd also like to hear more about your views on what you mean by supporting such a model computationally. On the group theory wiki, I'm using MediaWiki's basic tools as well as Semantic MediaWiki to help in property-theoretic exploration. This isn't ''smart'' in any sense on the machine's part, but it meets some of the practical needs. Nonetheless, if you have ideas for something more sophisticated, I'd be glad to hear. [[User:Vipul|Vipul]] 22:13, 25 February 2009 (UTC) JA: Sure, there is all that. My main interest at present is focused on using logic as a tool for practical knowledge management in mathematical domains. That means starting small, breaking off this or that chunk of suitably interesting or useful domains and representing them in such a way that you can not only get computers to help you reason about them but in such a way that human beings can grok what's going on. JA: Uh-oh, I'm being called to dinner, I will have to continue later tonight. [[User:Jon Awbrey|Jon Awbrey]] 23:08, 25 February 2009 (UTC) VN: Well, that's interesting. I'm myself interested in the practical aspects, too, though perhaps from a different angle. I'd like to hear more. I'll also go through some of your past writings to understand your perspective. If you have comments on the way things have been done on the subject wikis specifically, I'd like to hear those as well. [[User:Vipul|Vipul]] 00:00, 26 February 2009 (UTC) JA: Maybe it will be quickest just to list a few links. Most of the longer pieces at MyWikiBiz are unpublished papers and project workups of mine that I'm still in the process TeXing and Wikifying and redoing the graphics that got mushed up moving between different platforms over the years, so most of them have a point where I last left off reformatting or rewriting. At any rate, here are a few serving suggestions on the basic logic side of things: :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] : * [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Propositional_Calculus Differential Propositional Calculus] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Functional_Logic_:_Quantification_Theory Functional Logic : Quantification Theory] JA: [[User:Jon Awbrey|Jon Awbrey]] 03:26, 26 February 2009 (UTC) ===Scope and Purpose of a Logic Wiki=== ab76333c8c47817d818782c517927265dd18df3c 330 329 2009-02-27T18:10:12Z Jon Awbrey 3 /* Initial Discussion */ wikitext text/x-wiki This discussion page is intended for a discussion on using property theory as a means of organizing material in subject wikis, as well as the underlying issues of mathematics, logic, computation and philosophy. [[User:Vipul|Vipul]] 20:20, 25 February 2009 (UTC) ==Prospects for a Logic Wiki== ===Initial Discussion=== Previously on Buffer &hellip; * [http://blog.subwiki.org/?p=13 The History of Subject Wikis] * [http://blog.subwiki.org/?p=13&cpage=1#comment-2 Comments] <blockquote> <p>Vipul,</p> <p>From the initial sample of your writing distribution that I’ve read so far, your approach to mathematical objects by way of their properties is essentially a logical angle.</p> <p>If one elects to think in functional, computational terms from the very beginning, then it is convenient to think of a property <math>p\!</math> as being a mapping from a space <math>X,\!</math> the universe of discourse, to a space of 2 elements, say, <math>\mathbb{B} = \{ 0, 1 \}.</math> So a property is something of the form <math>p : X \to \mathbb{B}.</math> That level of consideration amounts to propositional calculus. Trivial as it may seem, it’s been my experience that getting efficient computational support at this level is key to many of the things we’d like to do at higher levels in the way of mathematical knowledge management.</p> <p>Well, enough for now, as I’m not even sure the above formatting will work here. Let’s look for a wiki discussion page where it will be easier to talk. Either at MyWikiBiz, one of your Subject Wikis, or if you like email archiving there is the Inquiry List that an e-friend set up for me.</p> <p>[http://blog.subwiki.org/?p=13&cpage=1#comment-6 Jon Awbrey, 24 Feb 2009, 20:42 UTC]</p> </blockquote> VN: Jon, I agree with your basic ideas. There's of course the issue that the space <math>X</math> under consideration is often ''not'' a ''set'' (it's too large to be one -- for instance, a [[Groupprops:group property|group property]] is defined as a ''function'' on the collection of all groups), so we cnanot simply treat it as a set map. Still, it behaves largely like a set map. And as you point out, properties behave a lot like propositions with the ''parameter'' being from the space <math>X</math>. At some point in time, I was interested in the mathematical aspects of properties. Properties form a Boolean algebra, inheriting conjunction and disjunction etc. (in set-theoretic jargon, this would just be the power set of <math>X</math>, except that <math>X</math> isn't a set). Then, we can use various partial binary operations on <math>X</math> to define corresponding binary operations on the property space. I believe this, too, has been explored by some people, who came up with structures such as quantales and gaggles. The broad idea, as far as I understand, is that the binary operation must distribute over disjunctions (ORs). Many of the operators that I discuss in the group theory wiki, such as the [[groupprops:composition operator for subgroup properties|composition operator for subgroup properties]] and the [[groupprops:join operator|join operator]], are quantalic operators. Some of the terminology that I use for some property-theoretic notation is borrowed from terminology used for such operators, for instance, the [[groupprops:left residual operator for composition|left residual]] and [[groupprops:right residual operator for composition|right residual]]. Some of the basic structural theorems and approaches are also taken from corresponding ideas that already existed in logic. (I arrived at some of them before becoming aware of the corresponding structures explored in logic, and later found that much of this had been done before). As far as I know, though, there has been no systematic application of the ideas developed in different parts of logic to this ''real world'' (?) application to understanding properties in different mathematical domains. This puzzled me since I think that using the ideas of logic and logical structure in this way can help achieve a better understanding of many disciplines, particularly those in mathematics and computer science. Perhaps it is because there are no deep theorems here? My own use of property theory as an organizational principle seems to rely very minimally on the logical ideas beneath and rests more on the practical results that are important within the specific subject. But perhaps there are deeper theoretical logic issues that should be explored. I'd be glad to hear about your perspective, logical, philosophical, or any other. I'd also like to hear more about your views on what you mean by supporting such a model computationally. On the group theory wiki, I'm using MediaWiki's basic tools as well as Semantic MediaWiki to help in property-theoretic exploration. This isn't ''smart'' in any sense on the machine's part, but it meets some of the practical needs. Nonetheless, if you have ideas for something more sophisticated, I'd be glad to hear. [[User:Vipul|Vipul]] 22:13, 25 February 2009 (UTC) JA: Sure, there is all that. My main interest at present is focused on using logic as a tool for practical knowledge management in mathematical domains. That means starting small, breaking off this or that chunk of suitably interesting or useful domains and representing them in such a way that you can not only get computers to help you reason about them but in such a way that human beings can grok what's going on. JA: Uh-oh, I'm being called to dinner, I will have to continue later tonight. [[User:Jon Awbrey|Jon Awbrey]] 23:08, 25 February 2009 (UTC) VN: Well, that's interesting. I'm myself interested in the practical aspects, too, though perhaps from a different angle. I'd like to hear more. I'll also go through some of your past writings to understand your perspective. If you have comments on the way things have been done on the subject wikis specifically, I'd like to hear those as well. [[User:Vipul|Vipul]] 00:00, 26 February 2009 (UTC) JA: Maybe it will be quickest just to list a few links. Most of the longer pieces at MyWikiBiz are unpublished papers and project workups of mine that I'm still in the process TeXing and Wikifying and redoing the graphics that got mushed up moving between different platforms over the years, so most of them have a point where I last left off reformatting or rewriting. At any rate, here are a few serving suggestions on the basic logic side of things: :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Propositional_Calculus Differential Propositional Calculus] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Functional_Logic_:_Quantification_Theory Functional Logic : Quantification Theory] JA: [[User:Jon Awbrey|Jon Awbrey]] 03:26, 26 February 2009 (UTC) ===Scope and Purpose of a Logic Wiki=== 363ba1d1336ef8fb275152263671ca84ff9b5d1a 331 330 2009-03-01T18:05:12Z Vipul 2 wikitext text/x-wiki This discussion page is intended for a discussion on using property theory as a means of organizing material in subject wikis, as well as the underlying issues of mathematics, logic, computation and philosophy. [[User:Vipul|Vipul]] 20:20, 25 February 2009 (UTC) ==Prospects for a Logic Wiki== ===Initial Discussion=== Previously on Buffer &hellip; * [http://blog.subwiki.org/?p=13 The History of Subject Wikis] * [http://blog.subwiki.org/?p=13&cpage=1#comment-2 Comments] <blockquote> <p>Vipul,</p> <p>From the initial sample of your writing distribution that I’ve read so far, your approach to mathematical objects by way of their properties is essentially a logical angle.</p> <p>If one elects to think in functional, computational terms from the very beginning, then it is convenient to think of a property <math>p\!</math> as being a mapping from a space <math>X,\!</math> the universe of discourse, to a space of 2 elements, say, <math>\mathbb{B} = \{ 0, 1 \}.</math> So a property is something of the form <math>p : X \to \mathbb{B}.</math> That level of consideration amounts to propositional calculus. Trivial as it may seem, it’s been my experience that getting efficient computational support at this level is key to many of the things we’d like to do at higher levels in the way of mathematical knowledge management.</p> <p>Well, enough for now, as I’m not even sure the above formatting will work here. Let’s look for a wiki discussion page where it will be easier to talk. Either at MyWikiBiz, one of your Subject Wikis, or if you like email archiving there is the Inquiry List that an e-friend set up for me.</p> <p>[http://blog.subwiki.org/?p=13&cpage=1#comment-6 Jon Awbrey, 24 Feb 2009, 20:42 UTC]</p> </blockquote> VN: Jon, I agree with your basic ideas. There's of course the issue that the space <math>X</math> under consideration is often ''not'' a ''set'' (it's too large to be one -- for instance, a [[Groupprops:group property|group property]] is defined as a ''function'' on the collection of all groups), so we cnanot simply treat it as a set map. Still, it behaves largely like a set map. And as you point out, properties behave a lot like propositions with the ''parameter'' being from the space <math>X</math>. At some point in time, I was interested in the mathematical aspects of properties. Properties form a Boolean algebra, inheriting conjunction and disjunction etc. (in set-theoretic jargon, this would just be the power set of <math>X</math>, except that <math>X</math> isn't a set). Then, we can use various partial binary operations on <math>X</math> to define corresponding binary operations on the property space. I believe this, too, has been explored by some people, who came up with structures such as quantales and gaggles. The broad idea, as far as I understand, is that the binary operation must distribute over disjunctions (ORs). Many of the operators that I discuss in the group theory wiki, such as the [[groupprops:composition operator for subgroup properties|composition operator for subgroup properties]] and the [[groupprops:join operator|join operator]], are quantalic operators. Some of the terminology that I use for some property-theoretic notation is borrowed from terminology used for such operators, for instance, the [[groupprops:left residual operator for composition|left residual]] and [[groupprops:right residual operator for composition|right residual]]. Some of the basic structural theorems and approaches are also taken from corresponding ideas that already existed in logic. (I arrived at some of them before becoming aware of the corresponding structures explored in logic, and later found that much of this had been done before). As far as I know, though, there has been no systematic application of the ideas developed in different parts of logic to this ''real world'' (?) application to understanding properties in different mathematical domains. This puzzled me since I think that using the ideas of logic and logical structure in this way can help achieve a better understanding of many disciplines, particularly those in mathematics and computer science. Perhaps it is because there are no deep theorems here? My own use of property theory as an organizational principle seems to rely very minimally on the logical ideas beneath and rests more on the practical results that are important within the specific subject. But perhaps there are deeper theoretical logic issues that should be explored. I'd be glad to hear about your perspective, logical, philosophical, or any other. I'd also like to hear more about your views on what you mean by supporting such a model computationally. On the group theory wiki, I'm using MediaWiki's basic tools as well as Semantic MediaWiki to help in property-theoretic exploration. This isn't ''smart'' in any sense on the machine's part, but it meets some of the practical needs. Nonetheless, if you have ideas for something more sophisticated, I'd be glad to hear. [[User:Vipul|Vipul]] 22:13, 25 February 2009 (UTC) JA: Sure, there is all that. My main interest at present is focused on using logic as a tool for practical knowledge management in mathematical domains. That means starting small, breaking off this or that chunk of suitably interesting or useful domains and representing them in such a way that you can not only get computers to help you reason about them but in such a way that human beings can grok what's going on. JA: Uh-oh, I'm being called to dinner, I will have to continue later tonight. [[User:Jon Awbrey|Jon Awbrey]] 23:08, 25 February 2009 (UTC) VN: Well, that's interesting. I'm myself interested in the practical aspects, too, though perhaps from a different angle. I'd like to hear more. I'll also go through some of your past writings to understand your perspective. If you have comments on the way things have been done on the subject wikis specifically, I'd like to hear those as well. [[User:Vipul|Vipul]] 00:00, 26 February 2009 (UTC) JA: Maybe it will be quickest just to list a few links. Most of the longer pieces at MyWikiBiz are unpublished papers and project workups of mine that I'm still in the process TeXing and Wikifying and redoing the graphics that got mushed up moving between different platforms over the years, so most of them have a point where I last left off reformatting or rewriting. At any rate, here are a few serving suggestions on the basic logic side of things: :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Propositional_Calculus Differential Propositional Calculus] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Functional_Logic_:_Quantification_Theory Functional Logic : Quantification Theory] JA: [[User:Jon Awbrey|Jon Awbrey]] 03:26, 26 February 2009 (UTC) VN: Jon, I've started looking at your work. Your language and approach are new to me, so it'll take me some time to go through it. Other work has also come up. I hope to get back to you in a couple of weeks. [[User:Vipul|Vipul]] 18:05, 1 March 2009 (UTC) ===Scope and Purpose of a Logic Wiki=== 3e5bf0e54d9fb955761e8179eca2e031b9b4bf68 332 331 2009-03-01T22:40:54Z Jon Awbrey 3 /* Initial Discussion */ me too ... wikitext text/x-wiki This discussion page is intended for a discussion on using property theory as a means of organizing material in subject wikis, as well as the underlying issues of mathematics, logic, computation and philosophy. [[User:Vipul|Vipul]] 20:20, 25 February 2009 (UTC) ==Prospects for a Logic Wiki== ===Initial Discussion=== Previously on Buffer &hellip; * [http://blog.subwiki.org/?p=13 The History of Subject Wikis] * [http://blog.subwiki.org/?p=13&cpage=1#comment-2 Comments] <blockquote> <p>Vipul,</p> <p>From the initial sample of your writing distribution that I’ve read so far, your approach to mathematical objects by way of their properties is essentially a logical angle.</p> <p>If one elects to think in functional, computational terms from the very beginning, then it is convenient to think of a property <math>p\!</math> as being a mapping from a space <math>X,\!</math> the universe of discourse, to a space of 2 elements, say, <math>\mathbb{B} = \{ 0, 1 \}.</math> So a property is something of the form <math>p : X \to \mathbb{B}.</math> That level of consideration amounts to propositional calculus. Trivial as it may seem, it’s been my experience that getting efficient computational support at this level is key to many of the things we’d like to do at higher levels in the way of mathematical knowledge management.</p> <p>Well, enough for now, as I’m not even sure the above formatting will work here. Let’s look for a wiki discussion page where it will be easier to talk. Either at MyWikiBiz, one of your Subject Wikis, or if you like email archiving there is the Inquiry List that an e-friend set up for me.</p> <p>[http://blog.subwiki.org/?p=13&cpage=1#comment-6 Jon Awbrey, 24 Feb 2009, 20:42 UTC]</p> </blockquote> VN: Jon, I agree with your basic ideas. There's of course the issue that the space <math>X</math> under consideration is often ''not'' a ''set'' (it's too large to be one -- for instance, a [[Groupprops:group property|group property]] is defined as a ''function'' on the collection of all groups), so we cnanot simply treat it as a set map. Still, it behaves largely like a set map. And as you point out, properties behave a lot like propositions with the ''parameter'' being from the space <math>X</math>. At some point in time, I was interested in the mathematical aspects of properties. Properties form a Boolean algebra, inheriting conjunction and disjunction etc. (in set-theoretic jargon, this would just be the power set of <math>X</math>, except that <math>X</math> isn't a set). Then, we can use various partial binary operations on <math>X</math> to define corresponding binary operations on the property space. I believe this, too, has been explored by some people, who came up with structures such as quantales and gaggles. The broad idea, as far as I understand, is that the binary operation must distribute over disjunctions (ORs). Many of the operators that I discuss in the group theory wiki, such as the [[groupprops:composition operator for subgroup properties|composition operator for subgroup properties]] and the [[groupprops:join operator|join operator]], are quantalic operators. Some of the terminology that I use for some property-theoretic notation is borrowed from terminology used for such operators, for instance, the [[groupprops:left residual operator for composition|left residual]] and [[groupprops:right residual operator for composition|right residual]]. Some of the basic structural theorems and approaches are also taken from corresponding ideas that already existed in logic. (I arrived at some of them before becoming aware of the corresponding structures explored in logic, and later found that much of this had been done before). As far as I know, though, there has been no systematic application of the ideas developed in different parts of logic to this ''real world'' (?) application to understanding properties in different mathematical domains. This puzzled me since I think that using the ideas of logic and logical structure in this way can help achieve a better understanding of many disciplines, particularly those in mathematics and computer science. Perhaps it is because there are no deep theorems here? My own use of property theory as an organizational principle seems to rely very minimally on the logical ideas beneath and rests more on the practical results that are important within the specific subject. But perhaps there are deeper theoretical logic issues that should be explored. I'd be glad to hear about your perspective, logical, philosophical, or any other. I'd also like to hear more about your views on what you mean by supporting such a model computationally. On the group theory wiki, I'm using MediaWiki's basic tools as well as Semantic MediaWiki to help in property-theoretic exploration. This isn't ''smart'' in any sense on the machine's part, but it meets some of the practical needs. Nonetheless, if you have ideas for something more sophisticated, I'd be glad to hear. [[User:Vipul|Vipul]] 22:13, 25 February 2009 (UTC) JA: Sure, there is all that. My main interest at present is focused on using logic as a tool for practical knowledge management in mathematical domains. That means starting small, breaking off this or that chunk of suitably interesting or useful domains and representing them in such a way that you can not only get computers to help you reason about them but in such a way that human beings can grok what's going on. JA: Uh-oh, I'm being called to dinner, I will have to continue later tonight. [[User:Jon Awbrey|Jon Awbrey]] 23:08, 25 February 2009 (UTC) VN: Well, that's interesting. I'm myself interested in the practical aspects, too, though perhaps from a different angle. I'd like to hear more. I'll also go through some of your past writings to understand your perspective. If you have comments on the way things have been done on the subject wikis specifically, I'd like to hear those as well. [[User:Vipul|Vipul]] 00:00, 26 February 2009 (UTC) JA: Maybe it will be quickest just to list a few links. Most of the longer pieces at MyWikiBiz are unpublished papers and project workups of mine that I'm still in the process TeXing and Wikifying and redoing the graphics that got mushed up moving between different platforms over the years, so most of them have a point where I last left off reformatting or rewriting. At any rate, here are a few serving suggestions on the basic logic side of things: :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Propositional_Calculus Differential Propositional Calculus] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Functional_Logic_:_Quantification_Theory Functional Logic : Quantification Theory] JA: [[User:Jon Awbrey|Jon Awbrey]] 03:26, 26 February 2009 (UTC) VN: Jon, I've started looking at your work. Your language and approach are new to me, so it'll take me some time to go through it. Other work has also come up. I hope to get back to you in a couple of weeks. [[User:Vipul|Vipul]] 18:05, 1 March 2009 (UTC) JA: No particular rush. I went looking for some of my old e-lectures where I worked up much more concrete examples, but they need to be converted from ASCII to HTML-TeX-Wiki formats to be readable at all, and that will take me a while. I will probably keep outlining the occasional notes here in the interval. Later, [[User:Jon Awbrey|Jon Awbrey]] 22:40, 1 March 2009 (UTC) ===Scope and Purpose of a Logic Wiki=== 2e15d3ce93813071a28ebc6515418d740b55c525 333 332 2009-03-01T22:42:27Z Jon Awbrey 3 /* Initial Discussion */ wikitext text/x-wiki This discussion page is intended for a discussion on using property theory as a means of organizing material in subject wikis, as well as the underlying issues of mathematics, logic, computation and philosophy. [[User:Vipul|Vipul]] 20:20, 25 February 2009 (UTC) ==Prospects for a Logic Wiki== ===Initial Discussion=== Previously on Buffer &hellip; * [http://blog.subwiki.org/?p=13 The History of Subject Wikis] * [http://blog.subwiki.org/?p=13&cpage=1#comment-2 Comments] <blockquote> <p>Vipul,</p> <p>From the initial sample of your writing distribution that I’ve read so far, your approach to mathematical objects by way of their properties is essentially a logical angle.</p> <p>If one elects to think in functional, computational terms from the very beginning, then it is convenient to think of a property <math>p\!</math> as being a mapping from a space <math>X,\!</math> the universe of discourse, to a space of 2 elements, say, <math>\mathbb{B} = \{ 0, 1 \}.</math> So a property is something of the form <math>p : X \to \mathbb{B}.</math> That level of consideration amounts to propositional calculus. Trivial as it may seem, it’s been my experience that getting efficient computational support at this level is key to many of the things we’d like to do at higher levels in the way of mathematical knowledge management.</p> <p>Well, enough for now, as I’m not even sure the above formatting will work here. Let’s look for a wiki discussion page where it will be easier to talk. Either at MyWikiBiz, one of your Subject Wikis, or if you like email archiving there is the Inquiry List that an e-friend set up for me.</p> <p>[http://blog.subwiki.org/?p=13&cpage=1#comment-6 Jon Awbrey, 24 Feb 2009, 20:42 UTC]</p> </blockquote> VN: Jon, I agree with your basic ideas. There's of course the issue that the space <math>X</math> under consideration is often ''not'' a ''set'' (it's too large to be one -- for instance, a [[Groupprops:group property|group property]] is defined as a ''function'' on the collection of all groups), so we cnanot simply treat it as a set map. Still, it behaves largely like a set map. And as you point out, properties behave a lot like propositions with the ''parameter'' being from the space <math>X</math>. At some point in time, I was interested in the mathematical aspects of properties. Properties form a Boolean algebra, inheriting conjunction and disjunction etc. (in set-theoretic jargon, this would just be the power set of <math>X</math>, except that <math>X</math> isn't a set). Then, we can use various partial binary operations on <math>X</math> to define corresponding binary operations on the property space. I believe this, too, has been explored by some people, who came up with structures such as quantales and gaggles. The broad idea, as far as I understand, is that the binary operation must distribute over disjunctions (ORs). Many of the operators that I discuss in the group theory wiki, such as the [[groupprops:composition operator for subgroup properties|composition operator for subgroup properties]] and the [[groupprops:join operator|join operator]], are quantalic operators. Some of the terminology that I use for some property-theoretic notation is borrowed from terminology used for such operators, for instance, the [[groupprops:left residual operator for composition|left residual]] and [[groupprops:right residual operator for composition|right residual]]. Some of the basic structural theorems and approaches are also taken from corresponding ideas that already existed in logic. (I arrived at some of them before becoming aware of the corresponding structures explored in logic, and later found that much of this had been done before). As far as I know, though, there has been no systematic application of the ideas developed in different parts of logic to this ''real world'' (?) application to understanding properties in different mathematical domains. This puzzled me since I think that using the ideas of logic and logical structure in this way can help achieve a better understanding of many disciplines, particularly those in mathematics and computer science. Perhaps it is because there are no deep theorems here? My own use of property theory as an organizational principle seems to rely very minimally on the logical ideas beneath and rests more on the practical results that are important within the specific subject. But perhaps there are deeper theoretical logic issues that should be explored. I'd be glad to hear about your perspective, logical, philosophical, or any other. I'd also like to hear more about your views on what you mean by supporting such a model computationally. On the group theory wiki, I'm using MediaWiki's basic tools as well as Semantic MediaWiki to help in property-theoretic exploration. This isn't ''smart'' in any sense on the machine's part, but it meets some of the practical needs. Nonetheless, if you have ideas for something more sophisticated, I'd be glad to hear. [[User:Vipul|Vipul]] 22:13, 25 February 2009 (UTC) JA: Sure, there is all that. My main interest at present is focused on using logic as a tool for practical knowledge management in mathematical domains. That means starting small, breaking off this or that chunk of suitably interesting or useful domains and representing them in such a way that you can not only get computers to help you reason about them but in such a way that human beings can grok what's going on. JA: Uh-oh, I'm being called to dinner, I will have to continue later tonight. [[User:Jon Awbrey|Jon Awbrey]] 23:08, 25 February 2009 (UTC) VN: Well, that's interesting. I'm myself interested in the practical aspects, too, though perhaps from a different angle. I'd like to hear more. I'll also go through some of your past writings to understand your perspective. If you have comments on the way things have been done on the subject wikis specifically, I'd like to hear those as well. [[User:Vipul|Vipul]] 00:00, 26 February 2009 (UTC) JA: Maybe it will be quickest just to list a few links. Most of the longer pieces at MyWikiBiz are unpublished papers and project workups of mine that I'm still in the process TeXing and Wikifying and redoing the graphics that got mushed up moving between different platforms over the years, so most of them have a point where I last left off reformatting or rewriting. At any rate, here are a few serving suggestions on the basic logic side of things: :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Propositional_Calculus Differential Propositional Calculus] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Functional_Logic_:_Quantification_Theory Functional Logic : Quantification Theory] JA: [[User:Jon Awbrey|Jon Awbrey]] 03:26, 26 February 2009 (UTC) VN: Jon, I've started looking at your work. Your language and approach are new to me, so it'll take me some time to go through it. Other work has also come up. I hope to get back to you in a couple of weeks. [[User:Vipul|Vipul]] 18:05, 1 March 2009 (UTC) JA: No particular rush. I went looking for some old e-lectures of mine where I worked up much more concrete examples, but they need to be converted from ASCII to HTML-TeX-Wiki formats to be readable at all, and that will take me a while. I will probably keep outlining the occasional notes here in the interval. Later, [[User:Jon Awbrey|Jon Awbrey]] 22:40, 1 March 2009 (UTC) ===Scope and Purpose of a Logic Wiki=== 380d17506a04cd679b0120b92cf9b9db6c0deece 334 333 2009-03-02T02:48:39Z Jon Awbrey 3 /* Scope and Purpose of a Logic Wiki */ + notes wikitext text/x-wiki This discussion page is intended for a discussion on using property theory as a means of organizing material in subject wikis, as well as the underlying issues of mathematics, logic, computation and philosophy. [[User:Vipul|Vipul]] 20:20, 25 February 2009 (UTC) ==Prospects for a Logic Wiki== ===Initial Discussion=== Previously on Buffer &hellip; * [http://blog.subwiki.org/?p=13 The History of Subject Wikis] * [http://blog.subwiki.org/?p=13&cpage=1#comment-2 Comments] <blockquote> <p>Vipul,</p> <p>From the initial sample of your writing distribution that I’ve read so far, your approach to mathematical objects by way of their properties is essentially a logical angle.</p> <p>If one elects to think in functional, computational terms from the very beginning, then it is convenient to think of a property <math>p\!</math> as being a mapping from a space <math>X,\!</math> the universe of discourse, to a space of 2 elements, say, <math>\mathbb{B} = \{ 0, 1 \}.</math> So a property is something of the form <math>p : X \to \mathbb{B}.</math> That level of consideration amounts to propositional calculus. Trivial as it may seem, it’s been my experience that getting efficient computational support at this level is key to many of the things we’d like to do at higher levels in the way of mathematical knowledge management.</p> <p>Well, enough for now, as I’m not even sure the above formatting will work here. Let’s look for a wiki discussion page where it will be easier to talk. Either at MyWikiBiz, one of your Subject Wikis, or if you like email archiving there is the Inquiry List that an e-friend set up for me.</p> <p>[http://blog.subwiki.org/?p=13&cpage=1#comment-6 Jon Awbrey, 24 Feb 2009, 20:42 UTC]</p> </blockquote> VN: Jon, I agree with your basic ideas. There's of course the issue that the space <math>X</math> under consideration is often ''not'' a ''set'' (it's too large to be one -- for instance, a [[Groupprops:group property|group property]] is defined as a ''function'' on the collection of all groups), so we cnanot simply treat it as a set map. Still, it behaves largely like a set map. And as you point out, properties behave a lot like propositions with the ''parameter'' being from the space <math>X</math>. At some point in time, I was interested in the mathematical aspects of properties. Properties form a Boolean algebra, inheriting conjunction and disjunction etc. (in set-theoretic jargon, this would just be the power set of <math>X</math>, except that <math>X</math> isn't a set). Then, we can use various partial binary operations on <math>X</math> to define corresponding binary operations on the property space. I believe this, too, has been explored by some people, who came up with structures such as quantales and gaggles. The broad idea, as far as I understand, is that the binary operation must distribute over disjunctions (ORs). Many of the operators that I discuss in the group theory wiki, such as the [[groupprops:composition operator for subgroup properties|composition operator for subgroup properties]] and the [[groupprops:join operator|join operator]], are quantalic operators. Some of the terminology that I use for some property-theoretic notation is borrowed from terminology used for such operators, for instance, the [[groupprops:left residual operator for composition|left residual]] and [[groupprops:right residual operator for composition|right residual]]. Some of the basic structural theorems and approaches are also taken from corresponding ideas that already existed in logic. (I arrived at some of them before becoming aware of the corresponding structures explored in logic, and later found that much of this had been done before). As far as I know, though, there has been no systematic application of the ideas developed in different parts of logic to this ''real world'' (?) application to understanding properties in different mathematical domains. This puzzled me since I think that using the ideas of logic and logical structure in this way can help achieve a better understanding of many disciplines, particularly those in mathematics and computer science. Perhaps it is because there are no deep theorems here? My own use of property theory as an organizational principle seems to rely very minimally on the logical ideas beneath and rests more on the practical results that are important within the specific subject. But perhaps there are deeper theoretical logic issues that should be explored. I'd be glad to hear about your perspective, logical, philosophical, or any other. I'd also like to hear more about your views on what you mean by supporting such a model computationally. On the group theory wiki, I'm using MediaWiki's basic tools as well as Semantic MediaWiki to help in property-theoretic exploration. This isn't ''smart'' in any sense on the machine's part, but it meets some of the practical needs. Nonetheless, if you have ideas for something more sophisticated, I'd be glad to hear. [[User:Vipul|Vipul]] 22:13, 25 February 2009 (UTC) JA: Sure, there is all that. My main interest at present is focused on using logic as a tool for practical knowledge management in mathematical domains. That means starting small, breaking off this or that chunk of suitably interesting or useful domains and representing them in such a way that you can not only get computers to help you reason about them but in such a way that human beings can grok what's going on. JA: Uh-oh, I'm being called to dinner, I will have to continue later tonight. [[User:Jon Awbrey|Jon Awbrey]] 23:08, 25 February 2009 (UTC) VN: Well, that's interesting. I'm myself interested in the practical aspects, too, though perhaps from a different angle. I'd like to hear more. I'll also go through some of your past writings to understand your perspective. If you have comments on the way things have been done on the subject wikis specifically, I'd like to hear those as well. [[User:Vipul|Vipul]] 00:00, 26 February 2009 (UTC) JA: Maybe it will be quickest just to list a few links. Most of the longer pieces at MyWikiBiz are unpublished papers and project workups of mine that I'm still in the process TeXing and Wikifying and redoing the graphics that got mushed up moving between different platforms over the years, so most of them have a point where I last left off reformatting or rewriting. At any rate, here are a few serving suggestions on the basic logic side of things: :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Propositional_Calculus Differential Propositional Calculus] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Functional_Logic_:_Quantification_Theory Functional Logic : Quantification Theory] JA: [[User:Jon Awbrey|Jon Awbrey]] 03:26, 26 February 2009 (UTC) VN: Jon, I've started looking at your work. Your language and approach are new to me, so it'll take me some time to go through it. Other work has also come up. I hope to get back to you in a couple of weeks. [[User:Vipul|Vipul]] 18:05, 1 March 2009 (UTC) JA: No particular rush. I went looking for some old e-lectures of mine where I worked up much more concrete examples, but they need to be converted from ASCII to HTML-TeX-Wiki formats to be readable at all, and that will take me a while. I will probably keep outlining the occasional notes here in the interval. Later, [[User:Jon Awbrey|Jon Awbrey]] 22:40, 1 March 2009 (UTC) ===Scope and Purpose of a Logic Wiki=== * Objectives ** Maintain a focus on active methods &mdash; for example, computational tools, heuristics, and proofs &mdash; that is parallel to the focus on static content. ca3c15d79662caf2b74692d273e6ab8313968af2 335 334 2009-03-02T02:50:41Z Jon Awbrey 3 /* Scope and Purpose of a Logic Wiki */ wikitext text/x-wiki This discussion page is intended for a discussion on using property theory as a means of organizing material in subject wikis, as well as the underlying issues of mathematics, logic, computation and philosophy. [[User:Vipul|Vipul]] 20:20, 25 February 2009 (UTC) ==Prospects for a Logic Wiki== ===Initial Discussion=== Previously on Buffer &hellip; * [http://blog.subwiki.org/?p=13 The History of Subject Wikis] * [http://blog.subwiki.org/?p=13&cpage=1#comment-2 Comments] <blockquote> <p>Vipul,</p> <p>From the initial sample of your writing distribution that I’ve read so far, your approach to mathematical objects by way of their properties is essentially a logical angle.</p> <p>If one elects to think in functional, computational terms from the very beginning, then it is convenient to think of a property <math>p\!</math> as being a mapping from a space <math>X,\!</math> the universe of discourse, to a space of 2 elements, say, <math>\mathbb{B} = \{ 0, 1 \}.</math> So a property is something of the form <math>p : X \to \mathbb{B}.</math> That level of consideration amounts to propositional calculus. Trivial as it may seem, it’s been my experience that getting efficient computational support at this level is key to many of the things we’d like to do at higher levels in the way of mathematical knowledge management.</p> <p>Well, enough for now, as I’m not even sure the above formatting will work here. Let’s look for a wiki discussion page where it will be easier to talk. Either at MyWikiBiz, one of your Subject Wikis, or if you like email archiving there is the Inquiry List that an e-friend set up for me.</p> <p>[http://blog.subwiki.org/?p=13&cpage=1#comment-6 Jon Awbrey, 24 Feb 2009, 20:42 UTC]</p> </blockquote> VN: Jon, I agree with your basic ideas. There's of course the issue that the space <math>X</math> under consideration is often ''not'' a ''set'' (it's too large to be one -- for instance, a [[Groupprops:group property|group property]] is defined as a ''function'' on the collection of all groups), so we cnanot simply treat it as a set map. Still, it behaves largely like a set map. And as you point out, properties behave a lot like propositions with the ''parameter'' being from the space <math>X</math>. At some point in time, I was interested in the mathematical aspects of properties. Properties form a Boolean algebra, inheriting conjunction and disjunction etc. (in set-theoretic jargon, this would just be the power set of <math>X</math>, except that <math>X</math> isn't a set). Then, we can use various partial binary operations on <math>X</math> to define corresponding binary operations on the property space. I believe this, too, has been explored by some people, who came up with structures such as quantales and gaggles. The broad idea, as far as I understand, is that the binary operation must distribute over disjunctions (ORs). Many of the operators that I discuss in the group theory wiki, such as the [[groupprops:composition operator for subgroup properties|composition operator for subgroup properties]] and the [[groupprops:join operator|join operator]], are quantalic operators. Some of the terminology that I use for some property-theoretic notation is borrowed from terminology used for such operators, for instance, the [[groupprops:left residual operator for composition|left residual]] and [[groupprops:right residual operator for composition|right residual]]. Some of the basic structural theorems and approaches are also taken from corresponding ideas that already existed in logic. (I arrived at some of them before becoming aware of the corresponding structures explored in logic, and later found that much of this had been done before). As far as I know, though, there has been no systematic application of the ideas developed in different parts of logic to this ''real world'' (?) application to understanding properties in different mathematical domains. This puzzled me since I think that using the ideas of logic and logical structure in this way can help achieve a better understanding of many disciplines, particularly those in mathematics and computer science. Perhaps it is because there are no deep theorems here? My own use of property theory as an organizational principle seems to rely very minimally on the logical ideas beneath and rests more on the practical results that are important within the specific subject. But perhaps there are deeper theoretical logic issues that should be explored. I'd be glad to hear about your perspective, logical, philosophical, or any other. I'd also like to hear more about your views on what you mean by supporting such a model computationally. On the group theory wiki, I'm using MediaWiki's basic tools as well as Semantic MediaWiki to help in property-theoretic exploration. This isn't ''smart'' in any sense on the machine's part, but it meets some of the practical needs. Nonetheless, if you have ideas for something more sophisticated, I'd be glad to hear. [[User:Vipul|Vipul]] 22:13, 25 February 2009 (UTC) JA: Sure, there is all that. My main interest at present is focused on using logic as a tool for practical knowledge management in mathematical domains. That means starting small, breaking off this or that chunk of suitably interesting or useful domains and representing them in such a way that you can not only get computers to help you reason about them but in such a way that human beings can grok what's going on. JA: Uh-oh, I'm being called to dinner, I will have to continue later tonight. [[User:Jon Awbrey|Jon Awbrey]] 23:08, 25 February 2009 (UTC) VN: Well, that's interesting. I'm myself interested in the practical aspects, too, though perhaps from a different angle. I'd like to hear more. I'll also go through some of your past writings to understand your perspective. If you have comments on the way things have been done on the subject wikis specifically, I'd like to hear those as well. [[User:Vipul|Vipul]] 00:00, 26 February 2009 (UTC) JA: Maybe it will be quickest just to list a few links. Most of the longer pieces at MyWikiBiz are unpublished papers and project workups of mine that I'm still in the process TeXing and Wikifying and redoing the graphics that got mushed up moving between different platforms over the years, so most of them have a point where I last left off reformatting or rewriting. At any rate, here are a few serving suggestions on the basic logic side of things: :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Propositional_Calculus Differential Propositional Calculus] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Functional_Logic_:_Quantification_Theory Functional Logic : Quantification Theory] JA: [[User:Jon Awbrey|Jon Awbrey]] 03:26, 26 February 2009 (UTC) VN: Jon, I've started looking at your work. Your language and approach are new to me, so it'll take me some time to go through it. Other work has also come up. I hope to get back to you in a couple of weeks. [[User:Vipul|Vipul]] 18:05, 1 March 2009 (UTC) JA: No particular rush. I went looking for some old e-lectures of mine where I worked up much more concrete examples, but they need to be converted from ASCII to HTML-TeX-Wiki formats to be readable at all, and that will take me a while. I will probably keep outlining the occasional notes here in the interval. Later, [[User:Jon Awbrey|Jon Awbrey]] 22:40, 1 March 2009 (UTC) ===Scope and Purpose of a Logic Wiki=== ; Objectives : Maintain a focus on active methods &mdash; computational tools, heuristics, and proofs &mdash; that is parallel to the focus on static content. 4747312d0acc9a9c93865a53632a73e4560f1aa8 336 335 2009-03-02T03:36:18Z Jon Awbrey 3 /* Scope and Purpose of a Logic Wiki */ wikitext text/x-wiki This discussion page is intended for a discussion on using property theory as a means of organizing material in subject wikis, as well as the underlying issues of mathematics, logic, computation and philosophy. [[User:Vipul|Vipul]] 20:20, 25 February 2009 (UTC) ==Prospects for a Logic Wiki== ===Initial Discussion=== Previously on Buffer &hellip; * [http://blog.subwiki.org/?p=13 The History of Subject Wikis] * [http://blog.subwiki.org/?p=13&cpage=1#comment-2 Comments] <blockquote> <p>Vipul,</p> <p>From the initial sample of your writing distribution that I’ve read so far, your approach to mathematical objects by way of their properties is essentially a logical angle.</p> <p>If one elects to think in functional, computational terms from the very beginning, then it is convenient to think of a property <math>p\!</math> as being a mapping from a space <math>X,\!</math> the universe of discourse, to a space of 2 elements, say, <math>\mathbb{B} = \{ 0, 1 \}.</math> So a property is something of the form <math>p : X \to \mathbb{B}.</math> That level of consideration amounts to propositional calculus. Trivial as it may seem, it’s been my experience that getting efficient computational support at this level is key to many of the things we’d like to do at higher levels in the way of mathematical knowledge management.</p> <p>Well, enough for now, as I’m not even sure the above formatting will work here. Let’s look for a wiki discussion page where it will be easier to talk. Either at MyWikiBiz, one of your Subject Wikis, or if you like email archiving there is the Inquiry List that an e-friend set up for me.</p> <p>[http://blog.subwiki.org/?p=13&cpage=1#comment-6 Jon Awbrey, 24 Feb 2009, 20:42 UTC]</p> </blockquote> VN: Jon, I agree with your basic ideas. There's of course the issue that the space <math>X</math> under consideration is often ''not'' a ''set'' (it's too large to be one -- for instance, a [[Groupprops:group property|group property]] is defined as a ''function'' on the collection of all groups), so we cnanot simply treat it as a set map. Still, it behaves largely like a set map. And as you point out, properties behave a lot like propositions with the ''parameter'' being from the space <math>X</math>. At some point in time, I was interested in the mathematical aspects of properties. Properties form a Boolean algebra, inheriting conjunction and disjunction etc. (in set-theoretic jargon, this would just be the power set of <math>X</math>, except that <math>X</math> isn't a set). Then, we can use various partial binary operations on <math>X</math> to define corresponding binary operations on the property space. I believe this, too, has been explored by some people, who came up with structures such as quantales and gaggles. The broad idea, as far as I understand, is that the binary operation must distribute over disjunctions (ORs). Many of the operators that I discuss in the group theory wiki, such as the [[groupprops:composition operator for subgroup properties|composition operator for subgroup properties]] and the [[groupprops:join operator|join operator]], are quantalic operators. Some of the terminology that I use for some property-theoretic notation is borrowed from terminology used for such operators, for instance, the [[groupprops:left residual operator for composition|left residual]] and [[groupprops:right residual operator for composition|right residual]]. Some of the basic structural theorems and approaches are also taken from corresponding ideas that already existed in logic. (I arrived at some of them before becoming aware of the corresponding structures explored in logic, and later found that much of this had been done before). As far as I know, though, there has been no systematic application of the ideas developed in different parts of logic to this ''real world'' (?) application to understanding properties in different mathematical domains. This puzzled me since I think that using the ideas of logic and logical structure in this way can help achieve a better understanding of many disciplines, particularly those in mathematics and computer science. Perhaps it is because there are no deep theorems here? My own use of property theory as an organizational principle seems to rely very minimally on the logical ideas beneath and rests more on the practical results that are important within the specific subject. But perhaps there are deeper theoretical logic issues that should be explored. I'd be glad to hear about your perspective, logical, philosophical, or any other. I'd also like to hear more about your views on what you mean by supporting such a model computationally. On the group theory wiki, I'm using MediaWiki's basic tools as well as Semantic MediaWiki to help in property-theoretic exploration. This isn't ''smart'' in any sense on the machine's part, but it meets some of the practical needs. Nonetheless, if you have ideas for something more sophisticated, I'd be glad to hear. [[User:Vipul|Vipul]] 22:13, 25 February 2009 (UTC) JA: Sure, there is all that. My main interest at present is focused on using logic as a tool for practical knowledge management in mathematical domains. That means starting small, breaking off this or that chunk of suitably interesting or useful domains and representing them in such a way that you can not only get computers to help you reason about them but in such a way that human beings can grok what's going on. JA: Uh-oh, I'm being called to dinner, I will have to continue later tonight. [[User:Jon Awbrey|Jon Awbrey]] 23:08, 25 February 2009 (UTC) VN: Well, that's interesting. I'm myself interested in the practical aspects, too, though perhaps from a different angle. I'd like to hear more. I'll also go through some of your past writings to understand your perspective. If you have comments on the way things have been done on the subject wikis specifically, I'd like to hear those as well. [[User:Vipul|Vipul]] 00:00, 26 February 2009 (UTC) JA: Maybe it will be quickest just to list a few links. Most of the longer pieces at MyWikiBiz are unpublished papers and project workups of mine that I'm still in the process TeXing and Wikifying and redoing the graphics that got mushed up moving between different platforms over the years, so most of them have a point where I last left off reformatting or rewriting. At any rate, here are a few serving suggestions on the basic logic side of things: :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Propositional_Calculus Differential Propositional Calculus] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Functional_Logic_:_Quantification_Theory Functional Logic : Quantification Theory] JA: [[User:Jon Awbrey|Jon Awbrey]] 03:26, 26 February 2009 (UTC) VN: Jon, I've started looking at your work. Your language and approach are new to me, so it'll take me some time to go through it. Other work has also come up. I hope to get back to you in a couple of weeks. [[User:Vipul|Vipul]] 18:05, 1 March 2009 (UTC) JA: No particular rush. I went looking for some old e-lectures of mine where I worked up much more concrete examples, but they need to be converted from ASCII to HTML-TeX-Wiki formats to be readable at all, and that will take me a while. I will probably keep outlining the occasional notes here in the interval. Later, [[User:Jon Awbrey|Jon Awbrey]] 22:40, 1 March 2009 (UTC) ===Scope and Purpose of a Logic Wiki=== ; Objectives : Maintain a focus of active methods that is parallel to the focus on static content. :: Algorithms :: Computational tools :: Efficient Calculi :: Heuristics :: Proofs 08968258a39e047652011f2a630d0cdfb0797a09 337 336 2009-03-02T03:38:06Z Jon Awbrey 3 /* Scope and Purpose of a Logic Wiki */ wikitext text/x-wiki This discussion page is intended for a discussion on using property theory as a means of organizing material in subject wikis, as well as the underlying issues of mathematics, logic, computation and philosophy. [[User:Vipul|Vipul]] 20:20, 25 February 2009 (UTC) ==Prospects for a Logic Wiki== ===Initial Discussion=== Previously on Buffer &hellip; * [http://blog.subwiki.org/?p=13 The History of Subject Wikis] * [http://blog.subwiki.org/?p=13&cpage=1#comment-2 Comments] <blockquote> <p>Vipul,</p> <p>From the initial sample of your writing distribution that I’ve read so far, your approach to mathematical objects by way of their properties is essentially a logical angle.</p> <p>If one elects to think in functional, computational terms from the very beginning, then it is convenient to think of a property <math>p\!</math> as being a mapping from a space <math>X,\!</math> the universe of discourse, to a space of 2 elements, say, <math>\mathbb{B} = \{ 0, 1 \}.</math> So a property is something of the form <math>p : X \to \mathbb{B}.</math> That level of consideration amounts to propositional calculus. Trivial as it may seem, it’s been my experience that getting efficient computational support at this level is key to many of the things we’d like to do at higher levels in the way of mathematical knowledge management.</p> <p>Well, enough for now, as I’m not even sure the above formatting will work here. Let’s look for a wiki discussion page where it will be easier to talk. Either at MyWikiBiz, one of your Subject Wikis, or if you like email archiving there is the Inquiry List that an e-friend set up for me.</p> <p>[http://blog.subwiki.org/?p=13&cpage=1#comment-6 Jon Awbrey, 24 Feb 2009, 20:42 UTC]</p> </blockquote> VN: Jon, I agree with your basic ideas. There's of course the issue that the space <math>X</math> under consideration is often ''not'' a ''set'' (it's too large to be one -- for instance, a [[Groupprops:group property|group property]] is defined as a ''function'' on the collection of all groups), so we cnanot simply treat it as a set map. Still, it behaves largely like a set map. And as you point out, properties behave a lot like propositions with the ''parameter'' being from the space <math>X</math>. At some point in time, I was interested in the mathematical aspects of properties. Properties form a Boolean algebra, inheriting conjunction and disjunction etc. (in set-theoretic jargon, this would just be the power set of <math>X</math>, except that <math>X</math> isn't a set). Then, we can use various partial binary operations on <math>X</math> to define corresponding binary operations on the property space. I believe this, too, has been explored by some people, who came up with structures such as quantales and gaggles. The broad idea, as far as I understand, is that the binary operation must distribute over disjunctions (ORs). Many of the operators that I discuss in the group theory wiki, such as the [[groupprops:composition operator for subgroup properties|composition operator for subgroup properties]] and the [[groupprops:join operator|join operator]], are quantalic operators. Some of the terminology that I use for some property-theoretic notation is borrowed from terminology used for such operators, for instance, the [[groupprops:left residual operator for composition|left residual]] and [[groupprops:right residual operator for composition|right residual]]. Some of the basic structural theorems and approaches are also taken from corresponding ideas that already existed in logic. (I arrived at some of them before becoming aware of the corresponding structures explored in logic, and later found that much of this had been done before). As far as I know, though, there has been no systematic application of the ideas developed in different parts of logic to this ''real world'' (?) application to understanding properties in different mathematical domains. This puzzled me since I think that using the ideas of logic and logical structure in this way can help achieve a better understanding of many disciplines, particularly those in mathematics and computer science. Perhaps it is because there are no deep theorems here? My own use of property theory as an organizational principle seems to rely very minimally on the logical ideas beneath and rests more on the practical results that are important within the specific subject. But perhaps there are deeper theoretical logic issues that should be explored. I'd be glad to hear about your perspective, logical, philosophical, or any other. I'd also like to hear more about your views on what you mean by supporting such a model computationally. On the group theory wiki, I'm using MediaWiki's basic tools as well as Semantic MediaWiki to help in property-theoretic exploration. This isn't ''smart'' in any sense on the machine's part, but it meets some of the practical needs. Nonetheless, if you have ideas for something more sophisticated, I'd be glad to hear. [[User:Vipul|Vipul]] 22:13, 25 February 2009 (UTC) JA: Sure, there is all that. My main interest at present is focused on using logic as a tool for practical knowledge management in mathematical domains. That means starting small, breaking off this or that chunk of suitably interesting or useful domains and representing them in such a way that you can not only get computers to help you reason about them but in such a way that human beings can grok what's going on. JA: Uh-oh, I'm being called to dinner, I will have to continue later tonight. [[User:Jon Awbrey|Jon Awbrey]] 23:08, 25 February 2009 (UTC) VN: Well, that's interesting. I'm myself interested in the practical aspects, too, though perhaps from a different angle. I'd like to hear more. I'll also go through some of your past writings to understand your perspective. If you have comments on the way things have been done on the subject wikis specifically, I'd like to hear those as well. [[User:Vipul|Vipul]] 00:00, 26 February 2009 (UTC) JA: Maybe it will be quickest just to list a few links. Most of the longer pieces at MyWikiBiz are unpublished papers and project workups of mine that I'm still in the process TeXing and Wikifying and redoing the graphics that got mushed up moving between different platforms over the years, so most of them have a point where I last left off reformatting or rewriting. At any rate, here are a few serving suggestions on the basic logic side of things: :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Propositional_Calculus Differential Propositional Calculus] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Functional_Logic_:_Quantification_Theory Functional Logic : Quantification Theory] JA: [[User:Jon Awbrey|Jon Awbrey]] 03:26, 26 February 2009 (UTC) VN: Jon, I've started looking at your work. Your language and approach are new to me, so it'll take me some time to go through it. Other work has also come up. I hope to get back to you in a couple of weeks. [[User:Vipul|Vipul]] 18:05, 1 March 2009 (UTC) JA: No particular rush. I went looking for some old e-lectures of mine where I worked up much more concrete examples, but they need to be converted from ASCII to HTML-TeX-Wiki formats to be readable at all, and that will take me a while. I will probably keep outlining the occasional notes here in the interval. Later, [[User:Jon Awbrey|Jon Awbrey]] 22:40, 1 March 2009 (UTC) ===Scope and Purpose of a Logic Wiki=== ; Objectives : Maintain a focus of active methods that is parallel to the focus on static content. :: Algorithms :: Computational tools :: Efficient calculi :: Heuristics :: Proofs 17a5e6ea407c33c0e8516fceadc47c865f2bdafc 338 337 2009-03-05T17:12:55Z Jon Awbrey 3 /* Scope and Purpose of a Logic Wiki */ + link wikitext text/x-wiki This discussion page is intended for a discussion on using property theory as a means of organizing material in subject wikis, as well as the underlying issues of mathematics, logic, computation and philosophy. [[User:Vipul|Vipul]] 20:20, 25 February 2009 (UTC) ==Prospects for a Logic Wiki== ===Initial Discussion=== Previously on Buffer &hellip; * [http://blog.subwiki.org/?p=13 The History of Subject Wikis] * [http://blog.subwiki.org/?p=13&cpage=1#comment-2 Comments] <blockquote> <p>Vipul,</p> <p>From the initial sample of your writing distribution that I’ve read so far, your approach to mathematical objects by way of their properties is essentially a logical angle.</p> <p>If one elects to think in functional, computational terms from the very beginning, then it is convenient to think of a property <math>p\!</math> as being a mapping from a space <math>X,\!</math> the universe of discourse, to a space of 2 elements, say, <math>\mathbb{B} = \{ 0, 1 \}.</math> So a property is something of the form <math>p : X \to \mathbb{B}.</math> That level of consideration amounts to propositional calculus. Trivial as it may seem, it’s been my experience that getting efficient computational support at this level is key to many of the things we’d like to do at higher levels in the way of mathematical knowledge management.</p> <p>Well, enough for now, as I’m not even sure the above formatting will work here. Let’s look for a wiki discussion page where it will be easier to talk. Either at MyWikiBiz, one of your Subject Wikis, or if you like email archiving there is the Inquiry List that an e-friend set up for me.</p> <p>[http://blog.subwiki.org/?p=13&cpage=1#comment-6 Jon Awbrey, 24 Feb 2009, 20:42 UTC]</p> </blockquote> VN: Jon, I agree with your basic ideas. There's of course the issue that the space <math>X</math> under consideration is often ''not'' a ''set'' (it's too large to be one -- for instance, a [[Groupprops:group property|group property]] is defined as a ''function'' on the collection of all groups), so we cnanot simply treat it as a set map. Still, it behaves largely like a set map. And as you point out, properties behave a lot like propositions with the ''parameter'' being from the space <math>X</math>. At some point in time, I was interested in the mathematical aspects of properties. Properties form a Boolean algebra, inheriting conjunction and disjunction etc. (in set-theoretic jargon, this would just be the power set of <math>X</math>, except that <math>X</math> isn't a set). Then, we can use various partial binary operations on <math>X</math> to define corresponding binary operations on the property space. I believe this, too, has been explored by some people, who came up with structures such as quantales and gaggles. The broad idea, as far as I understand, is that the binary operation must distribute over disjunctions (ORs). Many of the operators that I discuss in the group theory wiki, such as the [[groupprops:composition operator for subgroup properties|composition operator for subgroup properties]] and the [[groupprops:join operator|join operator]], are quantalic operators. Some of the terminology that I use for some property-theoretic notation is borrowed from terminology used for such operators, for instance, the [[groupprops:left residual operator for composition|left residual]] and [[groupprops:right residual operator for composition|right residual]]. Some of the basic structural theorems and approaches are also taken from corresponding ideas that already existed in logic. (I arrived at some of them before becoming aware of the corresponding structures explored in logic, and later found that much of this had been done before). As far as I know, though, there has been no systematic application of the ideas developed in different parts of logic to this ''real world'' (?) application to understanding properties in different mathematical domains. This puzzled me since I think that using the ideas of logic and logical structure in this way can help achieve a better understanding of many disciplines, particularly those in mathematics and computer science. Perhaps it is because there are no deep theorems here? My own use of property theory as an organizational principle seems to rely very minimally on the logical ideas beneath and rests more on the practical results that are important within the specific subject. But perhaps there are deeper theoretical logic issues that should be explored. I'd be glad to hear about your perspective, logical, philosophical, or any other. I'd also like to hear more about your views on what you mean by supporting such a model computationally. On the group theory wiki, I'm using MediaWiki's basic tools as well as Semantic MediaWiki to help in property-theoretic exploration. This isn't ''smart'' in any sense on the machine's part, but it meets some of the practical needs. Nonetheless, if you have ideas for something more sophisticated, I'd be glad to hear. [[User:Vipul|Vipul]] 22:13, 25 February 2009 (UTC) JA: Sure, there is all that. My main interest at present is focused on using logic as a tool for practical knowledge management in mathematical domains. That means starting small, breaking off this or that chunk of suitably interesting or useful domains and representing them in such a way that you can not only get computers to help you reason about them but in such a way that human beings can grok what's going on. JA: Uh-oh, I'm being called to dinner, I will have to continue later tonight. [[User:Jon Awbrey|Jon Awbrey]] 23:08, 25 February 2009 (UTC) VN: Well, that's interesting. I'm myself interested in the practical aspects, too, though perhaps from a different angle. I'd like to hear more. I'll also go through some of your past writings to understand your perspective. If you have comments on the way things have been done on the subject wikis specifically, I'd like to hear those as well. [[User:Vipul|Vipul]] 00:00, 26 February 2009 (UTC) JA: Maybe it will be quickest just to list a few links. Most of the longer pieces at MyWikiBiz are unpublished papers and project workups of mine that I'm still in the process TeXing and Wikifying and redoing the graphics that got mushed up moving between different platforms over the years, so most of them have a point where I last left off reformatting or rewriting. At any rate, here are a few serving suggestions on the basic logic side of things: :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Propositional_Calculus Differential Propositional Calculus] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Functional_Logic_:_Quantification_Theory Functional Logic : Quantification Theory] JA: [[User:Jon Awbrey|Jon Awbrey]] 03:26, 26 February 2009 (UTC) VN: Jon, I've started looking at your work. Your language and approach are new to me, so it'll take me some time to go through it. Other work has also come up. I hope to get back to you in a couple of weeks. [[User:Vipul|Vipul]] 18:05, 1 March 2009 (UTC) JA: No particular rush. I went looking for some old e-lectures of mine where I worked up much more concrete examples, but they need to be converted from ASCII to HTML-TeX-Wiki formats to be readable at all, and that will take me a while. I will probably keep outlining the occasional notes here in the interval. Later, [[User:Jon Awbrey|Jon Awbrey]] 22:40, 1 March 2009 (UTC) ===Scope and Purpose of a Logic Wiki=== ; Objectives : Maintain a focus of active methods that is parallel to the focus on static content. :: Algorithms :: Computational tools :: Efficient calculi :: Heuristics :: Proofs : Here's a project that looks interesting in this connection &mdash; :: [http://www.mat.univie.ac.at/~neum/FMathL.html Arnold Neumaier : Automated Mathematical Research Assistant] c3067124dfd53f8d818c0185c4d426c02ad507d8 339 338 2009-03-07T15:57:00Z Vipul 2 /* Scope and Purpose of a Logic Wiki */ wikitext text/x-wiki This discussion page is intended for a discussion on using property theory as a means of organizing material in subject wikis, as well as the underlying issues of mathematics, logic, computation and philosophy. [[User:Vipul|Vipul]] 20:20, 25 February 2009 (UTC) ==Prospects for a Logic Wiki== ===Initial Discussion=== Previously on Buffer &hellip; * [http://blog.subwiki.org/?p=13 The History of Subject Wikis] * [http://blog.subwiki.org/?p=13&cpage=1#comment-2 Comments] <blockquote> <p>Vipul,</p> <p>From the initial sample of your writing distribution that I’ve read so far, your approach to mathematical objects by way of their properties is essentially a logical angle.</p> <p>If one elects to think in functional, computational terms from the very beginning, then it is convenient to think of a property <math>p\!</math> as being a mapping from a space <math>X,\!</math> the universe of discourse, to a space of 2 elements, say, <math>\mathbb{B} = \{ 0, 1 \}.</math> So a property is something of the form <math>p : X \to \mathbb{B}.</math> That level of consideration amounts to propositional calculus. Trivial as it may seem, it’s been my experience that getting efficient computational support at this level is key to many of the things we’d like to do at higher levels in the way of mathematical knowledge management.</p> <p>Well, enough for now, as I’m not even sure the above formatting will work here. Let’s look for a wiki discussion page where it will be easier to talk. Either at MyWikiBiz, one of your Subject Wikis, or if you like email archiving there is the Inquiry List that an e-friend set up for me.</p> <p>[http://blog.subwiki.org/?p=13&cpage=1#comment-6 Jon Awbrey, 24 Feb 2009, 20:42 UTC]</p> </blockquote> VN: Jon, I agree with your basic ideas. There's of course the issue that the space <math>X</math> under consideration is often ''not'' a ''set'' (it's too large to be one -- for instance, a [[Groupprops:group property|group property]] is defined as a ''function'' on the collection of all groups), so we cnanot simply treat it as a set map. Still, it behaves largely like a set map. And as you point out, properties behave a lot like propositions with the ''parameter'' being from the space <math>X</math>. At some point in time, I was interested in the mathematical aspects of properties. Properties form a Boolean algebra, inheriting conjunction and disjunction etc. (in set-theoretic jargon, this would just be the power set of <math>X</math>, except that <math>X</math> isn't a set). Then, we can use various partial binary operations on <math>X</math> to define corresponding binary operations on the property space. I believe this, too, has been explored by some people, who came up with structures such as quantales and gaggles. The broad idea, as far as I understand, is that the binary operation must distribute over disjunctions (ORs). Many of the operators that I discuss in the group theory wiki, such as the [[groupprops:composition operator for subgroup properties|composition operator for subgroup properties]] and the [[groupprops:join operator|join operator]], are quantalic operators. Some of the terminology that I use for some property-theoretic notation is borrowed from terminology used for such operators, for instance, the [[groupprops:left residual operator for composition|left residual]] and [[groupprops:right residual operator for composition|right residual]]. Some of the basic structural theorems and approaches are also taken from corresponding ideas that already existed in logic. (I arrived at some of them before becoming aware of the corresponding structures explored in logic, and later found that much of this had been done before). As far as I know, though, there has been no systematic application of the ideas developed in different parts of logic to this ''real world'' (?) application to understanding properties in different mathematical domains. This puzzled me since I think that using the ideas of logic and logical structure in this way can help achieve a better understanding of many disciplines, particularly those in mathematics and computer science. Perhaps it is because there are no deep theorems here? My own use of property theory as an organizational principle seems to rely very minimally on the logical ideas beneath and rests more on the practical results that are important within the specific subject. But perhaps there are deeper theoretical logic issues that should be explored. I'd be glad to hear about your perspective, logical, philosophical, or any other. I'd also like to hear more about your views on what you mean by supporting such a model computationally. On the group theory wiki, I'm using MediaWiki's basic tools as well as Semantic MediaWiki to help in property-theoretic exploration. This isn't ''smart'' in any sense on the machine's part, but it meets some of the practical needs. Nonetheless, if you have ideas for something more sophisticated, I'd be glad to hear. [[User:Vipul|Vipul]] 22:13, 25 February 2009 (UTC) JA: Sure, there is all that. My main interest at present is focused on using logic as a tool for practical knowledge management in mathematical domains. That means starting small, breaking off this or that chunk of suitably interesting or useful domains and representing them in such a way that you can not only get computers to help you reason about them but in such a way that human beings can grok what's going on. JA: Uh-oh, I'm being called to dinner, I will have to continue later tonight. [[User:Jon Awbrey|Jon Awbrey]] 23:08, 25 February 2009 (UTC) VN: Well, that's interesting. I'm myself interested in the practical aspects, too, though perhaps from a different angle. I'd like to hear more. I'll also go through some of your past writings to understand your perspective. If you have comments on the way things have been done on the subject wikis specifically, I'd like to hear those as well. [[User:Vipul|Vipul]] 00:00, 26 February 2009 (UTC) JA: Maybe it will be quickest just to list a few links. Most of the longer pieces at MyWikiBiz are unpublished papers and project workups of mine that I'm still in the process TeXing and Wikifying and redoing the graphics that got mushed up moving between different platforms over the years, so most of them have a point where I last left off reformatting or rewriting. At any rate, here are a few serving suggestions on the basic logic side of things: :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Propositional_Calculus Differential Propositional Calculus] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Functional_Logic_:_Quantification_Theory Functional Logic : Quantification Theory] JA: [[User:Jon Awbrey|Jon Awbrey]] 03:26, 26 February 2009 (UTC) VN: Jon, I've started looking at your work. Your language and approach are new to me, so it'll take me some time to go through it. Other work has also come up. I hope to get back to you in a couple of weeks. [[User:Vipul|Vipul]] 18:05, 1 March 2009 (UTC) JA: No particular rush. I went looking for some old e-lectures of mine where I worked up much more concrete examples, but they need to be converted from ASCII to HTML-TeX-Wiki formats to be readable at all, and that will take me a while. I will probably keep outlining the occasional notes here in the interval. Later, [[User:Jon Awbrey|Jon Awbrey]] 22:40, 1 March 2009 (UTC) ===Scope and Purpose of a Logic Wiki=== ; Objectives : Maintain a focus of active methods that is parallel to the focus on static content. :: Algorithms :: Computational tools :: Efficient calculi :: Heuristics :: Proofs : Here's a project that looks interesting in this connection &mdash; :: [http://www.mat.univie.ac.at/~neum/FMathL.html Arnold Neumaier : Automated Mathematical Research Assistant] : VN: Jon, that certainly looks interesting. Thanks for the link. [[User:Vipul|Vipul]] 15:57, 7 March 2009 (UTC) 0c60a9395908797a3dfb2d600cca5f0c9c9476ad 340 339 2009-03-11T23:54:25Z Vipul 2 /* Scope and Purpose of a Logic Wiki */ wikitext text/x-wiki This discussion page is intended for a discussion on using property theory as a means of organizing material in subject wikis, as well as the underlying issues of mathematics, logic, computation and philosophy. [[User:Vipul|Vipul]] 20:20, 25 February 2009 (UTC) ==Prospects for a Logic Wiki== ===Initial Discussion=== Previously on Buffer &hellip; * [http://blog.subwiki.org/?p=13 The History of Subject Wikis] * [http://blog.subwiki.org/?p=13&cpage=1#comment-2 Comments] <blockquote> <p>Vipul,</p> <p>From the initial sample of your writing distribution that I’ve read so far, your approach to mathematical objects by way of their properties is essentially a logical angle.</p> <p>If one elects to think in functional, computational terms from the very beginning, then it is convenient to think of a property <math>p\!</math> as being a mapping from a space <math>X,\!</math> the universe of discourse, to a space of 2 elements, say, <math>\mathbb{B} = \{ 0, 1 \}.</math> So a property is something of the form <math>p : X \to \mathbb{B}.</math> That level of consideration amounts to propositional calculus. Trivial as it may seem, it’s been my experience that getting efficient computational support at this level is key to many of the things we’d like to do at higher levels in the way of mathematical knowledge management.</p> <p>Well, enough for now, as I’m not even sure the above formatting will work here. Let’s look for a wiki discussion page where it will be easier to talk. Either at MyWikiBiz, one of your Subject Wikis, or if you like email archiving there is the Inquiry List that an e-friend set up for me.</p> <p>[http://blog.subwiki.org/?p=13&cpage=1#comment-6 Jon Awbrey, 24 Feb 2009, 20:42 UTC]</p> </blockquote> VN: Jon, I agree with your basic ideas. There's of course the issue that the space <math>X</math> under consideration is often ''not'' a ''set'' (it's too large to be one -- for instance, a [[Groupprops:group property|group property]] is defined as a ''function'' on the collection of all groups), so we cnanot simply treat it as a set map. Still, it behaves largely like a set map. And as you point out, properties behave a lot like propositions with the ''parameter'' being from the space <math>X</math>. At some point in time, I was interested in the mathematical aspects of properties. Properties form a Boolean algebra, inheriting conjunction and disjunction etc. (in set-theoretic jargon, this would just be the power set of <math>X</math>, except that <math>X</math> isn't a set). Then, we can use various partial binary operations on <math>X</math> to define corresponding binary operations on the property space. I believe this, too, has been explored by some people, who came up with structures such as quantales and gaggles. The broad idea, as far as I understand, is that the binary operation must distribute over disjunctions (ORs). Many of the operators that I discuss in the group theory wiki, such as the [[groupprops:composition operator for subgroup properties|composition operator for subgroup properties]] and the [[groupprops:join operator|join operator]], are quantalic operators. Some of the terminology that I use for some property-theoretic notation is borrowed from terminology used for such operators, for instance, the [[groupprops:left residual operator for composition|left residual]] and [[groupprops:right residual operator for composition|right residual]]. Some of the basic structural theorems and approaches are also taken from corresponding ideas that already existed in logic. (I arrived at some of them before becoming aware of the corresponding structures explored in logic, and later found that much of this had been done before). As far as I know, though, there has been no systematic application of the ideas developed in different parts of logic to this ''real world'' (?) application to understanding properties in different mathematical domains. This puzzled me since I think that using the ideas of logic and logical structure in this way can help achieve a better understanding of many disciplines, particularly those in mathematics and computer science. Perhaps it is because there are no deep theorems here? My own use of property theory as an organizational principle seems to rely very minimally on the logical ideas beneath and rests more on the practical results that are important within the specific subject. But perhaps there are deeper theoretical logic issues that should be explored. I'd be glad to hear about your perspective, logical, philosophical, or any other. I'd also like to hear more about your views on what you mean by supporting such a model computationally. On the group theory wiki, I'm using MediaWiki's basic tools as well as Semantic MediaWiki to help in property-theoretic exploration. This isn't ''smart'' in any sense on the machine's part, but it meets some of the practical needs. Nonetheless, if you have ideas for something more sophisticated, I'd be glad to hear. [[User:Vipul|Vipul]] 22:13, 25 February 2009 (UTC) JA: Sure, there is all that. My main interest at present is focused on using logic as a tool for practical knowledge management in mathematical domains. That means starting small, breaking off this or that chunk of suitably interesting or useful domains and representing them in such a way that you can not only get computers to help you reason about them but in such a way that human beings can grok what's going on. JA: Uh-oh, I'm being called to dinner, I will have to continue later tonight. [[User:Jon Awbrey|Jon Awbrey]] 23:08, 25 February 2009 (UTC) VN: Well, that's interesting. I'm myself interested in the practical aspects, too, though perhaps from a different angle. I'd like to hear more. I'll also go through some of your past writings to understand your perspective. If you have comments on the way things have been done on the subject wikis specifically, I'd like to hear those as well. [[User:Vipul|Vipul]] 00:00, 26 February 2009 (UTC) JA: Maybe it will be quickest just to list a few links. Most of the longer pieces at MyWikiBiz are unpublished papers and project workups of mine that I'm still in the process TeXing and Wikifying and redoing the graphics that got mushed up moving between different platforms over the years, so most of them have a point where I last left off reformatting or rewriting. At any rate, here are a few serving suggestions on the basic logic side of things: :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Propositional_Calculus Differential Propositional Calculus] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Functional_Logic_:_Quantification_Theory Functional Logic : Quantification Theory] JA: [[User:Jon Awbrey|Jon Awbrey]] 03:26, 26 February 2009 (UTC) VN: Jon, I've started looking at your work. Your language and approach are new to me, so it'll take me some time to go through it. Other work has also come up. I hope to get back to you in a couple of weeks. [[User:Vipul|Vipul]] 18:05, 1 March 2009 (UTC) JA: No particular rush. I went looking for some old e-lectures of mine where I worked up much more concrete examples, but they need to be converted from ASCII to HTML-TeX-Wiki formats to be readable at all, and that will take me a while. I will probably keep outlining the occasional notes here in the interval. Later, [[User:Jon Awbrey|Jon Awbrey]] 22:40, 1 March 2009 (UTC) ===Scope and Purpose of a Logic Wiki=== ; Objectives : Maintain a focus of active methods that is parallel to the focus on static content. :: Algorithms :: Computational tools :: Efficient calculi :: Heuristics :: Proofs : Here's a project that looks interesting in this connection &mdash; :: [http://www.mat.univie.ac.at/~neum/FMathL.html Arnold Neumaier : Automated Mathematical Research Assistant] : VN: Jon, that certainly looks interesting. Thanks for the link. [[User:Vipul|Vipul]] 15:57, 7 March 2009 (UTC) : VN: I think the discussion has two parallel tracks here: what a logic wiki should contain and what logical paradigms can be used efficiently to organize content in all subject wikis (may be you're talking of only one of these right now). : VN: Regarding the organization of content across subject wikis. I had a quick look at the FMathL link. From what I can figure out, this is interesting, but still preliminary. What I'm interested in right now is something actionable that I can use to make the definitions and proofs easier and quicker to understand, easier to link to, etc. I'm sure projects such as FMathL can give some useful ideas in that connection, so I'd be glad to hear what ideas you've extracted from there. I [http://blog.subwiki.org/?p=25 wrote a recent blog post] describing some of the issues that I'm thinking about, or rather, the place I'm starting from. : VN: Regarding a specific logic wiki, it seems to me that logic is an extremely big subject. Your comments and past writings give me the impression that you're interested more in the proof theory/proof calculus part of the subject. I'm assuming you're not interested so much in the model theory side of things. Okay, so how about calling it a proof theory wiki or something like that? Of course, I might be misunderstanding the scope, so more clarification from you is welcome. I'm also not sure how Neumaier's automated research assistant connects up with the content of the proposed wiki. Are you sugesting that FMathL's ideas be used to organize the wiki, or that the wiki contain information about FMathL? [[User:Vipul|Vipul]] 23:54, 11 March 2009 (UTC) a65c1a34c063201740bfd32b5ec0831807eb26ca 341 340 2009-03-16T15:58:58Z Jon Awbrey 3 /* Scope and Purpose of a Logic Wiki */ wikitext text/x-wiki This discussion page is intended for a discussion on using property theory as a means of organizing material in subject wikis, as well as the underlying issues of mathematics, logic, computation and philosophy. [[User:Vipul|Vipul]] 20:20, 25 February 2009 (UTC) ==Prospects for a Logic Wiki== ===Initial Discussion=== Previously on Buffer &hellip; * [http://blog.subwiki.org/?p=13 The History of Subject Wikis] * [http://blog.subwiki.org/?p=13&cpage=1#comment-2 Comments] <blockquote> <p>Vipul,</p> <p>From the initial sample of your writing distribution that I’ve read so far, your approach to mathematical objects by way of their properties is essentially a logical angle.</p> <p>If one elects to think in functional, computational terms from the very beginning, then it is convenient to think of a property <math>p\!</math> as being a mapping from a space <math>X,\!</math> the universe of discourse, to a space of 2 elements, say, <math>\mathbb{B} = \{ 0, 1 \}.</math> So a property is something of the form <math>p : X \to \mathbb{B}.</math> That level of consideration amounts to propositional calculus. Trivial as it may seem, it’s been my experience that getting efficient computational support at this level is key to many of the things we’d like to do at higher levels in the way of mathematical knowledge management.</p> <p>Well, enough for now, as I’m not even sure the above formatting will work here. Let’s look for a wiki discussion page where it will be easier to talk. Either at MyWikiBiz, one of your Subject Wikis, or if you like email archiving there is the Inquiry List that an e-friend set up for me.</p> <p>[http://blog.subwiki.org/?p=13&cpage=1#comment-6 Jon Awbrey, 24 Feb 2009, 20:42 UTC]</p> </blockquote> VN: Jon, I agree with your basic ideas. There's of course the issue that the space <math>X</math> under consideration is often ''not'' a ''set'' (it's too large to be one -- for instance, a [[Groupprops:group property|group property]] is defined as a ''function'' on the collection of all groups), so we cnanot simply treat it as a set map. Still, it behaves largely like a set map. And as you point out, properties behave a lot like propositions with the ''parameter'' being from the space <math>X</math>. At some point in time, I was interested in the mathematical aspects of properties. Properties form a Boolean algebra, inheriting conjunction and disjunction etc. (in set-theoretic jargon, this would just be the power set of <math>X</math>, except that <math>X</math> isn't a set). Then, we can use various partial binary operations on <math>X</math> to define corresponding binary operations on the property space. I believe this, too, has been explored by some people, who came up with structures such as quantales and gaggles. The broad idea, as far as I understand, is that the binary operation must distribute over disjunctions (ORs). Many of the operators that I discuss in the group theory wiki, such as the [[groupprops:composition operator for subgroup properties|composition operator for subgroup properties]] and the [[groupprops:join operator|join operator]], are quantalic operators. Some of the terminology that I use for some property-theoretic notation is borrowed from terminology used for such operators, for instance, the [[groupprops:left residual operator for composition|left residual]] and [[groupprops:right residual operator for composition|right residual]]. Some of the basic structural theorems and approaches are also taken from corresponding ideas that already existed in logic. (I arrived at some of them before becoming aware of the corresponding structures explored in logic, and later found that much of this had been done before). As far as I know, though, there has been no systematic application of the ideas developed in different parts of logic to this ''real world'' (?) application to understanding properties in different mathematical domains. This puzzled me since I think that using the ideas of logic and logical structure in this way can help achieve a better understanding of many disciplines, particularly those in mathematics and computer science. Perhaps it is because there are no deep theorems here? My own use of property theory as an organizational principle seems to rely very minimally on the logical ideas beneath and rests more on the practical results that are important within the specific subject. But perhaps there are deeper theoretical logic issues that should be explored. I'd be glad to hear about your perspective, logical, philosophical, or any other. I'd also like to hear more about your views on what you mean by supporting such a model computationally. On the group theory wiki, I'm using MediaWiki's basic tools as well as Semantic MediaWiki to help in property-theoretic exploration. This isn't ''smart'' in any sense on the machine's part, but it meets some of the practical needs. Nonetheless, if you have ideas for something more sophisticated, I'd be glad to hear. [[User:Vipul|Vipul]] 22:13, 25 February 2009 (UTC) JA: Sure, there is all that. My main interest at present is focused on using logic as a tool for practical knowledge management in mathematical domains. That means starting small, breaking off this or that chunk of suitably interesting or useful domains and representing them in such a way that you can not only get computers to help you reason about them but in such a way that human beings can grok what's going on. JA: Uh-oh, I'm being called to dinner, I will have to continue later tonight. [[User:Jon Awbrey|Jon Awbrey]] 23:08, 25 February 2009 (UTC) VN: Well, that's interesting. I'm myself interested in the practical aspects, too, though perhaps from a different angle. I'd like to hear more. I'll also go through some of your past writings to understand your perspective. If you have comments on the way things have been done on the subject wikis specifically, I'd like to hear those as well. [[User:Vipul|Vipul]] 00:00, 26 February 2009 (UTC) JA: Maybe it will be quickest just to list a few links. Most of the longer pieces at MyWikiBiz are unpublished papers and project workups of mine that I'm still in the process TeXing and Wikifying and redoing the graphics that got mushed up moving between different platforms over the years, so most of them have a point where I last left off reformatting or rewriting. At any rate, here are a few serving suggestions on the basic logic side of things: :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Propositional_Calculus Differential Propositional Calculus] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Functional_Logic_:_Quantification_Theory Functional Logic : Quantification Theory] JA: [[User:Jon Awbrey|Jon Awbrey]] 03:26, 26 February 2009 (UTC) VN: Jon, I've started looking at your work. Your language and approach are new to me, so it'll take me some time to go through it. Other work has also come up. I hope to get back to you in a couple of weeks. [[User:Vipul|Vipul]] 18:05, 1 March 2009 (UTC) JA: No particular rush. I went looking for some old e-lectures of mine where I worked up much more concrete examples, but they need to be converted from ASCII to HTML-TeX-Wiki formats to be readable at all, and that will take me a while. I will probably keep outlining the occasional notes here in the interval. Later, [[User:Jon Awbrey|Jon Awbrey]] 22:40, 1 March 2009 (UTC) ===Scope and Purpose of a Logic Wiki=== ; Objectives : Maintain a focus of active methods that is parallel to the focus on static content. :: Algorithms :: Computational tools :: Efficient calculi :: Heuristics :: Proofs : Here's a project that looks interesting in this connection &mdash; :: [http://www.mat.univie.ac.at/~neum/FMathL.html Arnold Neumaier : Automated Mathematical Research Assistant] VN: Jon, that certainly looks interesting. Thanks for the link. [[User:Vipul|Vipul]] 15:57, 7 March 2009 (UTC) VN: I think the discussion has two parallel tracks here: what a logic wiki should contain and what logical paradigms can be used efficiently to organize content in all subject wikis (may be you're talking of only one of these right now). VN: Regarding the organization of content across subject wikis. I had a quick look at the FMathL link. From what I can figure out, this is interesting, but still preliminary. What I'm interested in right now is something actionable that I can use to make the definitions and proofs easier and quicker to understand, easier to link to, etc. I'm sure projects such as FMathL can give some useful ideas in that connection, so I'd be glad to hear what ideas you've extracted from there. I [http://blog.subwiki.org/?p=25 wrote a recent blog post] describing some of the issues that I'm thinking about, or rather, the place I'm starting from. VN: Regarding a specific logic wiki, it seems to me that logic is an extremely big subject. Your comments and past writings give me the impression that you're interested more in the proof theory/proof calculus part of the subject. I'm assuming you're not interested so much in the model theory side of things. Okay, so how about calling it a proof theory wiki or something like that? Of course, I might be misunderstanding the scope, so more clarification from you is welcome. I'm also not sure how Neumaier's automated research assistant connects up with the content of the proposed wiki. Are you sugesting that FMathL's ideas be used to organize the wiki, or that the wiki contain information about FMathL? [[User:Vipul|Vipul]] 23:54, 11 March 2009 (UTC) JA: First time I've stopped by this way in a while. It will take me a while longer to work up the concrete example that I had in mind, which is currently in progress [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Analytic_Turing_Automata|here]. JA: I will have to leave the whole of logic to others. The most I can hope to do with my time here is maybe to grease the wheels of our practical vehicles at critical points. JA: Lunch''!'' Back later &hellip; 6d9f46eb29a2b3e0e0858d793717e5e97596cc12 342 341 2009-03-16T15:59:39Z Jon Awbrey 3 /* Scope and Purpose of a Logic Wiki */ wikitext text/x-wiki This discussion page is intended for a discussion on using property theory as a means of organizing material in subject wikis, as well as the underlying issues of mathematics, logic, computation and philosophy. [[User:Vipul|Vipul]] 20:20, 25 February 2009 (UTC) ==Prospects for a Logic Wiki== ===Initial Discussion=== Previously on Buffer &hellip; * [http://blog.subwiki.org/?p=13 The History of Subject Wikis] * [http://blog.subwiki.org/?p=13&cpage=1#comment-2 Comments] <blockquote> <p>Vipul,</p> <p>From the initial sample of your writing distribution that I’ve read so far, your approach to mathematical objects by way of their properties is essentially a logical angle.</p> <p>If one elects to think in functional, computational terms from the very beginning, then it is convenient to think of a property <math>p\!</math> as being a mapping from a space <math>X,\!</math> the universe of discourse, to a space of 2 elements, say, <math>\mathbb{B} = \{ 0, 1 \}.</math> So a property is something of the form <math>p : X \to \mathbb{B}.</math> That level of consideration amounts to propositional calculus. Trivial as it may seem, it’s been my experience that getting efficient computational support at this level is key to many of the things we’d like to do at higher levels in the way of mathematical knowledge management.</p> <p>Well, enough for now, as I’m not even sure the above formatting will work here. Let’s look for a wiki discussion page where it will be easier to talk. Either at MyWikiBiz, one of your Subject Wikis, or if you like email archiving there is the Inquiry List that an e-friend set up for me.</p> <p>[http://blog.subwiki.org/?p=13&cpage=1#comment-6 Jon Awbrey, 24 Feb 2009, 20:42 UTC]</p> </blockquote> VN: Jon, I agree with your basic ideas. There's of course the issue that the space <math>X</math> under consideration is often ''not'' a ''set'' (it's too large to be one -- for instance, a [[Groupprops:group property|group property]] is defined as a ''function'' on the collection of all groups), so we cnanot simply treat it as a set map. Still, it behaves largely like a set map. And as you point out, properties behave a lot like propositions with the ''parameter'' being from the space <math>X</math>. At some point in time, I was interested in the mathematical aspects of properties. Properties form a Boolean algebra, inheriting conjunction and disjunction etc. (in set-theoretic jargon, this would just be the power set of <math>X</math>, except that <math>X</math> isn't a set). Then, we can use various partial binary operations on <math>X</math> to define corresponding binary operations on the property space. I believe this, too, has been explored by some people, who came up with structures such as quantales and gaggles. The broad idea, as far as I understand, is that the binary operation must distribute over disjunctions (ORs). Many of the operators that I discuss in the group theory wiki, such as the [[groupprops:composition operator for subgroup properties|composition operator for subgroup properties]] and the [[groupprops:join operator|join operator]], are quantalic operators. Some of the terminology that I use for some property-theoretic notation is borrowed from terminology used for such operators, for instance, the [[groupprops:left residual operator for composition|left residual]] and [[groupprops:right residual operator for composition|right residual]]. Some of the basic structural theorems and approaches are also taken from corresponding ideas that already existed in logic. (I arrived at some of them before becoming aware of the corresponding structures explored in logic, and later found that much of this had been done before). As far as I know, though, there has been no systematic application of the ideas developed in different parts of logic to this ''real world'' (?) application to understanding properties in different mathematical domains. This puzzled me since I think that using the ideas of logic and logical structure in this way can help achieve a better understanding of many disciplines, particularly those in mathematics and computer science. Perhaps it is because there are no deep theorems here? My own use of property theory as an organizational principle seems to rely very minimally on the logical ideas beneath and rests more on the practical results that are important within the specific subject. But perhaps there are deeper theoretical logic issues that should be explored. I'd be glad to hear about your perspective, logical, philosophical, or any other. I'd also like to hear more about your views on what you mean by supporting such a model computationally. On the group theory wiki, I'm using MediaWiki's basic tools as well as Semantic MediaWiki to help in property-theoretic exploration. This isn't ''smart'' in any sense on the machine's part, but it meets some of the practical needs. Nonetheless, if you have ideas for something more sophisticated, I'd be glad to hear. [[User:Vipul|Vipul]] 22:13, 25 February 2009 (UTC) JA: Sure, there is all that. My main interest at present is focused on using logic as a tool for practical knowledge management in mathematical domains. That means starting small, breaking off this or that chunk of suitably interesting or useful domains and representing them in such a way that you can not only get computers to help you reason about them but in such a way that human beings can grok what's going on. JA: Uh-oh, I'm being called to dinner, I will have to continue later tonight. [[User:Jon Awbrey|Jon Awbrey]] 23:08, 25 February 2009 (UTC) VN: Well, that's interesting. I'm myself interested in the practical aspects, too, though perhaps from a different angle. I'd like to hear more. I'll also go through some of your past writings to understand your perspective. If you have comments on the way things have been done on the subject wikis specifically, I'd like to hear those as well. [[User:Vipul|Vipul]] 00:00, 26 February 2009 (UTC) JA: Maybe it will be quickest just to list a few links. Most of the longer pieces at MyWikiBiz are unpublished papers and project workups of mine that I'm still in the process TeXing and Wikifying and redoing the graphics that got mushed up moving between different platforms over the years, so most of them have a point where I last left off reformatting or rewriting. At any rate, here are a few serving suggestions on the basic logic side of things: :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Propositional_Calculus Differential Propositional Calculus] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Functional_Logic_:_Quantification_Theory Functional Logic : Quantification Theory] JA: [[User:Jon Awbrey|Jon Awbrey]] 03:26, 26 February 2009 (UTC) VN: Jon, I've started looking at your work. Your language and approach are new to me, so it'll take me some time to go through it. Other work has also come up. I hope to get back to you in a couple of weeks. [[User:Vipul|Vipul]] 18:05, 1 March 2009 (UTC) JA: No particular rush. I went looking for some old e-lectures of mine where I worked up much more concrete examples, but they need to be converted from ASCII to HTML-TeX-Wiki formats to be readable at all, and that will take me a while. I will probably keep outlining the occasional notes here in the interval. Later, [[User:Jon Awbrey|Jon Awbrey]] 22:40, 1 March 2009 (UTC) ===Scope and Purpose of a Logic Wiki=== ; Objectives : Maintain a focus of active methods that is parallel to the focus on static content. :: Algorithms :: Computational tools :: Efficient calculi :: Heuristics :: Proofs : Here's a project that looks interesting in this connection &mdash; :: [http://www.mat.univie.ac.at/~neum/FMathL.html Arnold Neumaier : Automated Mathematical Research Assistant] VN: Jon, that certainly looks interesting. Thanks for the link. [[User:Vipul|Vipul]] 15:57, 7 March 2009 (UTC) VN: I think the discussion has two parallel tracks here: what a logic wiki should contain and what logical paradigms can be used efficiently to organize content in all subject wikis (may be you're talking of only one of these right now). VN: Regarding the organization of content across subject wikis. I had a quick look at the FMathL link. From what I can figure out, this is interesting, but still preliminary. What I'm interested in right now is something actionable that I can use to make the definitions and proofs easier and quicker to understand, easier to link to, etc. I'm sure projects such as FMathL can give some useful ideas in that connection, so I'd be glad to hear what ideas you've extracted from there. I [http://blog.subwiki.org/?p=25 wrote a recent blog post] describing some of the issues that I'm thinking about, or rather, the place I'm starting from. VN: Regarding a specific logic wiki, it seems to me that logic is an extremely big subject. Your comments and past writings give me the impression that you're interested more in the proof theory/proof calculus part of the subject. I'm assuming you're not interested so much in the model theory side of things. Okay, so how about calling it a proof theory wiki or something like that? Of course, I might be misunderstanding the scope, so more clarification from you is welcome. I'm also not sure how Neumaier's automated research assistant connects up with the content of the proposed wiki. Are you sugesting that FMathL's ideas be used to organize the wiki, or that the wiki contain information about FMathL? [[User:Vipul|Vipul]] 23:54, 11 March 2009 (UTC) JA: First time I've stopped by this way in a while. It will take me a while longer to work up the concrete example that I had in mind, which is currently in progress [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Analytic_Turing_Automata here]. JA: I will have to leave the whole of logic to others. The most I can hope to do with my time here is maybe to grease the wheels of our practical vehicles at critical points. JA: Lunch''!'' Back later &hellip; 1fd90737d230e462f018f93a16e924baa5cb988e 343 342 2009-03-16T16:02:24Z Jon Awbrey 3 /* Scope and Purpose of a Logic Wiki */ forgot sig wikitext text/x-wiki This discussion page is intended for a discussion on using property theory as a means of organizing material in subject wikis, as well as the underlying issues of mathematics, logic, computation and philosophy. [[User:Vipul|Vipul]] 20:20, 25 February 2009 (UTC) ==Prospects for a Logic Wiki== ===Initial Discussion=== Previously on Buffer &hellip; * [http://blog.subwiki.org/?p=13 The History of Subject Wikis] * [http://blog.subwiki.org/?p=13&cpage=1#comment-2 Comments] <blockquote> <p>Vipul,</p> <p>From the initial sample of your writing distribution that I’ve read so far, your approach to mathematical objects by way of their properties is essentially a logical angle.</p> <p>If one elects to think in functional, computational terms from the very beginning, then it is convenient to think of a property <math>p\!</math> as being a mapping from a space <math>X,\!</math> the universe of discourse, to a space of 2 elements, say, <math>\mathbb{B} = \{ 0, 1 \}.</math> So a property is something of the form <math>p : X \to \mathbb{B}.</math> That level of consideration amounts to propositional calculus. Trivial as it may seem, it’s been my experience that getting efficient computational support at this level is key to many of the things we’d like to do at higher levels in the way of mathematical knowledge management.</p> <p>Well, enough for now, as I’m not even sure the above formatting will work here. Let’s look for a wiki discussion page where it will be easier to talk. Either at MyWikiBiz, one of your Subject Wikis, or if you like email archiving there is the Inquiry List that an e-friend set up for me.</p> <p>[http://blog.subwiki.org/?p=13&cpage=1#comment-6 Jon Awbrey, 24 Feb 2009, 20:42 UTC]</p> </blockquote> VN: Jon, I agree with your basic ideas. There's of course the issue that the space <math>X</math> under consideration is often ''not'' a ''set'' (it's too large to be one -- for instance, a [[Groupprops:group property|group property]] is defined as a ''function'' on the collection of all groups), so we cnanot simply treat it as a set map. Still, it behaves largely like a set map. And as you point out, properties behave a lot like propositions with the ''parameter'' being from the space <math>X</math>. At some point in time, I was interested in the mathematical aspects of properties. Properties form a Boolean algebra, inheriting conjunction and disjunction etc. (in set-theoretic jargon, this would just be the power set of <math>X</math>, except that <math>X</math> isn't a set). Then, we can use various partial binary operations on <math>X</math> to define corresponding binary operations on the property space. I believe this, too, has been explored by some people, who came up with structures such as quantales and gaggles. The broad idea, as far as I understand, is that the binary operation must distribute over disjunctions (ORs). Many of the operators that I discuss in the group theory wiki, such as the [[groupprops:composition operator for subgroup properties|composition operator for subgroup properties]] and the [[groupprops:join operator|join operator]], are quantalic operators. Some of the terminology that I use for some property-theoretic notation is borrowed from terminology used for such operators, for instance, the [[groupprops:left residual operator for composition|left residual]] and [[groupprops:right residual operator for composition|right residual]]. Some of the basic structural theorems and approaches are also taken from corresponding ideas that already existed in logic. (I arrived at some of them before becoming aware of the corresponding structures explored in logic, and later found that much of this had been done before). As far as I know, though, there has been no systematic application of the ideas developed in different parts of logic to this ''real world'' (?) application to understanding properties in different mathematical domains. This puzzled me since I think that using the ideas of logic and logical structure in this way can help achieve a better understanding of many disciplines, particularly those in mathematics and computer science. Perhaps it is because there are no deep theorems here? My own use of property theory as an organizational principle seems to rely very minimally on the logical ideas beneath and rests more on the practical results that are important within the specific subject. But perhaps there are deeper theoretical logic issues that should be explored. I'd be glad to hear about your perspective, logical, philosophical, or any other. I'd also like to hear more about your views on what you mean by supporting such a model computationally. On the group theory wiki, I'm using MediaWiki's basic tools as well as Semantic MediaWiki to help in property-theoretic exploration. This isn't ''smart'' in any sense on the machine's part, but it meets some of the practical needs. Nonetheless, if you have ideas for something more sophisticated, I'd be glad to hear. [[User:Vipul|Vipul]] 22:13, 25 February 2009 (UTC) JA: Sure, there is all that. My main interest at present is focused on using logic as a tool for practical knowledge management in mathematical domains. That means starting small, breaking off this or that chunk of suitably interesting or useful domains and representing them in such a way that you can not only get computers to help you reason about them but in such a way that human beings can grok what's going on. JA: Uh-oh, I'm being called to dinner, I will have to continue later tonight. [[User:Jon Awbrey|Jon Awbrey]] 23:08, 25 February 2009 (UTC) VN: Well, that's interesting. I'm myself interested in the practical aspects, too, though perhaps from a different angle. I'd like to hear more. I'll also go through some of your past writings to understand your perspective. If you have comments on the way things have been done on the subject wikis specifically, I'd like to hear those as well. [[User:Vipul|Vipul]] 00:00, 26 February 2009 (UTC) JA: Maybe it will be quickest just to list a few links. Most of the longer pieces at MyWikiBiz are unpublished papers and project workups of mine that I'm still in the process TeXing and Wikifying and redoing the graphics that got mushed up moving between different platforms over the years, so most of them have a point where I last left off reformatting or rewriting. At any rate, here are a few serving suggestions on the basic logic side of things: :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Propositional_Calculus Differential Propositional Calculus] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Functional_Logic_:_Quantification_Theory Functional Logic : Quantification Theory] JA: [[User:Jon Awbrey|Jon Awbrey]] 03:26, 26 February 2009 (UTC) VN: Jon, I've started looking at your work. Your language and approach are new to me, so it'll take me some time to go through it. Other work has also come up. I hope to get back to you in a couple of weeks. [[User:Vipul|Vipul]] 18:05, 1 March 2009 (UTC) JA: No particular rush. I went looking for some old e-lectures of mine where I worked up much more concrete examples, but they need to be converted from ASCII to HTML-TeX-Wiki formats to be readable at all, and that will take me a while. I will probably keep outlining the occasional notes here in the interval. Later, [[User:Jon Awbrey|Jon Awbrey]] 22:40, 1 March 2009 (UTC) ===Scope and Purpose of a Logic Wiki=== ; Objectives : Maintain a focus of active methods that is parallel to the focus on static content. :: Algorithms :: Computational tools :: Efficient calculi :: Heuristics :: Proofs : Here's a project that looks interesting in this connection &mdash; :: [http://www.mat.univie.ac.at/~neum/FMathL.html Arnold Neumaier : Automated Mathematical Research Assistant] VN: Jon, that certainly looks interesting. Thanks for the link. [[User:Vipul|Vipul]] 15:57, 7 March 2009 (UTC) VN: I think the discussion has two parallel tracks here: what a logic wiki should contain and what logical paradigms can be used efficiently to organize content in all subject wikis (may be you're talking of only one of these right now). VN: Regarding the organization of content across subject wikis. I had a quick look at the FMathL link. From what I can figure out, this is interesting, but still preliminary. What I'm interested in right now is something actionable that I can use to make the definitions and proofs easier and quicker to understand, easier to link to, etc. I'm sure projects such as FMathL can give some useful ideas in that connection, so I'd be glad to hear what ideas you've extracted from there. I [http://blog.subwiki.org/?p=25 wrote a recent blog post] describing some of the issues that I'm thinking about, or rather, the place I'm starting from. VN: Regarding a specific logic wiki, it seems to me that logic is an extremely big subject. Your comments and past writings give me the impression that you're interested more in the proof theory/proof calculus part of the subject. I'm assuming you're not interested so much in the model theory side of things. Okay, so how about calling it a proof theory wiki or something like that? Of course, I might be misunderstanding the scope, so more clarification from you is welcome. I'm also not sure how Neumaier's automated research assistant connects up with the content of the proposed wiki. Are you sugesting that FMathL's ideas be used to organize the wiki, or that the wiki contain information about FMathL? [[User:Vipul|Vipul]] 23:54, 11 March 2009 (UTC) JA: First time I've stopped by this way in a while. It will take me a while longer to work up the concrete example that I had in mind, which is currently in progress [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Analytic_Turing_Automata here]. JA: I will have to leave the whole of logic to others. The most I can hope to do with my time here is maybe to grease the wheels of our practical vehicles at critical points. JA: Lunch''!'' Back later &hellip; [[User:Jon Awbrey|Jon Awbrey]] 16:02, 16 March 2009 (UTC) a63c9fe455ee786d4359f1dd0483bb40d4e6630f 344 343 2009-03-16T20:16:28Z Jon Awbrey 3 /* Prospects for a Logic Wiki */ anamnesis wikitext text/x-wiki This discussion page is intended for a discussion on using property theory as a means of organizing material in subject wikis, as well as the underlying issues of mathematics, logic, computation and philosophy. [[User:Vipul|Vipul]] 20:20, 25 February 2009 (UTC) ==Prospects for a Logic Wiki== ===Initial Discussion=== Previously on Buffer &hellip; * [http://blog.subwiki.org/?p=13 The History of Subject Wikis] * [http://blog.subwiki.org/?p=13&cpage=1#comment-2 Comments] <blockquote> <p>Vipul,</p> <p>From the initial sample of your writing distribution that I’ve read so far, your approach to mathematical objects by way of their properties is essentially a logical angle.</p> <p>If one elects to think in functional, computational terms from the very beginning, then it is convenient to think of a property <math>p\!</math> as being a mapping from a space <math>X,\!</math> the universe of discourse, to a space of 2 elements, say, <math>\mathbb{B} = \{ 0, 1 \}.</math> So a property is something of the form <math>p : X \to \mathbb{B}.</math> That level of consideration amounts to propositional calculus. Trivial as it may seem, it’s been my experience that getting efficient computational support at this level is key to many of the things we’d like to do at higher levels in the way of mathematical knowledge management.</p> <p>Well, enough for now, as I’m not even sure the above formatting will work here. Let’s look for a wiki discussion page where it will be easier to talk. Either at MyWikiBiz, one of your Subject Wikis, or if you like email archiving there is the Inquiry List that an e-friend set up for me.</p> <p>[http://blog.subwiki.org/?p=13&cpage=1#comment-6 Jon Awbrey, 24 Feb 2009, 20:42 UTC]</p> </blockquote> VN: Jon, I agree with your basic ideas. There's of course the issue that the space <math>X</math> under consideration is often ''not'' a ''set'' (it's too large to be one -- for instance, a [[Groupprops:group property|group property]] is defined as a ''function'' on the collection of all groups), so we cnanot simply treat it as a set map. Still, it behaves largely like a set map. And as you point out, properties behave a lot like propositions with the ''parameter'' being from the space <math>X</math>. At some point in time, I was interested in the mathematical aspects of properties. Properties form a Boolean algebra, inheriting conjunction and disjunction etc. (in set-theoretic jargon, this would just be the power set of <math>X</math>, except that <math>X</math> isn't a set). Then, we can use various partial binary operations on <math>X</math> to define corresponding binary operations on the property space. I believe this, too, has been explored by some people, who came up with structures such as quantales and gaggles. The broad idea, as far as I understand, is that the binary operation must distribute over disjunctions (ORs). Many of the operators that I discuss in the group theory wiki, such as the [[groupprops:composition operator for subgroup properties|composition operator for subgroup properties]] and the [[groupprops:join operator|join operator]], are quantalic operators. Some of the terminology that I use for some property-theoretic notation is borrowed from terminology used for such operators, for instance, the [[groupprops:left residual operator for composition|left residual]] and [[groupprops:right residual operator for composition|right residual]]. Some of the basic structural theorems and approaches are also taken from corresponding ideas that already existed in logic. (I arrived at some of them before becoming aware of the corresponding structures explored in logic, and later found that much of this had been done before). As far as I know, though, there has been no systematic application of the ideas developed in different parts of logic to this ''real world'' (?) application to understanding properties in different mathematical domains. This puzzled me since I think that using the ideas of logic and logical structure in this way can help achieve a better understanding of many disciplines, particularly those in mathematics and computer science. Perhaps it is because there are no deep theorems here? My own use of property theory as an organizational principle seems to rely very minimally on the logical ideas beneath and rests more on the practical results that are important within the specific subject. But perhaps there are deeper theoretical logic issues that should be explored. I'd be glad to hear about your perspective, logical, philosophical, or any other. I'd also like to hear more about your views on what you mean by supporting such a model computationally. On the group theory wiki, I'm using MediaWiki's basic tools as well as Semantic MediaWiki to help in property-theoretic exploration. This isn't ''smart'' in any sense on the machine's part, but it meets some of the practical needs. Nonetheless, if you have ideas for something more sophisticated, I'd be glad to hear. [[User:Vipul|Vipul]] 22:13, 25 February 2009 (UTC) JA: Sure, there is all that. My main interest at present is focused on using logic as a tool for practical knowledge management in mathematical domains. That means starting small, breaking off this or that chunk of suitably interesting or useful domains and representing them in such a way that you can not only get computers to help you reason about them but in such a way that human beings can grok what's going on. JA: Uh-oh, I'm being called to dinner, I will have to continue later tonight. [[User:Jon Awbrey|Jon Awbrey]] 23:08, 25 February 2009 (UTC) VN: Well, that's interesting. I'm myself interested in the practical aspects, too, though perhaps from a different angle. I'd like to hear more. I'll also go through some of your past writings to understand your perspective. If you have comments on the way things have been done on the subject wikis specifically, I'd like to hear those as well. [[User:Vipul|Vipul]] 00:00, 26 February 2009 (UTC) JA: Maybe it will be quickest just to list a few links. Most of the longer pieces at MyWikiBiz are unpublished papers and project workups of mine that I'm still in the process TeXing and Wikifying and redoing the graphics that got mushed up moving between different platforms over the years, so most of them have a point where I last left off reformatting or rewriting. At any rate, here are a few serving suggestions on the basic logic side of things: :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Propositional_Calculus Differential Propositional Calculus] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Functional_Logic_:_Quantification_Theory Functional Logic : Quantification Theory] JA: [[User:Jon Awbrey|Jon Awbrey]] 03:26, 26 February 2009 (UTC) VN: Jon, I've started looking at your work. Your language and approach are new to me, so it'll take me some time to go through it. Other work has also come up. I hope to get back to you in a couple of weeks. [[User:Vipul|Vipul]] 18:05, 1 March 2009 (UTC) JA: No particular rush. I went looking for some old e-lectures of mine where I worked up much more concrete examples, but they need to be converted from ASCII to HTML-TeX-Wiki formats to be readable at all, and that will take me a while. I will probably keep outlining the occasional notes here in the interval. Later, [[User:Jon Awbrey|Jon Awbrey]] 22:40, 1 March 2009 (UTC) ===Scope and Purpose of a Logic Wiki=== ; Objectives : Maintain a focus of active methods that is parallel to the focus on static content. :: Algorithms :: Computational tools :: Efficient calculi :: Heuristics :: Proofs : Here's a project that looks interesting in this connection &mdash; :: [http://www.mat.univie.ac.at/~neum/FMathL.html Arnold Neumaier : Automated Mathematical Research Assistant] VN: Jon, that certainly looks interesting. Thanks for the link. [[User:Vipul|Vipul]] 15:57, 7 March 2009 (UTC) VN: I think the discussion has two parallel tracks here: what a logic wiki should contain and what logical paradigms can be used efficiently to organize content in all subject wikis (may be you're talking of only one of these right now). VN: Regarding the organization of content across subject wikis. I had a quick look at the FMathL link. From what I can figure out, this is interesting, but still preliminary. What I'm interested in right now is something actionable that I can use to make the definitions and proofs easier and quicker to understand, easier to link to, etc. I'm sure projects such as FMathL can give some useful ideas in that connection, so I'd be glad to hear what ideas you've extracted from there. I [http://blog.subwiki.org/?p=25 wrote a recent blog post] describing some of the issues that I'm thinking about, or rather, the place I'm starting from. VN: Regarding a specific logic wiki, it seems to me that logic is an extremely big subject. Your comments and past writings give me the impression that you're interested more in the proof theory/proof calculus part of the subject. I'm assuming you're not interested so much in the model theory side of things. Okay, so how about calling it a proof theory wiki or something like that? Of course, I might be misunderstanding the scope, so more clarification from you is welcome. I'm also not sure how Neumaier's automated research assistant connects up with the content of the proposed wiki. Are you sugesting that FMathL's ideas be used to organize the wiki, or that the wiki contain information about FMathL? [[User:Vipul|Vipul]] 23:54, 11 March 2009 (UTC) JA: First time I've stopped by this way in a while. It will take me a while longer to work up the concrete example that I had in mind, which is currently in progress [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Analytic_Turing_Automata here]. JA: I will have to leave the whole of logic to others. The most I can hope to do with my time here is maybe to grease the wheels of our practical vehicles at critical points. JA: Lunch''!'' Back later &hellip; [[User:Jon Awbrey|Jon Awbrey]] 16:02, 16 March 2009 (UTC) ===Narrative Memoir=== JA: I thought it might help to switch from didactic to personal narrative. Recent discussions on various lists reminded of why I first started work on "automatic theorem proving" (ATM) or "mechanized mathematical reasoning", as they called it back then. I had long been interested in the interplay between algebra, geometry, and logic &mdash; what one of my professors later called "the lambda point" &mdash; and when I started grad school in math in the late 70's this interest congealed into a study of the linkages between group theory, number theory, graph theory, and logic. On the graph theory face of it I got bitten by the Reconstruction Conjecture bug and eventually worked out a group-theoretic (orbit counting) way of looking at it. But it was all getting to be a bit too much for my unaided brain, and so one of my office-mates suggested that I try to bring the mainframe to bear on it. I was more than hesitant at first, but had been taking courses in Lisp, so I eventually said <math>y_0?\!</math> and went to work on trying to write a theorem prover. [[User:Jon Awbrey|Jon Awbrey]] 20:16, 16 March 2009 (UTC) b512bff65f5e5ed343ffde49f59f5c8a1dcfb55c 345 344 2009-03-16T20:22:44Z Jon Awbrey 3 /* Narrative Memoir */ to be continued ... wikitext text/x-wiki This discussion page is intended for a discussion on using property theory as a means of organizing material in subject wikis, as well as the underlying issues of mathematics, logic, computation and philosophy. [[User:Vipul|Vipul]] 20:20, 25 February 2009 (UTC) ==Prospects for a Logic Wiki== ===Initial Discussion=== Previously on Buffer &hellip; * [http://blog.subwiki.org/?p=13 The History of Subject Wikis] * [http://blog.subwiki.org/?p=13&cpage=1#comment-2 Comments] <blockquote> <p>Vipul,</p> <p>From the initial sample of your writing distribution that I’ve read so far, your approach to mathematical objects by way of their properties is essentially a logical angle.</p> <p>If one elects to think in functional, computational terms from the very beginning, then it is convenient to think of a property <math>p\!</math> as being a mapping from a space <math>X,\!</math> the universe of discourse, to a space of 2 elements, say, <math>\mathbb{B} = \{ 0, 1 \}.</math> So a property is something of the form <math>p : X \to \mathbb{B}.</math> That level of consideration amounts to propositional calculus. Trivial as it may seem, it’s been my experience that getting efficient computational support at this level is key to many of the things we’d like to do at higher levels in the way of mathematical knowledge management.</p> <p>Well, enough for now, as I’m not even sure the above formatting will work here. Let’s look for a wiki discussion page where it will be easier to talk. Either at MyWikiBiz, one of your Subject Wikis, or if you like email archiving there is the Inquiry List that an e-friend set up for me.</p> <p>[http://blog.subwiki.org/?p=13&cpage=1#comment-6 Jon Awbrey, 24 Feb 2009, 20:42 UTC]</p> </blockquote> VN: Jon, I agree with your basic ideas. There's of course the issue that the space <math>X</math> under consideration is often ''not'' a ''set'' (it's too large to be one -- for instance, a [[Groupprops:group property|group property]] is defined as a ''function'' on the collection of all groups), so we cnanot simply treat it as a set map. Still, it behaves largely like a set map. And as you point out, properties behave a lot like propositions with the ''parameter'' being from the space <math>X</math>. At some point in time, I was interested in the mathematical aspects of properties. Properties form a Boolean algebra, inheriting conjunction and disjunction etc. (in set-theoretic jargon, this would just be the power set of <math>X</math>, except that <math>X</math> isn't a set). Then, we can use various partial binary operations on <math>X</math> to define corresponding binary operations on the property space. I believe this, too, has been explored by some people, who came up with structures such as quantales and gaggles. The broad idea, as far as I understand, is that the binary operation must distribute over disjunctions (ORs). Many of the operators that I discuss in the group theory wiki, such as the [[groupprops:composition operator for subgroup properties|composition operator for subgroup properties]] and the [[groupprops:join operator|join operator]], are quantalic operators. Some of the terminology that I use for some property-theoretic notation is borrowed from terminology used for such operators, for instance, the [[groupprops:left residual operator for composition|left residual]] and [[groupprops:right residual operator for composition|right residual]]. Some of the basic structural theorems and approaches are also taken from corresponding ideas that already existed in logic. (I arrived at some of them before becoming aware of the corresponding structures explored in logic, and later found that much of this had been done before). As far as I know, though, there has been no systematic application of the ideas developed in different parts of logic to this ''real world'' (?) application to understanding properties in different mathematical domains. This puzzled me since I think that using the ideas of logic and logical structure in this way can help achieve a better understanding of many disciplines, particularly those in mathematics and computer science. Perhaps it is because there are no deep theorems here? My own use of property theory as an organizational principle seems to rely very minimally on the logical ideas beneath and rests more on the practical results that are important within the specific subject. But perhaps there are deeper theoretical logic issues that should be explored. I'd be glad to hear about your perspective, logical, philosophical, or any other. I'd also like to hear more about your views on what you mean by supporting such a model computationally. On the group theory wiki, I'm using MediaWiki's basic tools as well as Semantic MediaWiki to help in property-theoretic exploration. This isn't ''smart'' in any sense on the machine's part, but it meets some of the practical needs. Nonetheless, if you have ideas for something more sophisticated, I'd be glad to hear. [[User:Vipul|Vipul]] 22:13, 25 February 2009 (UTC) JA: Sure, there is all that. My main interest at present is focused on using logic as a tool for practical knowledge management in mathematical domains. That means starting small, breaking off this or that chunk of suitably interesting or useful domains and representing them in such a way that you can not only get computers to help you reason about them but in such a way that human beings can grok what's going on. JA: Uh-oh, I'm being called to dinner, I will have to continue later tonight. [[User:Jon Awbrey|Jon Awbrey]] 23:08, 25 February 2009 (UTC) VN: Well, that's interesting. I'm myself interested in the practical aspects, too, though perhaps from a different angle. I'd like to hear more. I'll also go through some of your past writings to understand your perspective. If you have comments on the way things have been done on the subject wikis specifically, I'd like to hear those as well. [[User:Vipul|Vipul]] 00:00, 26 February 2009 (UTC) JA: Maybe it will be quickest just to list a few links. Most of the longer pieces at MyWikiBiz are unpublished papers and project workups of mine that I'm still in the process TeXing and Wikifying and redoing the graphics that got mushed up moving between different platforms over the years, so most of them have a point where I last left off reformatting or rewriting. At any rate, here are a few serving suggestions on the basic logic side of things: :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Propositional_Calculus Differential Propositional Calculus] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Functional_Logic_:_Quantification_Theory Functional Logic : Quantification Theory] JA: [[User:Jon Awbrey|Jon Awbrey]] 03:26, 26 February 2009 (UTC) VN: Jon, I've started looking at your work. Your language and approach are new to me, so it'll take me some time to go through it. Other work has also come up. I hope to get back to you in a couple of weeks. [[User:Vipul|Vipul]] 18:05, 1 March 2009 (UTC) JA: No particular rush. I went looking for some old e-lectures of mine where I worked up much more concrete examples, but they need to be converted from ASCII to HTML-TeX-Wiki formats to be readable at all, and that will take me a while. I will probably keep outlining the occasional notes here in the interval. Later, [[User:Jon Awbrey|Jon Awbrey]] 22:40, 1 March 2009 (UTC) ===Scope and Purpose of a Logic Wiki=== ; Objectives : Maintain a focus of active methods that is parallel to the focus on static content. :: Algorithms :: Computational tools :: Efficient calculi :: Heuristics :: Proofs : Here's a project that looks interesting in this connection &mdash; :: [http://www.mat.univie.ac.at/~neum/FMathL.html Arnold Neumaier : Automated Mathematical Research Assistant] VN: Jon, that certainly looks interesting. Thanks for the link. [[User:Vipul|Vipul]] 15:57, 7 March 2009 (UTC) VN: I think the discussion has two parallel tracks here: what a logic wiki should contain and what logical paradigms can be used efficiently to organize content in all subject wikis (may be you're talking of only one of these right now). VN: Regarding the organization of content across subject wikis. I had a quick look at the FMathL link. From what I can figure out, this is interesting, but still preliminary. What I'm interested in right now is something actionable that I can use to make the definitions and proofs easier and quicker to understand, easier to link to, etc. I'm sure projects such as FMathL can give some useful ideas in that connection, so I'd be glad to hear what ideas you've extracted from there. I [http://blog.subwiki.org/?p=25 wrote a recent blog post] describing some of the issues that I'm thinking about, or rather, the place I'm starting from. VN: Regarding a specific logic wiki, it seems to me that logic is an extremely big subject. Your comments and past writings give me the impression that you're interested more in the proof theory/proof calculus part of the subject. I'm assuming you're not interested so much in the model theory side of things. Okay, so how about calling it a proof theory wiki or something like that? Of course, I might be misunderstanding the scope, so more clarification from you is welcome. I'm also not sure how Neumaier's automated research assistant connects up with the content of the proposed wiki. Are you sugesting that FMathL's ideas be used to organize the wiki, or that the wiki contain information about FMathL? [[User:Vipul|Vipul]] 23:54, 11 March 2009 (UTC) JA: First time I've stopped by this way in a while. It will take me a while longer to work up the concrete example that I had in mind, which is currently in progress [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Analytic_Turing_Automata here]. JA: I will have to leave the whole of logic to others. The most I can hope to do with my time here is maybe to grease the wheels of our practical vehicles at critical points. JA: Lunch''!'' Back later &hellip; [[User:Jon Awbrey|Jon Awbrey]] 16:02, 16 March 2009 (UTC) ===Narrative Memoir=== JA: I thought it might help to switch from didactic to personal narrative. Recent discussions on various lists reminded of why I first started work on "automatic theorem proving" (ATM) or "mechanized mathematical reasoning", as they called it back then. I had long been interested in the interplay between algebra, geometry, and logic &mdash; what one of my professors later called "the lambda point" &mdash; and when I started grad school in math in the late 70's this interest congealed into a study of the linkages between group theory, number theory, graph theory, and logic. On the graph theory face of it I got bitten by the Reconstruction Conjecture bug and eventually worked out a group-theoretic (orbit counting) way of looking at the problem. But it was all getting to be a bit too much for my unaided brain, and so one of my office-mates suggested that I try to bring the mainframe to bear on it. I was more than hesitant at first, but had been taking courses in Lisp, so I eventually said <math>y_0?\!</math> and went to work on trying to write a theorem prover. [[User:Jon Awbrey|Jon Awbrey]] 20:16, 16 March 2009 (UTC) JA: To be continued &hellip; 3a550dd622f87ca13453c5c2fa9143cafbbca854 346 345 2009-03-16T20:24:13Z Jon Awbrey 3 /* Narrative Memoir */ phrasing wikitext text/x-wiki This discussion page is intended for a discussion on using property theory as a means of organizing material in subject wikis, as well as the underlying issues of mathematics, logic, computation and philosophy. [[User:Vipul|Vipul]] 20:20, 25 February 2009 (UTC) ==Prospects for a Logic Wiki== ===Initial Discussion=== Previously on Buffer &hellip; * [http://blog.subwiki.org/?p=13 The History of Subject Wikis] * [http://blog.subwiki.org/?p=13&cpage=1#comment-2 Comments] <blockquote> <p>Vipul,</p> <p>From the initial sample of your writing distribution that I’ve read so far, your approach to mathematical objects by way of their properties is essentially a logical angle.</p> <p>If one elects to think in functional, computational terms from the very beginning, then it is convenient to think of a property <math>p\!</math> as being a mapping from a space <math>X,\!</math> the universe of discourse, to a space of 2 elements, say, <math>\mathbb{B} = \{ 0, 1 \}.</math> So a property is something of the form <math>p : X \to \mathbb{B}.</math> That level of consideration amounts to propositional calculus. Trivial as it may seem, it’s been my experience that getting efficient computational support at this level is key to many of the things we’d like to do at higher levels in the way of mathematical knowledge management.</p> <p>Well, enough for now, as I’m not even sure the above formatting will work here. Let’s look for a wiki discussion page where it will be easier to talk. Either at MyWikiBiz, one of your Subject Wikis, or if you like email archiving there is the Inquiry List that an e-friend set up for me.</p> <p>[http://blog.subwiki.org/?p=13&cpage=1#comment-6 Jon Awbrey, 24 Feb 2009, 20:42 UTC]</p> </blockquote> VN: Jon, I agree with your basic ideas. There's of course the issue that the space <math>X</math> under consideration is often ''not'' a ''set'' (it's too large to be one -- for instance, a [[Groupprops:group property|group property]] is defined as a ''function'' on the collection of all groups), so we cnanot simply treat it as a set map. Still, it behaves largely like a set map. And as you point out, properties behave a lot like propositions with the ''parameter'' being from the space <math>X</math>. At some point in time, I was interested in the mathematical aspects of properties. Properties form a Boolean algebra, inheriting conjunction and disjunction etc. (in set-theoretic jargon, this would just be the power set of <math>X</math>, except that <math>X</math> isn't a set). Then, we can use various partial binary operations on <math>X</math> to define corresponding binary operations on the property space. I believe this, too, has been explored by some people, who came up with structures such as quantales and gaggles. The broad idea, as far as I understand, is that the binary operation must distribute over disjunctions (ORs). Many of the operators that I discuss in the group theory wiki, such as the [[groupprops:composition operator for subgroup properties|composition operator for subgroup properties]] and the [[groupprops:join operator|join operator]], are quantalic operators. Some of the terminology that I use for some property-theoretic notation is borrowed from terminology used for such operators, for instance, the [[groupprops:left residual operator for composition|left residual]] and [[groupprops:right residual operator for composition|right residual]]. Some of the basic structural theorems and approaches are also taken from corresponding ideas that already existed in logic. (I arrived at some of them before becoming aware of the corresponding structures explored in logic, and later found that much of this had been done before). As far as I know, though, there has been no systematic application of the ideas developed in different parts of logic to this ''real world'' (?) application to understanding properties in different mathematical domains. This puzzled me since I think that using the ideas of logic and logical structure in this way can help achieve a better understanding of many disciplines, particularly those in mathematics and computer science. Perhaps it is because there are no deep theorems here? My own use of property theory as an organizational principle seems to rely very minimally on the logical ideas beneath and rests more on the practical results that are important within the specific subject. But perhaps there are deeper theoretical logic issues that should be explored. I'd be glad to hear about your perspective, logical, philosophical, or any other. I'd also like to hear more about your views on what you mean by supporting such a model computationally. On the group theory wiki, I'm using MediaWiki's basic tools as well as Semantic MediaWiki to help in property-theoretic exploration. This isn't ''smart'' in any sense on the machine's part, but it meets some of the practical needs. Nonetheless, if you have ideas for something more sophisticated, I'd be glad to hear. [[User:Vipul|Vipul]] 22:13, 25 February 2009 (UTC) JA: Sure, there is all that. My main interest at present is focused on using logic as a tool for practical knowledge management in mathematical domains. That means starting small, breaking off this or that chunk of suitably interesting or useful domains and representing them in such a way that you can not only get computers to help you reason about them but in such a way that human beings can grok what's going on. JA: Uh-oh, I'm being called to dinner, I will have to continue later tonight. [[User:Jon Awbrey|Jon Awbrey]] 23:08, 25 February 2009 (UTC) VN: Well, that's interesting. I'm myself interested in the practical aspects, too, though perhaps from a different angle. I'd like to hear more. I'll also go through some of your past writings to understand your perspective. If you have comments on the way things have been done on the subject wikis specifically, I'd like to hear those as well. [[User:Vipul|Vipul]] 00:00, 26 February 2009 (UTC) JA: Maybe it will be quickest just to list a few links. Most of the longer pieces at MyWikiBiz are unpublished papers and project workups of mine that I'm still in the process TeXing and Wikifying and redoing the graphics that got mushed up moving between different platforms over the years, so most of them have a point where I last left off reformatting or rewriting. At any rate, here are a few serving suggestions on the basic logic side of things: :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Propositional_Calculus Differential Propositional Calculus] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Functional_Logic_:_Quantification_Theory Functional Logic : Quantification Theory] JA: [[User:Jon Awbrey|Jon Awbrey]] 03:26, 26 February 2009 (UTC) VN: Jon, I've started looking at your work. Your language and approach are new to me, so it'll take me some time to go through it. Other work has also come up. I hope to get back to you in a couple of weeks. [[User:Vipul|Vipul]] 18:05, 1 March 2009 (UTC) JA: No particular rush. I went looking for some old e-lectures of mine where I worked up much more concrete examples, but they need to be converted from ASCII to HTML-TeX-Wiki formats to be readable at all, and that will take me a while. I will probably keep outlining the occasional notes here in the interval. Later, [[User:Jon Awbrey|Jon Awbrey]] 22:40, 1 March 2009 (UTC) ===Scope and Purpose of a Logic Wiki=== ; Objectives : Maintain a focus of active methods that is parallel to the focus on static content. :: Algorithms :: Computational tools :: Efficient calculi :: Heuristics :: Proofs : Here's a project that looks interesting in this connection &mdash; :: [http://www.mat.univie.ac.at/~neum/FMathL.html Arnold Neumaier : Automated Mathematical Research Assistant] VN: Jon, that certainly looks interesting. Thanks for the link. [[User:Vipul|Vipul]] 15:57, 7 March 2009 (UTC) VN: I think the discussion has two parallel tracks here: what a logic wiki should contain and what logical paradigms can be used efficiently to organize content in all subject wikis (may be you're talking of only one of these right now). VN: Regarding the organization of content across subject wikis. I had a quick look at the FMathL link. From what I can figure out, this is interesting, but still preliminary. What I'm interested in right now is something actionable that I can use to make the definitions and proofs easier and quicker to understand, easier to link to, etc. I'm sure projects such as FMathL can give some useful ideas in that connection, so I'd be glad to hear what ideas you've extracted from there. I [http://blog.subwiki.org/?p=25 wrote a recent blog post] describing some of the issues that I'm thinking about, or rather, the place I'm starting from. VN: Regarding a specific logic wiki, it seems to me that logic is an extremely big subject. Your comments and past writings give me the impression that you're interested more in the proof theory/proof calculus part of the subject. I'm assuming you're not interested so much in the model theory side of things. Okay, so how about calling it a proof theory wiki or something like that? Of course, I might be misunderstanding the scope, so more clarification from you is welcome. I'm also not sure how Neumaier's automated research assistant connects up with the content of the proposed wiki. Are you sugesting that FMathL's ideas be used to organize the wiki, or that the wiki contain information about FMathL? [[User:Vipul|Vipul]] 23:54, 11 March 2009 (UTC) JA: First time I've stopped by this way in a while. It will take me a while longer to work up the concrete example that I had in mind, which is currently in progress [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Analytic_Turing_Automata here]. JA: I will have to leave the whole of logic to others. The most I can hope to do with my time here is maybe to grease the wheels of our practical vehicles at critical points. JA: Lunch''!'' Back later &hellip; [[User:Jon Awbrey|Jon Awbrey]] 16:02, 16 March 2009 (UTC) ===Narrative Memoir=== JA: I thought it might help to switch from didactic to personal narrative. Recent discussions on various lists reminded of why I first started work on "automatic theorem proving" (ATM) or "mechanized mathematical reasoning", as they called it back then. I had long been interested in the interplay between algebra, geometry, and logic &mdash; what one of my professors later called "the lambda point" &mdash; and when I started grad school in math in the late 70's this interest congealed into a study of the linkages between group theory, number theory, graph theory, and logic. On the graph theory face of it I got bitten by the Reconstruction Conjecture bug and eventually worked out a group-theoretic (orbit counting) way of looking at the problem. But it was all getting to be a bit too much for my unaided brain, and so one of my office-mates suggested that I try bringing the mainframe to bear on it. I was more than hesitant at first, but had been taking courses in Lisp, so I eventually said <math>y_0?\!</math> and went to work on trying to write a theorem prover. [[User:Jon Awbrey|Jon Awbrey]] 20:16, 16 March 2009 (UTC) JA: To be continued &hellip; 6634283b856085f05e387c04de2b163bf79c18c9 349 346 2009-03-21T14:32:09Z Jon Awbrey 3 /* Scope and Purpose of a Logic Wiki */ link label wikitext text/x-wiki This discussion page is intended for a discussion on using property theory as a means of organizing material in subject wikis, as well as the underlying issues of mathematics, logic, computation and philosophy. [[User:Vipul|Vipul]] 20:20, 25 February 2009 (UTC) ==Prospects for a Logic Wiki== ===Initial Discussion=== Previously on Buffer &hellip; * [http://blog.subwiki.org/?p=13 The History of Subject Wikis] * [http://blog.subwiki.org/?p=13&cpage=1#comment-2 Comments] <blockquote> <p>Vipul,</p> <p>From the initial sample of your writing distribution that I’ve read so far, your approach to mathematical objects by way of their properties is essentially a logical angle.</p> <p>If one elects to think in functional, computational terms from the very beginning, then it is convenient to think of a property <math>p\!</math> as being a mapping from a space <math>X,\!</math> the universe of discourse, to a space of 2 elements, say, <math>\mathbb{B} = \{ 0, 1 \}.</math> So a property is something of the form <math>p : X \to \mathbb{B}.</math> That level of consideration amounts to propositional calculus. Trivial as it may seem, it’s been my experience that getting efficient computational support at this level is key to many of the things we’d like to do at higher levels in the way of mathematical knowledge management.</p> <p>Well, enough for now, as I’m not even sure the above formatting will work here. Let’s look for a wiki discussion page where it will be easier to talk. Either at MyWikiBiz, one of your Subject Wikis, or if you like email archiving there is the Inquiry List that an e-friend set up for me.</p> <p>[http://blog.subwiki.org/?p=13&cpage=1#comment-6 Jon Awbrey, 24 Feb 2009, 20:42 UTC]</p> </blockquote> VN: Jon, I agree with your basic ideas. There's of course the issue that the space <math>X</math> under consideration is often ''not'' a ''set'' (it's too large to be one -- for instance, a [[Groupprops:group property|group property]] is defined as a ''function'' on the collection of all groups), so we cnanot simply treat it as a set map. Still, it behaves largely like a set map. And as you point out, properties behave a lot like propositions with the ''parameter'' being from the space <math>X</math>. At some point in time, I was interested in the mathematical aspects of properties. Properties form a Boolean algebra, inheriting conjunction and disjunction etc. (in set-theoretic jargon, this would just be the power set of <math>X</math>, except that <math>X</math> isn't a set). Then, we can use various partial binary operations on <math>X</math> to define corresponding binary operations on the property space. I believe this, too, has been explored by some people, who came up with structures such as quantales and gaggles. The broad idea, as far as I understand, is that the binary operation must distribute over disjunctions (ORs). Many of the operators that I discuss in the group theory wiki, such as the [[groupprops:composition operator for subgroup properties|composition operator for subgroup properties]] and the [[groupprops:join operator|join operator]], are quantalic operators. Some of the terminology that I use for some property-theoretic notation is borrowed from terminology used for such operators, for instance, the [[groupprops:left residual operator for composition|left residual]] and [[groupprops:right residual operator for composition|right residual]]. Some of the basic structural theorems and approaches are also taken from corresponding ideas that already existed in logic. (I arrived at some of them before becoming aware of the corresponding structures explored in logic, and later found that much of this had been done before). As far as I know, though, there has been no systematic application of the ideas developed in different parts of logic to this ''real world'' (?) application to understanding properties in different mathematical domains. This puzzled me since I think that using the ideas of logic and logical structure in this way can help achieve a better understanding of many disciplines, particularly those in mathematics and computer science. Perhaps it is because there are no deep theorems here? My own use of property theory as an organizational principle seems to rely very minimally on the logical ideas beneath and rests more on the practical results that are important within the specific subject. But perhaps there are deeper theoretical logic issues that should be explored. I'd be glad to hear about your perspective, logical, philosophical, or any other. I'd also like to hear more about your views on what you mean by supporting such a model computationally. On the group theory wiki, I'm using MediaWiki's basic tools as well as Semantic MediaWiki to help in property-theoretic exploration. This isn't ''smart'' in any sense on the machine's part, but it meets some of the practical needs. Nonetheless, if you have ideas for something more sophisticated, I'd be glad to hear. [[User:Vipul|Vipul]] 22:13, 25 February 2009 (UTC) JA: Sure, there is all that. My main interest at present is focused on using logic as a tool for practical knowledge management in mathematical domains. That means starting small, breaking off this or that chunk of suitably interesting or useful domains and representing them in such a way that you can not only get computers to help you reason about them but in such a way that human beings can grok what's going on. JA: Uh-oh, I'm being called to dinner, I will have to continue later tonight. [[User:Jon Awbrey|Jon Awbrey]] 23:08, 25 February 2009 (UTC) VN: Well, that's interesting. I'm myself interested in the practical aspects, too, though perhaps from a different angle. I'd like to hear more. I'll also go through some of your past writings to understand your perspective. If you have comments on the way things have been done on the subject wikis specifically, I'd like to hear those as well. [[User:Vipul|Vipul]] 00:00, 26 February 2009 (UTC) JA: Maybe it will be quickest just to list a few links. Most of the longer pieces at MyWikiBiz are unpublished papers and project workups of mine that I'm still in the process TeXing and Wikifying and redoing the graphics that got mushed up moving between different platforms over the years, so most of them have a point where I last left off reformatting or rewriting. At any rate, here are a few serving suggestions on the basic logic side of things: :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Propositional_Calculus Differential Propositional Calculus] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Functional_Logic_:_Quantification_Theory Functional Logic : Quantification Theory] JA: [[User:Jon Awbrey|Jon Awbrey]] 03:26, 26 February 2009 (UTC) VN: Jon, I've started looking at your work. Your language and approach are new to me, so it'll take me some time to go through it. Other work has also come up. I hope to get back to you in a couple of weeks. [[User:Vipul|Vipul]] 18:05, 1 March 2009 (UTC) JA: No particular rush. I went looking for some old e-lectures of mine where I worked up much more concrete examples, but they need to be converted from ASCII to HTML-TeX-Wiki formats to be readable at all, and that will take me a while. I will probably keep outlining the occasional notes here in the interval. Later, [[User:Jon Awbrey|Jon Awbrey]] 22:40, 1 March 2009 (UTC) ===Scope and Purpose of a Logic Wiki=== ; Objectives : Maintain a focus of active methods that is parallel to the focus on static content. :: Algorithms :: Computational tools :: Efficient calculi :: Heuristics :: Proofs : Here's a project that looks interesting in this connection &mdash; :: [http://www.mat.univie.ac.at/~neum/FMathL.html Arnold Neumaier : Automated Mathematical Research Assistant] VN: Jon, that certainly looks interesting. Thanks for the link. [[User:Vipul|Vipul]] 15:57, 7 March 2009 (UTC) VN: I think the discussion has two parallel tracks here: what a logic wiki should contain and what logical paradigms can be used efficiently to organize content in all subject wikis (may be you're talking of only one of these right now). VN: Regarding the organization of content across subject wikis. I had a quick look at the FMathL link. From what I can figure out, this is interesting, but still preliminary. What I'm interested in right now is something actionable that I can use to make the definitions and proofs easier and quicker to understand, easier to link to, etc. I'm sure projects such as FMathL can give some useful ideas in that connection, so I'd be glad to hear what ideas you've extracted from there. I [http://blog.subwiki.org/?p=25 wrote a recent blog post] describing some of the issues that I'm thinking about, or rather, the place I'm starting from. VN: Regarding a specific logic wiki, it seems to me that logic is an extremely big subject. Your comments and past writings give me the impression that you're interested more in the proof theory/proof calculus part of the subject. I'm assuming you're not interested so much in the model theory side of things. Okay, so how about calling it a proof theory wiki or something like that? Of course, I might be misunderstanding the scope, so more clarification from you is welcome. I'm also not sure how Neumaier's automated research assistant connects up with the content of the proposed wiki. Are you sugesting that FMathL's ideas be used to organize the wiki, or that the wiki contain information about FMathL? [[User:Vipul|Vipul]] 23:54, 11 March 2009 (UTC) JA: First time I've stopped by this way in a while. It will take me a while longer to work up the concrete example that I had in mind, which is currently in progress here: :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Analytic_Turing_Automata Differential Analytic Turing Automata (DATA)]. JA: I will have to leave the whole of logic to others. The most I can hope to do with my time here is maybe to grease the wheels of our practical vehicles at critical points. JA: Lunch''!'' Back later &hellip; [[User:Jon Awbrey|Jon Awbrey]] 16:02, 16 March 2009 (UTC) ===Narrative Memoir=== JA: I thought it might help to switch from didactic to personal narrative. Recent discussions on various lists reminded of why I first started work on "automatic theorem proving" (ATM) or "mechanized mathematical reasoning", as they called it back then. I had long been interested in the interplay between algebra, geometry, and logic &mdash; what one of my professors later called "the lambda point" &mdash; and when I started grad school in math in the late 70's this interest congealed into a study of the linkages between group theory, number theory, graph theory, and logic. On the graph theory face of it I got bitten by the Reconstruction Conjecture bug and eventually worked out a group-theoretic (orbit counting) way of looking at the problem. But it was all getting to be a bit too much for my unaided brain, and so one of my office-mates suggested that I try bringing the mainframe to bear on it. I was more than hesitant at first, but had been taking courses in Lisp, so I eventually said <math>y_0?\!</math> and went to work on trying to write a theorem prover. [[User:Jon Awbrey|Jon Awbrey]] 20:16, 16 March 2009 (UTC) JA: To be continued &hellip; fcd84099aea05822b436a4ff39fbd3fb3f7ce766 350 349 2009-03-31T14:02:34Z Jon Awbrey 3 /* Scope and Purpose of a Logic Wiki */ wikitext text/x-wiki This discussion page is intended for a discussion on using property theory as a means of organizing material in subject wikis, as well as the underlying issues of mathematics, logic, computation and philosophy. [[User:Vipul|Vipul]] 20:20, 25 February 2009 (UTC) ==Prospects for a Logic Wiki== ===Initial Discussion=== Previously on Buffer &hellip; * [http://blog.subwiki.org/?p=13 The History of Subject Wikis] * [http://blog.subwiki.org/?p=13&cpage=1#comment-2 Comments] <blockquote> <p>Vipul,</p> <p>From the initial sample of your writing distribution that I’ve read so far, your approach to mathematical objects by way of their properties is essentially a logical angle.</p> <p>If one elects to think in functional, computational terms from the very beginning, then it is convenient to think of a property <math>p\!</math> as being a mapping from a space <math>X,\!</math> the universe of discourse, to a space of 2 elements, say, <math>\mathbb{B} = \{ 0, 1 \}.</math> So a property is something of the form <math>p : X \to \mathbb{B}.</math> That level of consideration amounts to propositional calculus. Trivial as it may seem, it’s been my experience that getting efficient computational support at this level is key to many of the things we’d like to do at higher levels in the way of mathematical knowledge management.</p> <p>Well, enough for now, as I’m not even sure the above formatting will work here. Let’s look for a wiki discussion page where it will be easier to talk. Either at MyWikiBiz, one of your Subject Wikis, or if you like email archiving there is the Inquiry List that an e-friend set up for me.</p> <p>[http://blog.subwiki.org/?p=13&cpage=1#comment-6 Jon Awbrey, 24 Feb 2009, 20:42 UTC]</p> </blockquote> VN: Jon, I agree with your basic ideas. There's of course the issue that the space <math>X</math> under consideration is often ''not'' a ''set'' (it's too large to be one -- for instance, a [[Groupprops:group property|group property]] is defined as a ''function'' on the collection of all groups), so we cnanot simply treat it as a set map. Still, it behaves largely like a set map. And as you point out, properties behave a lot like propositions with the ''parameter'' being from the space <math>X</math>. At some point in time, I was interested in the mathematical aspects of properties. Properties form a Boolean algebra, inheriting conjunction and disjunction etc. (in set-theoretic jargon, this would just be the power set of <math>X</math>, except that <math>X</math> isn't a set). Then, we can use various partial binary operations on <math>X</math> to define corresponding binary operations on the property space. I believe this, too, has been explored by some people, who came up with structures such as quantales and gaggles. The broad idea, as far as I understand, is that the binary operation must distribute over disjunctions (ORs). Many of the operators that I discuss in the group theory wiki, such as the [[groupprops:composition operator for subgroup properties|composition operator for subgroup properties]] and the [[groupprops:join operator|join operator]], are quantalic operators. Some of the terminology that I use for some property-theoretic notation is borrowed from terminology used for such operators, for instance, the [[groupprops:left residual operator for composition|left residual]] and [[groupprops:right residual operator for composition|right residual]]. Some of the basic structural theorems and approaches are also taken from corresponding ideas that already existed in logic. (I arrived at some of them before becoming aware of the corresponding structures explored in logic, and later found that much of this had been done before). As far as I know, though, there has been no systematic application of the ideas developed in different parts of logic to this ''real world'' (?) application to understanding properties in different mathematical domains. This puzzled me since I think that using the ideas of logic and logical structure in this way can help achieve a better understanding of many disciplines, particularly those in mathematics and computer science. Perhaps it is because there are no deep theorems here? My own use of property theory as an organizational principle seems to rely very minimally on the logical ideas beneath and rests more on the practical results that are important within the specific subject. But perhaps there are deeper theoretical logic issues that should be explored. I'd be glad to hear about your perspective, logical, philosophical, or any other. I'd also like to hear more about your views on what you mean by supporting such a model computationally. On the group theory wiki, I'm using MediaWiki's basic tools as well as Semantic MediaWiki to help in property-theoretic exploration. This isn't ''smart'' in any sense on the machine's part, but it meets some of the practical needs. Nonetheless, if you have ideas for something more sophisticated, I'd be glad to hear. [[User:Vipul|Vipul]] 22:13, 25 February 2009 (UTC) JA: Sure, there is all that. My main interest at present is focused on using logic as a tool for practical knowledge management in mathematical domains. That means starting small, breaking off this or that chunk of suitably interesting or useful domains and representing them in such a way that you can not only get computers to help you reason about them but in such a way that human beings can grok what's going on. JA: Uh-oh, I'm being called to dinner, I will have to continue later tonight. [[User:Jon Awbrey|Jon Awbrey]] 23:08, 25 February 2009 (UTC) VN: Well, that's interesting. I'm myself interested in the practical aspects, too, though perhaps from a different angle. I'd like to hear more. I'll also go through some of your past writings to understand your perspective. If you have comments on the way things have been done on the subject wikis specifically, I'd like to hear those as well. [[User:Vipul|Vipul]] 00:00, 26 February 2009 (UTC) JA: Maybe it will be quickest just to list a few links. Most of the longer pieces at MyWikiBiz are unpublished papers and project workups of mine that I'm still in the process TeXing and Wikifying and redoing the graphics that got mushed up moving between different platforms over the years, so most of them have a point where I last left off reformatting or rewriting. At any rate, here are a few serving suggestions on the basic logic side of things: :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Propositional_Calculus Differential Propositional Calculus] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Functional_Logic_:_Quantification_Theory Functional Logic : Quantification Theory] JA: [[User:Jon Awbrey|Jon Awbrey]] 03:26, 26 February 2009 (UTC) VN: Jon, I've started looking at your work. Your language and approach are new to me, so it'll take me some time to go through it. Other work has also come up. I hope to get back to you in a couple of weeks. [[User:Vipul|Vipul]] 18:05, 1 March 2009 (UTC) JA: No particular rush. I went looking for some old e-lectures of mine where I worked up much more concrete examples, but they need to be converted from ASCII to HTML-TeX-Wiki formats to be readable at all, and that will take me a while. I will probably keep outlining the occasional notes here in the interval. Later, [[User:Jon Awbrey|Jon Awbrey]] 22:40, 1 March 2009 (UTC) ===Scope and Purpose of a Logic Wiki=== ; Objectives : Maintain a focus of active methods that is parallel to the focus on static content. :: Algorithms :: Computational tools :: Efficient calculi :: Heuristics :: Proofs : Here's a project that looks interesting in this connection &mdash; :: [http://www.mat.univie.ac.at/~neum/FMathL.html Arnold Neumaier : Automated Mathematical Research Assistant] VN: Jon, that certainly looks interesting. Thanks for the link. [[User:Vipul|Vipul]] 15:57, 7 March 2009 (UTC) VN: I think the discussion has two parallel tracks here: what a logic wiki should contain and what logical paradigms can be used efficiently to organize content in all subject wikis (may be you're talking of only one of these right now). VN: Regarding the organization of content across subject wikis. I had a quick look at the FMathL link. From what I can figure out, this is interesting, but still preliminary. What I'm interested in right now is something actionable that I can use to make the definitions and proofs easier and quicker to understand, easier to link to, etc. I'm sure projects such as FMathL can give some useful ideas in that connection, so I'd be glad to hear what ideas you've extracted from there. I [http://blog.subwiki.org/?p=25 wrote a recent blog post] describing some of the issues that I'm thinking about, or rather, the place I'm starting from. VN: Regarding a specific logic wiki, it seems to me that logic is an extremely big subject. Your comments and past writings give me the impression that you're interested more in the proof theory/proof calculus part of the subject. I'm assuming you're not interested so much in the model theory side of things. Okay, so how about calling it a proof theory wiki or something like that? Of course, I might be misunderstanding the scope, so more clarification from you is welcome. I'm also not sure how Neumaier's automated research assistant connects up with the content of the proposed wiki. Are you sugesting that FMathL's ideas be used to organize the wiki, or that the wiki contain information about FMathL? [[User:Vipul|Vipul]] 23:54, 11 March 2009 (UTC) JA: First time I've stopped by this way in a while. It will take me a while longer to work up the concrete example that I had in mind, which is currently in progress here: :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Analytic_Turing_Automata Differential Analytic Turing Automata (DATA)]. JA: I will have to leave the whole of logic to others. The most I can hope to do with my time here is maybe to grease the wheels of our practical vehicles at critical points. JA: Lunch''!'' Back later &hellip; [[User:Jon Awbrey|Jon Awbrey]] 16:02, 16 March 2009 (UTC) JA: Just passing through. Currently reviving some old work I did on the Propositions As Types Analogy, located here: :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositions_As_Types Propositions As Types] ===Narrative Memoir=== JA: I thought it might help to switch from didactic to personal narrative. Recent discussions on various lists reminded of why I first started work on "automatic theorem proving" (ATM) or "mechanized mathematical reasoning", as they called it back then. I had long been interested in the interplay between algebra, geometry, and logic &mdash; what one of my professors later called "the lambda point" &mdash; and when I started grad school in math in the late 70's this interest congealed into a study of the linkages between group theory, number theory, graph theory, and logic. On the graph theory face of it I got bitten by the Reconstruction Conjecture bug and eventually worked out a group-theoretic (orbit counting) way of looking at the problem. But it was all getting to be a bit too much for my unaided brain, and so one of my office-mates suggested that I try bringing the mainframe to bear on it. I was more than hesitant at first, but had been taking courses in Lisp, so I eventually said <math>y_0?\!</math> and went to work on trying to write a theorem prover. [[User:Jon Awbrey|Jon Awbrey]] 20:16, 16 March 2009 (UTC) JA: To be continued &hellip; c619e2ea84d36e3eca6c00e9264124a861fa98cf 0 9 347 53 2009-03-21T01:16:03Z Vipul 2 wikitext text/x-wiki <noinclude>[[Status::Basic definition| ]][[Topic::Group theory| ]][[Primary wiki::Groupprops| ]]</noinclude> '''Normal subgroup''': A subgroup of a group that occurs as the kernel of a homomorphism, or equivalently, such that every left coset and right coset are equal. Related terms: Normality (the property of a subgroup being normal), [[Groupprops:Normal core|normal core]] (the largest normal subgroup contained in a given subgroup), [[Groupprops:Normal closure|normal closure]] (the smallest normal subgroup containing a given subgroup), [[Groupprops:Normalizer|normalizer]] (the largest subgroup containing a given subgroup, in which it is normal), [[Groupprops:Normal automorphism|normal automorphism]] (an automorphism that preserves every normal subgroup) Term variations: [[Groupprops:Subnormal subgroup|subnormal subgroup]], [[Groupprops:Abnormal subgroup|abnormal subgroup]], [[Groupprops:Quasinormal subgroup|quasinormal subgroup]], and others. See [[Groupprops:Category:Variations of normality]], [[Groupprops:Category:Opposites of normality]], and [[Groupprops:Category:Analogues of normality]]. Primary subject wiki entry: [[Groupprops:Normal subgroup]]. Other subject wiki entries: [[Diffgeom:Normal subgroup]] Coverage in guided tours: Groupprops guided tour for beginners; section not yet prepared. Also located at: [[Wikipedia:Normal subgroup]], [[Planetmath:NormalSubgroup]], [[Mathworld:NormalSubgroup]], [[Sor:N/n067690|Springer Online Reference Works]], [[Citizendium:Normal subgroup]] 5069c7efe17f4cacd64cd2689b65071146da26fd 348 347 2009-03-21T01:16:35Z Vipul 2 wikitext text/x-wiki <noinclude>[[Status::Basic definition| ]][[Topic::Group theory| ]][[Primary wiki::Groupprops| ]]</noinclude> '''Normal subgroup''': A subgroup of a group that occurs as the kernel of a homomorphism, or equivalently, such that every left coset and right coset are equal. Related terms: Normality (the property of a subgroup being normal), [[Groupprops:Normal core|normal core]] (the largest normal subgroup contained in a given subgroup), [[Groupprops:Normal closure|normal closure]] (the smallest normal subgroup containing a given subgroup), [[Groupprops:Normalizer|normalizer]] (the largest subgroup containing a given subgroup, in which it is normal), [[Groupprops:Normal automorphism|normal automorphism]] (an automorphism that restricts to an automorphism on every normal subgroup). Term variations: [[Groupprops:Subnormal subgroup|subnormal subgroup]], [[Groupprops:Abnormal subgroup|abnormal subgroup]], [[Groupprops:Quasinormal subgroup|quasinormal subgroup]], and others. See [[Groupprops:Category:Variations of normality]], [[Groupprops:Category:Opposites of normality]], and [[Groupprops:Category:Analogues of normality]]. Primary subject wiki entry: [[Groupprops:Normal subgroup]]. Other subject wiki entries: [[Diffgeom:Normal subgroup]] Coverage in guided tours: Groupprops guided tour for beginners; section not yet prepared. Also located at: [[Wikipedia:Normal subgroup]], [[Planetmath:NormalSubgroup]], [[Mathworld:NormalSubgroup]], [[Sor:N/n067690|Springer Online Reference Works]], [[Citizendium:Normal subgroup]] 92fe4766d942a7fee17c36cb37a2f1d1dfd63d38 106 22 351 305 2009-04-25T18:23:07Z Vipul 2 wikitext text/x-wiki * [[Groupprops:Main Page|Groupprops]]: The Group Properties Wiki (currently, around 2000 pages). Main topic is group theory * [[Topospaces:Main Page|Topospaces]]: The Topology Wiki (currently, around 400 pages). Main topics are point-set topology and algebraic topology * [[Commalg:Main Page|Commalg]]: The Commutative Algebra Wiki (currently, around 300 pages). Main topic is commutative algebra, with some algebraic geometry. * [[Diffgeom:Main Page|Diffgeom]]: The Differential Geometry Wiki * [[Measure:Main Page|Measure]]: The Measure Theory Wiki (to be set up) * [[Noncommutative:Main Page|Noncommutative]]: The Noncommutative Algebra Wiki * [[Companal:Main Page|Companal]]: The Complex Analysis Wiki * [[Cattheory:Main Page|Cattheory]]: The Category Theory Wiki (just started) * [[Market:Main Page|Market]]: The Economic Theory of Markets, Choices and Prices Wiki * [[Number:Main page|Number]]: The Number Theory Wiki Some upcoming subject wikis: * Mech: The Classical Mechanics Wiki * Linear: The Theoretical and Computational Linear Algebra Wiki * Galois: The Field Theory and Galois Theory Wiki * Graph: The Graph Theory Wiki * Complexity: The Complexity Theory Wiki 847277ccc92384e8f5ce74bfe87cf16ea9837de7 389 351 2010-01-18T15:40:11Z Vipul 2 wikitext text/x-wiki * [[Groupprops:Main Page|Groupprops]]: The Group Properties Wiki (currently, around 2000 pages). Main topic is group theory * [[Topospaces:Main Page|Topospaces]]: The Topology Wiki (currently, around 400 pages). Main topics are point-set topology and algebraic topology * [[Commalg:Main Page|Commalg]]: The Commutative Algebra Wiki (currently, around 300 pages). Main topic is commutative algebra, with some algebraic geometry. * [[Diffgeom:Main Page|Diffgeom]]: The Differential Geometry Wiki * [[Measure:Main Page|Measure]]: The Measure Theory Wiki (to be set up) * [[Noncommutative:Main Page|Noncommutative]]: The Noncommutative Algebra Wiki * [[Companal:Main Page|Companal]]: The Complex Analysis Wiki * [[Cattheory:Main Page|Cattheory]]: The Category Theory Wiki (just started) * [[Market:Main Page|Market]]: The Economic Theory of Markets, Choices and Prices Wiki * [[Number:Main Page|Number]]: The Number Theory Wiki * [[Mech:Main Page|Mech]]: The Classical Mechanics Wiki * [[Galois:Main Page|Galois]]: The Galois Theory Wiki Some upcoming subject wikis: * Linear: The Theoretical and Computational Linear Algebra Wiki * Graph: The Graph Theory Wiki * Complexity: The Complexity Theory Wiki b4caa7e23005689048a4b8294ff85b1fab1cc7cf 0 77 352 154 2009-04-25T18:24:10Z Vipul 2 wikitext text/x-wiki <noinclude>[[Topic::Algebraic number theory| ]][[Primary wiki::Number| ]]</noinclude> '''Regular prime''': A regular prime is a prime number <math>p</math> that does not divide the class number of the cyclotomic field obtained by adjoining <math>p^{th}</math> roots of unity to the field of rational numbers. Pimary subject wiki entry: [[Number:Regular prime]] Also located at [[Wikipedia:Regular prime]], [[Mathworld:RegularPrime]], [[Planetmath:RegularPrime]] d20ed22c23c3f5b0373453202a0a555f04b09a31 0 54 353 109 2009-04-25T18:26:43Z Vipul 2 wikitext text/x-wiki <noinclude>[[Status::Basic definition| ]][[Topic::Number theory| ]][[Primary wiki::Number| ]]</noinclude> '''Perfect number''': A natural number that equals the sum of all its proper (positive) divisors. Primary subject wiki entry: [[Number:Perfect number]] Also located at: [[Wikipedia:Perfect number]], [[Mathworld:PerfectNumber]], [[Planetmath:PerfectNumber]] 3c1d5790d585edcc624732d44303f1c1ef4565e7 0 166 354 2009-05-05T23:37:38Z Vipul 2 Created page with 'Pierre de Fermat was a French lawyer and amateur mathematician. A number of different mathematical definitions, theorems, conjectures, and methods are named after him. {{:Fermat...' wikitext text/x-wiki Pierre de Fermat was a French lawyer and amateur mathematician. A number of different mathematical definitions, theorems, conjectures, and methods are named after him. {{:Fermat (mathematics)}} a12bb37cfcb62ff25e6e52f43a371ed6a0af38e3 361 354 2009-05-05T23:48:46Z Vipul 2 wikitext text/x-wiki Pierre de Fermat was a French lawyer and amateur mathematician. A number of different mathematical definitions, theorems, conjectures, and methods are named after him. ==Mathematics== {{:Fermat (mathematics)}} d145640d4d920a66d73c0c08c3a813c13c6fdbba 0 167 355 2009-05-05T23:38:38Z Vipul 2 Created page with '===In number theory=== {{:Fermat's little theorem}} {{:Fermat's last theorem}} {{:Fermat number}} {{:Fermat prime}} {{:Fermat pseudoprime}}' wikitext text/x-wiki ===In number theory=== {{:Fermat's little theorem}} {{:Fermat's last theorem}} {{:Fermat number}} {{:Fermat prime}} {{:Fermat pseudoprime}} 4d528c13183f77da8b106a9364a6409e1ba4442d 362 355 2009-05-05T23:51:07Z Vipul 2 wikitext text/x-wiki ===In number theory=== {{:Fermat's little theorem}} {{:Fermat's last theorem}} {{:Fermat number}} {{:Fermat prime}} {{:Fermat pseudoprime}} ===In geometry=== {{:Fermat point of a triangle}} ae075d80b383c329f98f7a4b0d22cb6af841c68f 0 168 356 2009-05-05T23:41:03Z Vipul 2 Created page with '<noinclude>[[Primary wiki::Number| ]][[Topic::Number theory| ]]</noinclude> '''Fermat's little theorem''': If <math>p</math> is a prime number and <math>a</math> is an integer th...' wikitext text/x-wiki <noinclude>[[Primary wiki::Number| ]][[Topic::Number theory| ]]</noinclude> '''Fermat's little theorem''': If <math>p</math> is a prime number and <math>a</math> is an integer that is not a multiple of <math>p</math>, then <math>p</math> divides <math>a^{p-1} - 1</math>. Primary subject wiki entry: [[Number:Fermat's little theorem]]. d0e8f4f74bcb6b9fbca83d8b9b4b27701beb9118 0 169 357 2009-05-05T23:42:30Z Vipul 2 Created page with '<noinclude>[[Topic::Number theory| ]][[Primary wiki::Number| ]]</noinclude> '''Fermat's last theorem''': The equation: <math>x^n + y^n = z ^n</math> has no positive integer sol...' wikitext text/x-wiki <noinclude>[[Topic::Number theory| ]][[Primary wiki::Number| ]]</noinclude> '''Fermat's last theorem''': The equation: <math>x^n + y^n = z ^n</math> has no positive integer solutions for <math>x,y,z,n</math> with <math>n > 2</math>. Primary subject wiki entry: [[Number:Fermat's last theorem]]. 3322e027a099d48364f311143fabcf57cbbfa1bd 0 170 358 2009-05-05T23:44:00Z Vipul 2 Created page with '<noinclude>[[Topic::Number theory| ]][[Primary wiki::Number| ]]</noinclude> '''Fermat number''': This is a natural number of the form <math>2^{2^n} + 1</math> where <math>n</math...' wikitext text/x-wiki <noinclude>[[Topic::Number theory| ]][[Primary wiki::Number| ]]</noinclude> '''Fermat number''': This is a natural number of the form <math>2^{2^n} + 1</math> where <math>n</math> is a nonnegative integer. Primary subject wiki entry: [[Number:Fermat number]]. Other subject wiki entries: [[Groupprops:Fermat number]]. cd97a91cb81a520954e4e8545d52e12be7bcb467 0 171 359 2009-05-05T23:45:50Z Vipul 2 Created page with '<noinclude>[[Topic::Number theory| ]][[Primary wiki::Number| ]]</noinclude> '''Fermat pseudoprime''': A composite natural number <math>n</math> is termed a Fermat pseudoprime to ...' wikitext text/x-wiki <noinclude>[[Topic::Number theory| ]][[Primary wiki::Number| ]]</noinclude> '''Fermat pseudoprime''': A composite natural number <math>n</math> is termed a Fermat pseudoprime to base <math>a</math>, for an integer <math>a</math>, if <math>a</math> and <math>n</math> are relatively prime and <math>a^{n-1} - 1</math> is a multiple of <math>n</math>. Primary subject wiki entry: [[Number:Fermat pseudoprime]]. 9551a56a0c9ed36a1e24e2fd44604add4c6ef326 0 172 360 2009-05-05T23:47:44Z Vipul 2 Created page with '<noinclude>[[Topic::Number theory| ]][[Primary wiki::Number| ]]</noinclude> '''Fermat prime''': This is a prime number of the form <math>2^{2^n} + 1</math>, where <math>n</math> ...' wikitext text/x-wiki <noinclude>[[Topic::Number theory| ]][[Primary wiki::Number| ]]</noinclude> '''Fermat prime''': This is a prime number of the form <math>2^{2^n} + 1</math>, where <math>n</math> is a nonnegative integer. Primary subject wiki entry: [[Number:Fermat prime]]. Other subject wiki entries: [[Groupprops:Fermat prime]]. 8bd405a1336252ce1f7b499808e7a96ac7bdea14 0 173 363 2009-05-05T23:52:16Z Vipul 2 Created page with '<noinclude>[[Topic::Planar geometry| ]]</noinclude> '''Fermat point of a triangle''': This is defined as the point that minimizes the sum of distances from all the three vertices...' wikitext text/x-wiki <noinclude>[[Topic::Planar geometry| ]]</noinclude> '''Fermat point of a triangle''': This is defined as the point that minimizes the sum of distances from all the three vertices of the triangle. baa834292a0d53fb1278396cd2dce3736e58e4db 0 174 364 2009-05-05T23:58:40Z Vipul 2 Created page with ''''Riemann hypothesis''': The statement that all nontrivial zeros of the [[Riemann zeta-function]] have real part <math>1/2</math>. Related terms: [[generalized Riemann hypothes...' wikitext text/x-wiki '''Riemann hypothesis''': The statement that all nontrivial zeros of the [[Riemann zeta-function]] have real part <math>1/2</math>. Related terms: [[generalized Riemann hypothesis]], [[extended Riemann hypothesis]] Primary subject wiki entry: [[Number:Riemann hypothesis]]. Other subject wiki entries: [[Companal:Riemann hypothesis]]. Also located at: [[Wikipedia:Riemann hypothesis]], [[Mathworld:RiemannHypothesis]] 96cc17db9dedde875bf6e9a287747f62bb011d4b 0 51 365 105 2009-05-14T23:25:40Z Vipul 2 wikitext text/x-wiki <noinclude>[[Status::Basic definition| ]][[Topic::Field theory| ]]</noinclude> '''Perfect field''': A field that either has characteristic zero, or has characteristic <math>p</math> and for which the map <math>x \mapsto x^p</math> is a surjective map. Equivalently, it is a field such that every algebraic extension field for it is separable. Primary subject wiki entry: [[Galois:Perfect field]] Also located at: [[Mathworld:PerfectField]], [[Planetmath:PerfectField]] 3b9963b131020b9470c572aa02d2b62127f93efa 0 45 366 90 2009-05-14T23:26:41Z Vipul 2 wikitext text/x-wiki <noinclude>[[Status::Basic definition| ]][[Topic::Field theory| ]]</noinclude> '''Normal field extension''': A field extension is termed normal if the fixed field under the automorphism group of the extension, is precisely the base field. Primary subject wiki entry: [[Galois:Normal extension]] Also located at: [[Planetmath:NormalExtension]], [[Wikipedia:Normal extension]], [[Mathworld:NormalExtension]] fbf15acc8d683eeb90b6febd80d20691c855e5d3 0 116 367 224 2009-08-11T20:52:42Z Vipul 2 moved [[Fully characteristic subgroup]] to [[Fully invariant subgroup]] wikitext text/x-wiki <noinclude>[[Status::Semi-basic definition| ]][[Primary wiki::Groupprops| ]][[Topic::Group theory| ]]</noinclude> '''Fully characteristic subgroup''': A subgroup of a group that is invariant under all endomorphisms of the group. Primary subject wiki entry: [[Groupprops:Fully characteristic subgroup]] Also located at: [[Wikipedia:Fully characteristic subgroup]], [[Planetmath:FullyInvariantSubgroup]] 6c6ed46fb3ae785e553596ca87208daa1a2f93a0 369 367 2009-08-11T20:53:13Z Vipul 2 wikitext text/x-wiki <noinclude>[[Status::Semi-basic definition| ]][[Primary wiki::Groupprops| ]][[Topic::Group theory| ]]</noinclude> '''Fully invariant subgroup''' or '''fully characteristic subgroup''': A subgroup of a group that is invariant under all endomorphisms of the group. Primary subject wiki entry: [[Groupprops:Fully invariant subgroup]] Also located at: [[Wikipedia:Fully characteristic subgroup]], [[Planetmath:FullyInvariantSubgroup]] 7b7cacfbafef49c62867b2764e6c575f3bc1dd88 0 175 368 2009-08-11T20:52:42Z Vipul 2 moved [[Fully characteristic subgroup]] to [[Fully invariant subgroup]] wikitext text/x-wiki #REDIRECT [[Fully invariant subgroup]] fdeb05f5e2cad001b9558b0efee50c1cfba53180 0 176 370 2009-08-11T20:55:29Z Vipul 2 Created page with '<noinclude>[[Status::Semi-basic definition| ]][[Primary wiki::Groupprops| ]][[Topic::Group theory| ]]</noinclude> '''Solvable group''': A group whose derived series terminates a…' wikitext text/x-wiki <noinclude>[[Status::Semi-basic definition| ]][[Primary wiki::Groupprops| ]][[Topic::Group theory| ]]</noinclude> '''Solvable group''': A group whose derived series terminates at the trivial subgroup in finitely many steps, or equivalently, a group with a normal series where all the quotient groups are abelian groups. '''Primary subject wiki entry''': [[Groupprops:Solvable group]] daa27b38f9c4dfe67dae466f9e146a7da71b84ad 371 370 2009-08-11T20:55:55Z Vipul 2 wikitext text/x-wiki <noinclude>[[Status::Semi-basic definition| ]][[Primary wiki::Groupprops| ]][[Topic::Group theory| ]]</noinclude> '''Solvable group''': A group whose derived series terminates at the trivial subgroup in finitely many steps, or equivalently, a group with a normal series where all the quotient groups are abelian groups. Primary subject wiki entry: [[Groupprops:Solvable group]] 8f9af560cd4d6988317c4149c4f2e7ff7938ee3e 0 177 372 2009-08-11T20:58:01Z Vipul 2 Created page with '<noinclude>[[Status::Basic definition| ]][[Topic::Group theory| ]][[Primary wiki::Groupprops| ]]</noinclude> '''Abelian group''': A group in which the group operation is commuta…' wikitext text/x-wiki <noinclude>[[Status::Basic definition| ]][[Topic::Group theory| ]][[Primary wiki::Groupprops| ]]</noinclude> '''Abelian group''': A group in which the group operation is commutative. Primary subject wiki entry: [[Groupprops:Abelian group]] Coverage in guided tours: Groupprops guided tour for beginners, in [[Groupprops:Tour:Introduction one (beginners)]]: see [[Groupprops:Tour:Abelian group]]. 65298485b56b4ceb2ad80cdd8c2da2ffdeb0e721 0 178 373 2009-08-11T21:01:50Z Vipul 2 Created page with '<noinclude>[[Topic::Group theory| ]][[Primary wiki::Groupprops| ]][[Status::Semi-basic definition| ]]</noinclude> '''Nilpotent group''': A group having a central series of finit…' wikitext text/x-wiki <noinclude>[[Topic::Group theory| ]][[Primary wiki::Groupprops| ]][[Status::Semi-basic definition| ]]</noinclude> '''Nilpotent group''': A group having a central series of finite length. Equivalently, a group where the upper central series terminates in finitely many steps at the whole group. Equivalently, a group where the lower central series terminates in finitely many steps at the trivial subgroup. Term variations: [[Nilpotency class]] (also called nilpotence class). Primary subject wiki entry: [[Groupprops:Nilpotent group]] 4aa81b2cd8f9faec6291cb79664ba132929c56f4 106 179 374 2009-08-21T15:18:25Z Vipul 2 Created page with 'This page discusses some important differences between subject wikis and Wikipedia. ==About Wikipedia and similarities between subject wikis and Wikipedia== ===Quick introducti…' wikitext text/x-wiki This page discusses some important differences between subject wikis and Wikipedia. ==About Wikipedia and similarities between subject wikis and Wikipedia== ===Quick introduction=== For many people using the Internet, "Wikipedia" is synonymous with "wiki" and also with the general appearance of a Wikipedia page. Thus, a lot of users who come to subject wiki pages simply think they have come to Wikipedia or some offshoot of Wikipedia. Before proceeding, it is important to clarify some distinctions. "Wikipedia" is a specific wiki-based [http://www.wikipedia.org multilingual encyclopedia] (since you're reading this, you have probably used the [[wp:Main Page|English-language version]] and possibly many other language versions). Each language Wikipedia can be viewed as a separate encyclopedia aiming to cover the entire range of encyclopedia-worthy knowledge. All the Wikipedias use a certain software called MediaWiki, which can be downloaded [[mediawikiwiki:Main Page|here]]. This software can also be used by other sister wikis to Wikipedia, such as [http://wikisource.orgwikis independent of Wikipedia, such as those hosted at [http://www.wikia.com Wikia] or [http://www.wiki-site.com Wiki-site] or [http://editthis.info Editthis.info] or independent wikis such as [http://wikileaks.org Wikileaks] or [http://www.sourcewatch.org SourceWatch]. Just because different wikis use the same underlying software and thus often have similar page layouts ''does not mean'' that they host similar content or are organized similarly. The differences between different wikis could be as small as the difference between different newspapers or as large as the difference between a newspaper and an academic journal. ===Distinctive features of Wikipedia=== Wikipedia is a wiki-based ''encyclopedia''. Both the choice of content and organization of content reflect the goal of being an encyclopedia. Specifically, pages on topics in Wikipedia have to meet criteria of [[wikipedia:wikipedia:notability|notability]] and facts in these pages must meet criteria of [[wikipedia:wikipedia:verifiability|verifiability]]. There are also guidelines against [[Wikipedia:Wikipedia:OR|putting original research on Wikipedia]], reflecting the fact that encyclopedias are secondary and tertiary sources of knowledge rather than primary sources. 668678f46b30f7f0767fc3b3cc058741db73a841 375 374 2009-08-21T15:20:25Z Vipul 2 /* About Wikipedia and similarities between subject wikis and Wikipedia */ wikitext text/x-wiki This page discusses some important differences between subject wikis and Wikipedia. ==About Wikipedia and similarities between subject wikis and Wikipedia== ===Quick introduction=== For many people using the Internet, "Wikipedia" is synonymous with "wiki" and also with the general appearance of a Wikipedia page. Thus, a lot of users who come to subject wiki pages simply think they have come to Wikipedia or some offshoot of Wikipedia. Before proceeding, it is important to clarify some distinctions. "Wikipedia" is a specific wiki-based [http://www.wikipedia.org multilingual encyclopedia] (since you're reading this, you have probably used the [[wp:Main Page|English-language version]] and possibly many other language versions). Each language Wikipedia can be viewed as a separate encyclopedia aiming to cover the entire range of encyclopedia-worthy knowledge. All the Wikipedias use a certain software called MediaWiki, which can be downloaded [[mediawikiwiki:Main Page|here]]. This software can also be used by other sister wikis to Wikipedia, such as [http://wikisource.org Wikisource], wikis independent of Wikipedia, such as those hosted at [http://www.wikia.com Wikia] or [http://www.wiki-site.com Wiki-site] or [http://editthis.info Editthis.info] or independent wikis such as [http://wikileaks.org Wikileaks] or [http://www.sourcewatch.org SourceWatch]. Just because different wikis use the same underlying software and thus often have similar page layouts ''does not mean'' that they host similar content or are organized similarly. The differences between different wikis could be as small as the difference between different newspapers or as large as the difference between a newspaper and an academic journal. ===Distinctive features of Wikipedia=== Wikipedia is a wiki-based ''encyclopedia''. Both the choice of content and organization of content reflect the goal of being an encyclopedia. Specifically, pages on topics in Wikipedia have to meet criteria of [[wikipedia:wikipedia:notability|notability]] and facts in these pages must meet criteria of [[wikipedia:wikipedia:verifiability|verifiability]]. There are also guidelines against [[Wikipedia:Wikipedia:OR|putting original research on Wikipedia]], reflecting the fact that encyclopedias are secondary and tertiary sources of knowledge rather than primary sources. afcc6b5b76b18049b0ad34ae2ef960da58905d7d 384 375 2009-12-30T12:24:57Z Vipul 2 wikitext text/x-wiki This page discusses some important differences between subject wikis and Wikipedia. ==About Wikipedia and similarities between subject wikis and Wikipedia== ===Quick introduction=== For many people using the Internet, "Wikipedia" is synonymous with "wiki" and also with the general appearance of a Wikipedia page. Thus, a lot of users who come to subject wiki pages simply think they have come to Wikipedia or some offshoot of Wikipedia. Before proceeding, it is important to clarify some distinctions. "Wikipedia" is a specific wiki-based [http://www.wikipedia.org multilingual encyclopedia] (since you're reading this, you have probably used the [[wp:Main Page|English-language version]] and possibly many other language versions). Each language Wikipedia can be viewed as a separate encyclopedia aiming to cover the entire range of encyclopedia-worthy knowledge. All the Wikipedias use a certain software called MediaWiki, which can be downloaded [[mediawikiwiki:Main Page|here]]. This software can also be used by other sister wikis to Wikipedia, such as [http://wikisource.org Wikisource], wikis independent of Wikipedia, such as those hosted at [http://www.wikia.com Wikia] or [http://www.wiki-site.com Wiki-site] or [http://editthis.info Editthis.info] or independent wikis such as [[wikitravel:Main Page|Wikitravel]], [http://wikileaks.org Wikileaks] or [http://www.sourcewatch.org SourceWatch]. Just because different wikis use the same underlying software and thus often have similar page layouts ''does not mean'' that they host similar content or are organized similarly. The differences between different wikis could be as small as the difference between different newspapers or as large as the difference between a newspaper and an academic journal. (To confuse matters further, MediaWiki is not the only software used to power wikis. You can see a fairly long list at the [[wikipedia:wiki software|wiki software page]] or the [[wikipedia:comparison of wiki software page]] on Wikipedia. One example is [http://www.pmwiki.org/wiki/PmWiki/PmWiki PmWiki]. ===Distinctive features of Wikipedia=== Wikipedia is a wiki-based ''encyclopedia''. Both the choice of content and organization of content reflect the goal of being an encyclopedia. Specifically, pages on topics in Wikipedia have to meet criteria of [[wikipedia:wikipedia:notability|notability]] and facts in these pages must meet criteria of [[wikipedia:wikipedia:verifiability|verifiability]]. There are also guidelines against [[Wikipedia:Wikipedia:OR|putting original research on Wikipedia]], reflecting the fact that encyclopedias are secondary and tertiary sources of knowledge rather than primary sources. ==Two related reasons for the use of MediaWiki or similar software to maintain linked content== There are many reasons for choosing MediaWiki to organize, store, and maintain linked content on the web, but we focus here on two of them: * The ''organizational advantage'', which is the advantage of being able to maintain and organize a large amount of related content and the way the pieces of the content are interrelated. * The ''collaboration advantage'', which is the idea that the content can be edited collaboratively by large numbers of people. These are both related. The organizational advantage includes significant modularization, which allows different people to edit different pieces, making collaboration easier. Similarly, more collaboration helps with more organization. However, it is important to note that these two kinds of advantages, though related, are distinct. Further, some projects use MediaWiki or other wiki software primarily for the collaborative advantage -- the fact that large numbers of people can edit the same page, create new pages, etc. Here, wikis compete with mailing lists, discussion forums, and blog discussion threads. They have their advantages and disadvantages in these respects. Some of the earliest wikis, including Ward Cunningham's [http://c2.com/cgi/wiki WikiWikiWeb] and [http://www.meatballwiki.org/wiki/ MeatBallWiki] were focused on carrying out such discussions as a process to create new ideas. Other projects use MediaWiki primarily for the organizational advantage. In some of these cases, the greater ease of collaboration is crucial but still instrumental and not the final goal. In others, large-scale collaboration may be desirable but not crucial, and in yet others, large-scale collaboration may not be desirable. For instance, in the case of Wikipedia, the reason for using MediaWiki was to create a better-organized encyclopedia by combining the ease of organizing with the possibility of large-scale collaboration. Something similar is true for [[Wikitravel:Main Page|Wikitravel]]. The extent to which large-scale collaboration is crucial to the success of something like Wikipedia, and whether the collaborative process as it has currently developed on Wikipedia is intrinsic to its mission, are moot points. f737d2d09da9cdf93406a88edcc6fffaa903d601 385 384 2009-12-30T12:26:24Z Vipul 2 /* About Wikipedia and similarities between subject wikis and Wikipedia */ wikitext text/x-wiki This page discusses some important differences between subject wikis and Wikipedia. ==About Wikipedia and similarities between subject wikis and Wikipedia== ===Quick introduction=== For many people using the Internet, "Wikipedia" is synonymous with "wiki" and also with the general appearance of a Wikipedia page. Thus, a lot of users who come to subject wiki pages simply think they have come to Wikipedia or some offshoot of Wikipedia. Before proceeding, it is important to clarify some distinctions. "Wikipedia" is a specific wiki-based [http://www.wikipedia.org multilingual encyclopedia] (since you're reading this, you have probably used the [[wp:Main Page|English-language version]] and possibly many other language versions). Each language Wikipedia can be viewed as a separate encyclopedia aiming to cover the entire range of encyclopedia-worthy knowledge. All the Wikipedias use a certain software called MediaWiki, which can be downloaded [[mediawikiwiki:Main Page|here]]. This software can also be used by other sister wikis to Wikipedia, such as [http://wikisource.org Wikisource], wikis independent of Wikipedia, such as those hosted at [http://www.wikia.com Wikia] or [http://www.wiki-site.com Wiki-site] or [http://editthis.info Editthis.info] or independent wikis such as [[wikitravel:Main Page|Wikitravel]], [http://wikileaks.org Wikileaks] or [http://www.sourcewatch.org SourceWatch]. Just because different wikis use the same underlying software and thus often have similar page layouts ''does not mean'' that they host similar content or are organized similarly. The differences between different wikis could be as small as the difference between different newspapers or as large as the difference between a newspaper and an academic journal. (To confuse matters further, MediaWiki is not the only software used to power wikis. You can see a fairly long list at the [[wikipedia:wiki software|wiki software page]] or the [[wikipedia:comparison of wiki software|comparison of wiki software page]] on Wikipedia. One example is [http://www.pmwiki.org/wiki/PmWiki/PmWiki PmWiki].) ===Distinctive features of Wikipedia=== Wikipedia is a wiki-based ''encyclopedia''. Both the choice of content and organization of content reflect the goal of being an encyclopedia. Specifically, pages on topics in Wikipedia have to meet criteria of [[wikipedia:wikipedia:notability|notability]] and facts in these pages must meet criteria of [[wikipedia:wikipedia:verifiability|verifiability]]. There are also guidelines against [[Wikipedia:Wikipedia:OR|putting original research on Wikipedia]], reflecting the fact that encyclopedias are secondary and tertiary sources of knowledge rather than primary sources. ==Two related reasons for the use of MediaWiki or similar software to maintain linked content== There are many reasons for choosing MediaWiki to organize, store, and maintain linked content on the web, but we focus here on two of them: * The ''organizational advantage'', which is the advantage of being able to maintain and organize a large amount of related content and the way the pieces of the content are interrelated. * The ''collaboration advantage'', which is the idea that the content can be edited collaboratively by large numbers of people. These are both related. The organizational advantage includes significant modularization, which allows different people to edit different pieces, making collaboration easier. Similarly, more collaboration helps with more organization. However, it is important to note that these two kinds of advantages, though related, are distinct. Further, some projects use MediaWiki or other wiki software primarily for the collaborative advantage -- the fact that large numbers of people can edit the same page, create new pages, etc. Here, wikis compete with mailing lists, discussion forums, and blog discussion threads. They have their advantages and disadvantages in these respects. Some of the earliest wikis, including Ward Cunningham's [http://c2.com/cgi/wiki WikiWikiWeb] and [http://www.meatballwiki.org/wiki/ MeatBallWiki] were focused on carrying out such discussions as a process to create new ideas. Other projects use MediaWiki primarily for the organizational advantage. In some of these cases, the greater ease of collaboration is crucial but still instrumental and not the final goal. In others, large-scale collaboration may be desirable but not crucial, and in yet others, large-scale collaboration may not be desirable. For instance, in the case of Wikipedia, the reason for using MediaWiki was to create a better-organized encyclopedia by combining the ease of organizing with the possibility of large-scale collaboration. Something similar is true for [[Wikitravel:Main Page|Wikitravel]]. The extent to which large-scale collaboration is crucial to the success of something like Wikipedia, and whether the collaborative process as it has currently developed on Wikipedia is intrinsic to its mission, are moot points. 62ca51e82857c9e6e15c343c244e500f2655dfcc 386 385 2009-12-30T12:30:00Z Vipul 2 wikitext text/x-wiki This page discusses some important differences between subject wikis and Wikipedia. ==About Wikipedia and similarities between subject wikis and Wikipedia== ===Quick introduction=== For many people using the Internet, "Wikipedia" is synonymous with "wiki" and also with the general appearance of a Wikipedia page. Thus, a lot of users who come to subject wiki pages simply think they have come to Wikipedia or some offshoot of Wikipedia. Before proceeding, it is important to clarify some distinctions. "Wikipedia" is a specific wiki-based [http://www.wikipedia.org multilingual encyclopedia] (since you're reading this, you have probably used the [[wp:Main Page|English-language version]] and possibly many other language versions). Each language Wikipedia can be viewed as a separate encyclopedia aiming to cover the entire range of encyclopedia-worthy knowledge. All the Wikipedias use a certain software called MediaWiki, which can be downloaded [[mediawikiwiki:Main Page|here]]. This software can also be used by other sister wikis to Wikipedia, such as [http://wikisource.org Wikisource], wikis independent of Wikipedia, such as those hosted at [http://www.wikia.com Wikia] or [http://www.wiki-site.com Wiki-site] or [http://editthis.info Editthis.info] or independent wikis such as [[wikitravel:Main Page|Wikitravel]], [http://wikileaks.org Wikileaks] or [http://www.sourcewatch.org SourceWatch]. Just because different wikis use the same underlying software and thus often have similar page layouts ''does not mean'' that they host similar content or are organized similarly. The differences between different wikis could be as small as the difference between different newspapers or as large as the difference between a newspaper and an academic journal. (To confuse matters further, MediaWiki is not the only software used to power wikis. You can see a fairly long list at the [[wikipedia:wiki software|wiki software page]] or the [[wikipedia:comparison of wiki software|comparison of wiki software page]] on Wikipedia. One example is [http://www.pmwiki.org/wiki/PmWiki/PmWiki PmWiki].) ===Distinctive features of Wikipedia=== Wikipedia is a wiki-based ''encyclopedia''. Both the choice of content and organization of content reflect the goal of being an encyclopedia. Specifically, pages on topics in Wikipedia have to meet criteria of [[wikipedia:wikipedia:notability|notability]] and facts in these pages must meet criteria of [[wikipedia:wikipedia:verifiability|verifiability]]. There are also guidelines against [[Wikipedia:Wikipedia:OR|putting original research on Wikipedia]], reflecting the fact that encyclopedias are secondary and tertiary sources of knowledge rather than primary sources. ==Two related reasons for the use of MediaWiki or similar software to maintain linked content== There are many reasons for choosing MediaWiki to organize, store, and maintain linked content on the web, but we focus here on two of them: * The ''organizational advantage'', which is the advantage of being able to maintain and organize a large amount of related content and the way the pieces of the content are interrelated. * The ''collaboration advantage'', which is the idea that the content can be edited collaboratively by large numbers of people. These are both related. The organizational advantage includes significant modularization, which allows different people to edit different pieces, making collaboration easier. Similarly, more collaboration helps with more organization. However, it is important to note that these two kinds of advantages, though related, are distinct. Further, some projects use MediaWiki or other wiki software primarily for the collaborative advantage -- the fact that large numbers of people can edit the same page, create new pages, etc. Here, wikis compete with mailing lists, discussion forums, and blog discussion threads. They have their advantages and disadvantages in these respects. Some of the earliest wikis, including Ward Cunningham's [http://c2.com/cgi/wiki WikiWikiWeb] and [http://www.meatballwiki.org/wiki/ MeatBallWiki] were focused on carrying out such discussions as a process to create new ideas. Other projects use MediaWiki primarily for the organizational advantage. In some of these cases, the greater ease of collaboration is crucial but still instrumental and not the final goal. In others, large-scale collaboration may be desirable but not crucial, and in yet others, large-scale collaboration may not be desirable. For instance, in the case of Wikipedia, the reason for using MediaWiki was to create a better-organized encyclopedia by combining the ease of organizing with the possibility of large-scale collaboration. Something similar is true for [[Wikitravel:Main Page|Wikitravel]]. The extent to which large-scale collaboration is crucial to the success of something like Wikipedia, and whether the collaborative process as it has currently developed on Wikipedia is intrinsic to its mission, are moot points. ==The key differences between subject wikis and Wikipedia== ===First impressions=== Wikipedia pages are typically designed well for ''accidental landing'' -- landing by people who aren't familiar with the topic and landed there by accident following links. In particular, Wikipedia pages start by setting the outermost context, then setting a further inner context, then setting a further inner context, and so on. Subject wiki pages are designed for people who have some familiarity with the topic area, so less effort is spent at the beginning of the page clarifying the general background. c4c6cdb0b2c1dca0d1ab070c35ef67a66e38b6f2 387 386 2009-12-30T12:56:25Z Vipul 2 /* The key differences between subject wikis and Wikipedia */ wikitext text/x-wiki This page discusses some important differences between subject wikis and Wikipedia. ==About Wikipedia and similarities between subject wikis and Wikipedia== ===Quick introduction=== For many people using the Internet, "Wikipedia" is synonymous with "wiki" and also with the general appearance of a Wikipedia page. Thus, a lot of users who come to subject wiki pages simply think they have come to Wikipedia or some offshoot of Wikipedia. Before proceeding, it is important to clarify some distinctions. "Wikipedia" is a specific wiki-based [http://www.wikipedia.org multilingual encyclopedia] (since you're reading this, you have probably used the [[wp:Main Page|English-language version]] and possibly many other language versions). Each language Wikipedia can be viewed as a separate encyclopedia aiming to cover the entire range of encyclopedia-worthy knowledge. All the Wikipedias use a certain software called MediaWiki, which can be downloaded [[mediawikiwiki:Main Page|here]]. This software can also be used by other sister wikis to Wikipedia, such as [http://wikisource.org Wikisource], wikis independent of Wikipedia, such as those hosted at [http://www.wikia.com Wikia] or [http://www.wiki-site.com Wiki-site] or [http://editthis.info Editthis.info] or independent wikis such as [[wikitravel:Main Page|Wikitravel]], [http://wikileaks.org Wikileaks] or [http://www.sourcewatch.org SourceWatch]. Just because different wikis use the same underlying software and thus often have similar page layouts ''does not mean'' that they host similar content or are organized similarly. The differences between different wikis could be as small as the difference between different newspapers or as large as the difference between a newspaper and an academic journal. (To confuse matters further, MediaWiki is not the only software used to power wikis. You can see a fairly long list at the [[wikipedia:wiki software|wiki software page]] or the [[wikipedia:comparison of wiki software|comparison of wiki software page]] on Wikipedia. One example is [http://www.pmwiki.org/wiki/PmWiki/PmWiki PmWiki].) ===Distinctive features of Wikipedia=== Wikipedia is a wiki-based ''encyclopedia''. Both the choice of content and organization of content reflect the goal of being an encyclopedia. Specifically, pages on topics in Wikipedia have to meet criteria of [[wikipedia:wikipedia:notability|notability]] and facts in these pages must meet criteria of [[wikipedia:wikipedia:verifiability|verifiability]]. There are also guidelines against [[Wikipedia:Wikipedia:OR|putting original research on Wikipedia]], reflecting the fact that encyclopedias are secondary and tertiary sources of knowledge rather than primary sources. ==Two related reasons for the use of MediaWiki or similar software to maintain linked content== There are many reasons for choosing MediaWiki to organize, store, and maintain linked content on the web, but we focus here on two of them: * The ''organizational advantage'', which is the advantage of being able to maintain and organize a large amount of related content and the way the pieces of the content are interrelated. * The ''collaboration advantage'', which is the idea that the content can be edited collaboratively by large numbers of people. These are both related. The organizational advantage includes significant modularization, which allows different people to edit different pieces, making collaboration easier. Similarly, more collaboration helps with more organization. However, it is important to note that these two kinds of advantages, though related, are distinct. Further, some projects use MediaWiki or other wiki software primarily for the collaborative advantage -- the fact that large numbers of people can edit the same page, create new pages, etc. Here, wikis compete with mailing lists, discussion forums, and blog discussion threads. They have their advantages and disadvantages in these respects. Some of the earliest wikis, including Ward Cunningham's [http://c2.com/cgi/wiki WikiWikiWeb] and [http://www.meatballwiki.org/wiki/ MeatBallWiki] were focused on carrying out such discussions as a process to create new ideas. Other projects use MediaWiki primarily for the organizational advantage. In some of these cases, the greater ease of collaboration is crucial but still instrumental and not the final goal. In others, large-scale collaboration may be desirable but not crucial, and in yet others, large-scale collaboration may not be desirable. For instance, in the case of Wikipedia, the reason for using MediaWiki was to create a better-organized encyclopedia by combining the ease of organizing with the possibility of large-scale collaboration. Something similar is true for [[Wikitravel:Main Page|Wikitravel]]. The extent to which large-scale collaboration is crucial to the success of something like Wikipedia, and whether the collaborative process as it has currently developed on Wikipedia is intrinsic to its mission, are moot points. ==The key differences between subject wikis and Wikipedia== ===First impressions=== Wikipedia pages are typically designed well for ''accidental landing'' -- landing by people who aren't familiar with the topic and landed there by accident following links. In particular, Wikipedia pages start by setting a broad context, and ''easing the reader into the subject''. Subject wiki pages are designed for people who have some familiarity with the topic area, so less effort is spent at the beginning of the page clarifying the general background. For instance, Wikipedia's page on [[wikipedia:normal subgroup|normal subgroup]] begins as follows: ''In [[wikipedia:abstract algebra|abstract algebra]], a '''normal subgroup''' is a [[wikipedia:subgroup|subgroup]] ...'' The initial ''In [[wikipedia:abstract algebra|abstract algebra]]'' phrase is needed to set a ''context''. A similar phrase is not employed in the group properties wiki definition of [[groupprops:normal subgroup|normal subgroup]]. ===Specific organizational paradigms=== Each subject wiki uses a number of organizational paradigms specifically suited for the needs of the subject matter, though general ideas of the [[subwiki:type-based paradigm|type-based paradigm]], [[subwiki:property-theoretic paradigm|property-theoretic paradigm]], and [[subwiki:relational paradigm|relational paradigm]] are shared by a number of subject wikis. Even these general paradigms, however, require a cohesive and reasonably narrow subject matter domain. The paradigms on subject wikis guide the use of semantic information and category information, and allow for specific automatically generated information (for instance, automatically listing examples related to a fact, or properties between two given properties). These paradigms are also responsible for very similar article structures for articles of the same ''type'', but the article structures differ quite a bit for articles of different types. Once again, it is the focus on a narrow subject domain that permits the use of reasonably standard templates for articles of a given type. No similar organizational paradigms are used on Wikipedia. ==Differences in goals and policies== ===General-purpose versus specific-purpose=== Wikipedia calls itself ''the free encyclopaedia that anybody can edit''([[wp:Main Page|Wikipedia Main Page]]). It plans to be an ''encyclopaedia of everything'' with a ''let's all get together and do it'' attitude. Of course, much of the structure and organization of the wiki is determined by a small core group, but much of the activity is also carried out by the large mass of ordinary users. While the aim of Wikipedia is to be a general-purpose encyclopaedia, the subject wikis aim to be neither general-purpose nor encyclopedic. Rather, it aims to cover a very specific area of mathematics in a particular fashion. ===What deserves an article=== As mentioned above, each subject wiki has a clear notion of what deserves an article, which tends to mean that there are in general lots of articles. The policy on subject wikis (within the framework of material relevant to the subject) is more in line with, though hardly the same as, [[wp:WP:Separatism|the separatism policy]] that some Wikipedians believe in, as opposed to the [[wp:WP:Mergism|mergism policy]] that seems to be the more accepted one in Wikipedia, as it stands today. ===Neutral point of view=== Wikipedia claims to adhere to a [[wp:WP:NPOV|neutral point of view policy]], which basically means that all views are represented fairly, and are ''attributed to their adherents''. Individual subject wikis follow no such policy. In fact, they usually represent definite and distinctive points of view, in terms of the choices made on how to structure individual articles and how to shape organizational paradigms for the subject wiki. ===Original research policies=== Wikipedia has an [[wp:WP:OR|explicit ban on original research]], with a clear emphasis that whatever is put should be from verifiable sources. As a general rule, subject wikis does not ''require'' all articles to cite sources. Rather, we first try to get the definition and the facts there, and gradually fill in the best references for those facts. 12875c912026f822233db2018ff3cb0001e072a2 388 387 2009-12-30T12:57:38Z Vipul 2 /* Specific organizational paradigms */ wikitext text/x-wiki This page discusses some important differences between subject wikis and Wikipedia. ==About Wikipedia and similarities between subject wikis and Wikipedia== ===Quick introduction=== For many people using the Internet, "Wikipedia" is synonymous with "wiki" and also with the general appearance of a Wikipedia page. Thus, a lot of users who come to subject wiki pages simply think they have come to Wikipedia or some offshoot of Wikipedia. Before proceeding, it is important to clarify some distinctions. "Wikipedia" is a specific wiki-based [http://www.wikipedia.org multilingual encyclopedia] (since you're reading this, you have probably used the [[wp:Main Page|English-language version]] and possibly many other language versions). Each language Wikipedia can be viewed as a separate encyclopedia aiming to cover the entire range of encyclopedia-worthy knowledge. All the Wikipedias use a certain software called MediaWiki, which can be downloaded [[mediawikiwiki:Main Page|here]]. This software can also be used by other sister wikis to Wikipedia, such as [http://wikisource.org Wikisource], wikis independent of Wikipedia, such as those hosted at [http://www.wikia.com Wikia] or [http://www.wiki-site.com Wiki-site] or [http://editthis.info Editthis.info] or independent wikis such as [[wikitravel:Main Page|Wikitravel]], [http://wikileaks.org Wikileaks] or [http://www.sourcewatch.org SourceWatch]. Just because different wikis use the same underlying software and thus often have similar page layouts ''does not mean'' that they host similar content or are organized similarly. The differences between different wikis could be as small as the difference between different newspapers or as large as the difference between a newspaper and an academic journal. (To confuse matters further, MediaWiki is not the only software used to power wikis. You can see a fairly long list at the [[wikipedia:wiki software|wiki software page]] or the [[wikipedia:comparison of wiki software|comparison of wiki software page]] on Wikipedia. One example is [http://www.pmwiki.org/wiki/PmWiki/PmWiki PmWiki].) ===Distinctive features of Wikipedia=== Wikipedia is a wiki-based ''encyclopedia''. Both the choice of content and organization of content reflect the goal of being an encyclopedia. Specifically, pages on topics in Wikipedia have to meet criteria of [[wikipedia:wikipedia:notability|notability]] and facts in these pages must meet criteria of [[wikipedia:wikipedia:verifiability|verifiability]]. There are also guidelines against [[Wikipedia:Wikipedia:OR|putting original research on Wikipedia]], reflecting the fact that encyclopedias are secondary and tertiary sources of knowledge rather than primary sources. ==Two related reasons for the use of MediaWiki or similar software to maintain linked content== There are many reasons for choosing MediaWiki to organize, store, and maintain linked content on the web, but we focus here on two of them: * The ''organizational advantage'', which is the advantage of being able to maintain and organize a large amount of related content and the way the pieces of the content are interrelated. * The ''collaboration advantage'', which is the idea that the content can be edited collaboratively by large numbers of people. These are both related. The organizational advantage includes significant modularization, which allows different people to edit different pieces, making collaboration easier. Similarly, more collaboration helps with more organization. However, it is important to note that these two kinds of advantages, though related, are distinct. Further, some projects use MediaWiki or other wiki software primarily for the collaborative advantage -- the fact that large numbers of people can edit the same page, create new pages, etc. Here, wikis compete with mailing lists, discussion forums, and blog discussion threads. They have their advantages and disadvantages in these respects. Some of the earliest wikis, including Ward Cunningham's [http://c2.com/cgi/wiki WikiWikiWeb] and [http://www.meatballwiki.org/wiki/ MeatBallWiki] were focused on carrying out such discussions as a process to create new ideas. Other projects use MediaWiki primarily for the organizational advantage. In some of these cases, the greater ease of collaboration is crucial but still instrumental and not the final goal. In others, large-scale collaboration may be desirable but not crucial, and in yet others, large-scale collaboration may not be desirable. For instance, in the case of Wikipedia, the reason for using MediaWiki was to create a better-organized encyclopedia by combining the ease of organizing with the possibility of large-scale collaboration. Something similar is true for [[Wikitravel:Main Page|Wikitravel]]. The extent to which large-scale collaboration is crucial to the success of something like Wikipedia, and whether the collaborative process as it has currently developed on Wikipedia is intrinsic to its mission, are moot points. ==The key differences between subject wikis and Wikipedia== ===First impressions=== Wikipedia pages are typically designed well for ''accidental landing'' -- landing by people who aren't familiar with the topic and landed there by accident following links. In particular, Wikipedia pages start by setting a broad context, and ''easing the reader into the subject''. Subject wiki pages are designed for people who have some familiarity with the topic area, so less effort is spent at the beginning of the page clarifying the general background. For instance, Wikipedia's page on [[wikipedia:normal subgroup|normal subgroup]] begins as follows: ''In [[wikipedia:abstract algebra|abstract algebra]], a '''normal subgroup''' is a [[wikipedia:subgroup|subgroup]] ...'' The initial ''In [[wikipedia:abstract algebra|abstract algebra]]'' phrase is needed to set a ''context''. A similar phrase is not employed in the group properties wiki definition of [[groupprops:normal subgroup|normal subgroup]]. ===Specific organizational paradigms=== Each subject wiki uses a number of organizational paradigms specifically suited for the needs of the subject matter, though general ideas of the [[subwiki:type-based organization|type-based paradigm]], [[subwiki:property-theoretic organization|property-theoretic paradigm]], and [[subwiki:relational organization|relational paradigm]] are shared by a number of subject wikis. Even these general paradigms, however, require a cohesive and reasonably narrow subject matter domain. The paradigms on subject wikis guide the use of semantic information and category information, and allow for specific automatically generated information (for instance, automatically listing examples related to a fact, or properties between two given properties). These paradigms are also responsible for very similar article structures for articles of the same ''type'', but the article structures differ quite a bit for articles of different types. Once again, it is the focus on a narrow subject domain that permits the use of reasonably standard templates for articles of a given type. No similar organizational paradigms are used on Wikipedia. ==Differences in goals and policies== ===General-purpose versus specific-purpose=== Wikipedia calls itself ''the free encyclopaedia that anybody can edit''([[wp:Main Page|Wikipedia Main Page]]). It plans to be an ''encyclopaedia of everything'' with a ''let's all get together and do it'' attitude. Of course, much of the structure and organization of the wiki is determined by a small core group, but much of the activity is also carried out by the large mass of ordinary users. While the aim of Wikipedia is to be a general-purpose encyclopaedia, the subject wikis aim to be neither general-purpose nor encyclopedic. Rather, it aims to cover a very specific area of mathematics in a particular fashion. ===What deserves an article=== As mentioned above, each subject wiki has a clear notion of what deserves an article, which tends to mean that there are in general lots of articles. The policy on subject wikis (within the framework of material relevant to the subject) is more in line with, though hardly the same as, [[wp:WP:Separatism|the separatism policy]] that some Wikipedians believe in, as opposed to the [[wp:WP:Mergism|mergism policy]] that seems to be the more accepted one in Wikipedia, as it stands today. ===Neutral point of view=== Wikipedia claims to adhere to a [[wp:WP:NPOV|neutral point of view policy]], which basically means that all views are represented fairly, and are ''attributed to their adherents''. Individual subject wikis follow no such policy. In fact, they usually represent definite and distinctive points of view, in terms of the choices made on how to structure individual articles and how to shape organizational paradigms for the subject wiki. ===Original research policies=== Wikipedia has an [[wp:WP:OR|explicit ban on original research]], with a clear emphasis that whatever is put should be from verifiable sources. As a general rule, subject wikis does not ''require'' all articles to cite sources. Rather, we first try to get the definition and the facts there, and gradually fill in the best references for those facts. f9b438df0bb1479c3e710276b64b9e59de88f675 2 163 376 318 2009-12-20T20:02:13Z Jon Awbrey 3 add web vita wikitext text/x-wiki <br> <p><font face="lucida calligraphy" size="7">Jon Awbrey</font></p> <br> __NOTOC__ ==Presently &hellip;== <center> [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems PERS] [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Information_%3D_Comprehension_%C3%97_Extension I = C &times; E] [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Peirce%27s_1870_Logic_Of_Relatives LOR 1870] [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositions_As_Types PAT Analogy] [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Analytic_Turing_Automata DATA Project] [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Cactus_Language Cactus Language] [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Inquiry Driven Systems] Logical Graphs : [http://knol.google.com/k/jon-awbrey/logical-graphs-1/3fkwvf69kridz/3 One] [http://knol.google.com/k/jon-awbrey/logical-graphs-2/3fkwvf69kridz/8 Two] [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Differential Logic : Introduction] [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Differential Propositional Calculus] [http://www.mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] </center> ==Recent Sightings== {| align="center" style="text-align:center" width="100%" |- | [http://stderr.org/cgi-bin/mailman/listinfo/inquiry Inquiry Project] | [http://stderr.org/pipermail/inquiry/ Inquiry Archive] |- | [http://www.mywikibiz.com/User:Jon_Awbrey MyWikiBiz Page] | [http://www.mywikibiz.com/User_talk:Jon_Awbrey MyWikiBiz Talk] |- | [http://www.mywikibiz.com/Directory:Jon_Awbrey MyWikiBiz Directory] | [http://www.mywikibiz.com/Directory_talk:Jon_Awbrey MyWikiBiz Dialogue] |- | [http://planetmath.org/ PlanetMath Project] | [http://planetmath.org/?op=getuser&id=15246 PlanetMath Profile] |- | [http://planetphysics.org/ PlanetPhysics Project] | [http://planetphysics.org/?op=getuser&id=513 PlanetPhysics Profile] |- | [http://knol.google.com/ Google Knol Project] | [http://knol.google.com/k/jon-awbrey/jon-awbrey/3fkwvf69kridz/1 Google Knol Profile] |- | [http://mathforum.org/kb/ Math Forum Project] | [http://mathforum.org/kb/accountView.jspa?userID=99854 Math Forum Profile] |- | [http://list.seqfan.eu/cgi-bin/mailman/listinfo/seqfan Fantasia Sequentia] | [http://list.seqfan.eu/pipermail/seqfan/ SeqFan Archive] |- | [http://www.research.att.com/~njas/sequences/Seis.html OEIS Land] | [http://www.research.att.com/~njas/sequences/?q=Awbrey Bolgia Mia] |- | [http://oeis.org/wiki/User:Jon_Awbrey OEIS Wiki Page] | [http://oeis.org/wiki/User_talk:Jon_Awbrey OEIS Wiki Talk] |- | [http://www.mathweb.org/wiki/User:Jon_Awbrey MathWeb Page] | [http://www.mathweb.org/wiki/User_talk:Jon_Awbrey MathWeb Talk] |- | [http://www.proofwiki.org/wiki/User:Jon_Awbrey ProofWiki Page] | [http://www.proofwiki.org/wiki/User_talk:Jon_Awbrey ProofWiki Talk] |- | [http://mathoverflow.net/ MathOverFlow] | [http://mathoverflow.net/users/1636/jon-awbrey MOFler Profile] |- | [http://www.p2pfoundation.net/User:JonAwbrey P2P Wiki Page] | [http://www.p2pfoundation.net/User_talk:JonAwbrey P2P Wiki Talk] |- | [http://vectors.usc.edu/ Vectors Project] | [http://vectors.usc.edu/thoughtmesh/ ThoughtMesh] |- | [http://altheim.4java.ca/ceryle/wiki/ Ceryle Project] | [http://altheim.4java.ca/ceryle/wiki/Wiki.jsp?page=JonAwbrey Ceryle Profile] |- | [http://forum.wolframscience.com/ NKS Forum] | [http://forum.wolframscience.com/member.php?s=&action=getinfo&userid=336 NKS Profile] |- | [http://www.wikinfo.org/index.php/User:Jon_Awbrey WikInfo Page] | [http://www.wikinfo.org/index.php/User_talk:Jon_Awbrey WikInfo Talk] |- | [http://www.getwiki.net/-User:Jon_Awbrey GetWiki Page] | [http://www.getwiki.net/-UserTalk:Jon_Awbrey GetWiki Talk] |- | [http://ontolog.cim3.net/ OntoLog Project] | [http://ontolog.cim3.net/cgi-bin/wiki.pl?JonAwbrey OntoLog Profile] |- | [http://semanticweb.org/wiki/User:Jon_Awbrey SemanticWeb Page] | [http://semanticweb.org/wiki/User_talk:Jon_Awbrey SemanticWeb Talk] |- | [http://zh.wikipedia.org/wiki/User:Jon_Awbrey 维基百科 : 用户页面] | [http://zh.wikipedia.org/wiki/User_talk:Jon_Awbrey 维基百科 : 讨论] |} ==Presentations and Publications== * Awbrey, S.M., and Awbrey, J.L. (May 2001), "Conceptual Barriers to Creating Integrative Universities", ''Organization : The Interdisciplinary Journal of Organization, Theory, and Society'' 8(2), Sage Publications, London, UK, pp. 269&ndash;284. [http://org.sagepub.com/cgi/content/abstract/8/2/269 Abstract]. * Awbrey, S.M., and Awbrey, J.L. (September 1999), "Organizations of Learning or Learning Organizations : The Challenge of Creating Integrative Universities for the Next Century", ''Second International Conference of the Journal ''Organization'' '', ''Re-Organizing Knowledge, Trans-Forming Institutions : Knowing, Knowledge, and the University in the 21st Century'', University of Massachusetts, Amherst, MA. [http://www.cspeirce.com/menu/library/aboutcsp/awbrey/integrat.htm Online]. * Awbrey, J.L., and Awbrey, S.M. (Autumn 1995), "Interpretation as Action : The Risk of Inquiry", ''Inquiry : Critical Thinking Across the Disciplines'' 15(1), pp. 40&ndash;52. [http://www.chss.montclair.edu/inquiry/fall95/awbrey.html Online]. * Awbrey, J.L., and Awbrey, S.M. (June 1992), "Interpretation as Action : The Risk of Inquiry", ''The Eleventh International Human Science Research Conference'', Oakland University, Rochester, Michigan. * Awbrey, S.M., and Awbrey, J.L. (May 1991), "An Architecture for Inquiry : Building Computer Platforms for Discovery", ''Proceedings of the Eighth International Conference on Technology and Education'', Toronto, Canada, pp. 874&ndash;875. [http://www.abccommunity.org/tmp-a.html Online]. * Awbrey, J.L., and Awbrey, S.M. (January 1991), "Exploring Research Data Interactively : Developing a Computer Architecture for Inquiry", Poster presented at the ''Annual Sigma Xi Research Forum'', University of Texas Medical Branch, Galveston, TX. * Awbrey, J.L., and Awbrey, S.M. (August 1990), "Exploring Research Data Interactively. Theme One : A Program of Inquiry", ''Proceedings of the Sixth Annual Conference on Applications of Artificial Intelligence and CD-ROM in Education and Training'', Society for Applied Learning Technology, Washington, DC, pp. 9&ndash;15. ==Education== * 1993&ndash;2003. [http://www2.oakland.edu/secs/dispprofile.asp?Fname=Fatma&Lname=Mili Graduate Study], [http://www2.oakland.edu/secs/dispprofile.asp?Fname=Mohamed&Lname=Zohdy Systems Engineering], [http://www4.oakland.edu/?id=3099&sid=87 Oakland University]. * '''1989. [http://web.archive.org/web/20001206101800/http://www.msu.edu/dig/msumap/psychology.html M.A. Psychology]''', [http://web.archive.org/web/20000902103838/http://www.msu.edu/dig/msumap/beaumont.html Michigan State University]. * 1985&ndash;1986. [http://quod.lib.umich.edu/cgi/i/image/image-idx?id=S-BHL-X-BL001808%5DBL001808 Graduate Study, Mathematics], [http://www.umich.edu/ University of Michigan]. * 1985. [http://web.archive.org/web/20061230174652/http://www.uiuc.edu/navigation/buildings/altgeld.top.html Graduate Study, Mathematics], [http://www.uiuc.edu/ University of Illinois at Urbana&ndash;Champaign]. * 1984. [http://www.psych.uiuc.edu/graduate/ Graduate Study, Psychology], [http://www.uiuc.edu/ University of Illinois at Champaign&ndash;Urbana]. * '''1980. [http://www.mth.msu.edu/images/wells_medium.jpg M.A. Mathematics]''', [http://www.msu.edu/~hvac/survey/BeaumontTower.html Michigan State University]. * '''1976. [http://web.archive.org/web/20001206050600/http://www.msu.edu/dig/msumap/phillips.html B.A. Mathematical and Philosophical Method]''', <br> [http://www.enolagaia.com/JMC.html Justin Morrill College], [http://www.msu.edu/ Michigan State University]. a7f59af282d8dc66612cd8800f3d57db277bf654 394 376 2010-06-09T17:58:53Z Jon Awbrey 3 update wikitext text/x-wiki <br> <p><font face="lucida calligraphy" size="7">Jon Awbrey</font></p> <br> __NOTOC__ ==Presently &hellip;== <center> [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Riffs_and_Rotes R&R] [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems PERS] [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Information_%3D_Comprehension_%C3%97_Extension I = C &times; E] [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Peirce%27s_1870_Logic_Of_Relatives LOR 1870] [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositions_As_Types PAT Analogy] [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Analytic_Turing_Automata DATA Project] [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Cactus_Language Cactus Language] [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Inquiry Driven Systems] Logical Graphs : [http://knol.google.com/k/jon-awbrey/logical-graphs-1/3fkwvf69kridz/3 One] [http://knol.google.com/k/jon-awbrey/logical-graphs-2/3fkwvf69kridz/8 Two] [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Differential Logic : Introduction] [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Differential Propositional Calculus] [http://www.mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] </center> ==Recent Sightings== {| align="center" style="text-align:center" width="100%" |- | [http://stderr.org/cgi-bin/mailman/listinfo/inquiry Inquiry Project] | [http://stderr.org/pipermail/inquiry/ Inquiry Archive] |- | [http://www.mywikibiz.com/User:Jon_Awbrey MyWikiBiz Page] | [http://www.mywikibiz.com/User_talk:Jon_Awbrey MyWikiBiz Talk] |- | [http://www.mywikibiz.com/Directory:Jon_Awbrey MyWikiBiz Directory] | [http://www.mywikibiz.com/Directory_talk:Jon_Awbrey MyWikiBiz Dialogue] |- | [http://planetmath.org/ PlanetMath Project] | [http://planetmath.org/?op=getuser&id=15246 PlanetMath Profile] |- | [http://planetphysics.org/ PlanetPhysics Project] | [http://planetphysics.org/?op=getuser&id=513 PlanetPhysics Profile] |- | [http://knol.google.com/ Google Knol Project] | [http://knol.google.com/k/jon-awbrey/jon-awbrey/3fkwvf69kridz/1 Google Knol Profile] |- | [http://mathforum.org/kb/ Math Forum Project] | [http://mathforum.org/kb/accountView.jspa?userID=99854 Math Forum Profile] |- | [http://list.seqfan.eu/cgi-bin/mailman/listinfo/seqfan Fantasia Sequentia] | [http://list.seqfan.eu/pipermail/seqfan/ SeqFan Archive] |- | [http://www.research.att.com/~njas/sequences/Seis.html OEIS Land] | [http://www.research.att.com/~njas/sequences/?q=Awbrey Bolgia Mia] |- | [http://oeis.org/wiki/User:Jon_Awbrey OEIS Wiki Page] | [http://oeis.org/wiki/User_talk:Jon_Awbrey OEIS Wiki Talk] |- | [http://www.mathweb.org/wiki/User:Jon_Awbrey MathWeb Page] | [http://www.mathweb.org/wiki/User_talk:Jon_Awbrey MathWeb Talk] |- | [http://www.proofwiki.org/wiki/User:Jon_Awbrey ProofWiki Page] | [http://www.proofwiki.org/wiki/User_talk:Jon_Awbrey ProofWiki Talk] |- | [http://mathoverflow.net/ MathOverFlow] | [http://mathoverflow.net/users/1636/jon-awbrey MOFler Profile] |- | [http://www.p2pfoundation.net/User:JonAwbrey P2P Wiki Page] | [http://www.p2pfoundation.net/User_talk:JonAwbrey P2P Wiki Talk] |- | [http://vectors.usc.edu/ Vectors Project] | [http://vectors.usc.edu/thoughtmesh/ ThoughtMesh] |- | [http://altheim.4java.ca/ceryle/wiki/ Ceryle Project] | [http://altheim.4java.ca/ceryle/wiki/Wiki.jsp?page=JonAwbrey Ceryle Profile] |- | [http://forum.wolframscience.com/ NKS Forum] | [http://forum.wolframscience.com/member.php?s=&action=getinfo&userid=336 NKS Profile] |- | [http://www.wikinfo.org/index.php/User:Jon_Awbrey WikInfo Page] | [http://www.wikinfo.org/index.php/User_talk:Jon_Awbrey WikInfo Talk] |- | [http://www.getwiki.net/-User:Jon_Awbrey GetWiki Page] | [http://www.getwiki.net/-UserTalk:Jon_Awbrey GetWiki Talk] |- | [http://ontolog.cim3.net/ OntoLog Project] | [http://ontolog.cim3.net/cgi-bin/wiki.pl?JonAwbrey OntoLog Profile] |- | [http://semanticweb.org/wiki/User:Jon_Awbrey SemanticWeb Page] | [http://semanticweb.org/wiki/User_talk:Jon_Awbrey SemanticWeb Talk] |- | [http://zh.wikipedia.org/wiki/User:Jon_Awbrey 维基百科 : 用户页面] | [http://zh.wikipedia.org/wiki/User_talk:Jon_Awbrey 维基百科 : 讨论] |} ==Presentations and Publications== * Awbrey, S.M., and Awbrey, J.L. (May 2001), "Conceptual Barriers to Creating Integrative Universities", ''Organization : The Interdisciplinary Journal of Organization, Theory, and Society'' 8(2), Sage Publications, London, UK, pp. 269&ndash;284. [http://org.sagepub.com/cgi/content/abstract/8/2/269 Abstract]. * Awbrey, S.M., and Awbrey, J.L. (September 1999), "Organizations of Learning or Learning Organizations : The Challenge of Creating Integrative Universities for the Next Century", ''Second International Conference of the Journal ''Organization'' '', ''Re-Organizing Knowledge, Trans-Forming Institutions : Knowing, Knowledge, and the University in the 21st Century'', University of Massachusetts, Amherst, MA. [http://www.cspeirce.com/menu/library/aboutcsp/awbrey/integrat.htm Online]. * Awbrey, J.L., and Awbrey, S.M. (Autumn 1995), "Interpretation as Action : The Risk of Inquiry", ''Inquiry : Critical Thinking Across the Disciplines'' 15(1), pp. 40&ndash;52. [http://www.chss.montclair.edu/inquiry/fall95/awbrey.html Online]. * Awbrey, J.L., and Awbrey, S.M. (June 1992), "Interpretation as Action : The Risk of Inquiry", ''The Eleventh International Human Science Research Conference'', Oakland University, Rochester, Michigan. * Awbrey, S.M., and Awbrey, J.L. (May 1991), "An Architecture for Inquiry : Building Computer Platforms for Discovery", ''Proceedings of the Eighth International Conference on Technology and Education'', Toronto, Canada, pp. 874&ndash;875. [http://www.abccommunity.org/tmp-a.html Online]. * Awbrey, J.L., and Awbrey, S.M. (January 1991), "Exploring Research Data Interactively : Developing a Computer Architecture for Inquiry", Poster presented at the ''Annual Sigma Xi Research Forum'', University of Texas Medical Branch, Galveston, TX. * Awbrey, J.L., and Awbrey, S.M. (August 1990), "Exploring Research Data Interactively. Theme One : A Program of Inquiry", ''Proceedings of the Sixth Annual Conference on Applications of Artificial Intelligence and CD-ROM in Education and Training'', Society for Applied Learning Technology, Washington, DC, pp. 9&ndash;15. ==Education== * 1993&ndash;2003. [http://www2.oakland.edu/secs/dispprofile.asp?Fname=Fatma&Lname=Mili Graduate Study], [http://www2.oakland.edu/secs/dispprofile.asp?Fname=Mohamed&Lname=Zohdy Systems Engineering], [http://www.oakland.edu/?id=3099&sid=87 Oakland University]. * '''1989. [http://web.archive.org/web/20001206101800/http://www.msu.edu/dig/msumap/psychology.html M.A. Psychology]''', [http://web.archive.org/web/20000902103838/http://www.msu.edu/dig/msumap/beaumont.html Michigan State University]. * 1985&ndash;1986. [http://quod.lib.umich.edu/cgi/i/image/image-idx?id=S-BHL-X-BL001808%5DBL001808 Graduate Study, Mathematics], [http://www.umich.edu/ University of Michigan]. * 1985. [http://web.archive.org/web/20061230174652/http://www.uiuc.edu/navigation/buildings/altgeld.top.html Graduate Study, Mathematics], [http://www.uiuc.edu/ University of Illinois at Urbana&ndash;Champaign]. * 1984. [http://www.psych.uiuc.edu/graduate/ Graduate Study, Psychology], [http://www.uiuc.edu/ University of Illinois at Champaign&ndash;Urbana]. * '''1980. [http://www.mth.msu.edu/images/wells_medium.jpg M.A. Mathematics]''', [http://www.msu.edu/~hvac/survey/BeaumontTower.html Michigan State University]. * '''1976. [http://web.archive.org/web/20001206050600/http://www.msu.edu/dig/msumap/phillips.html B.A. Mathematical and Philosophical Method]''', <br> [http://www.enolagaia.com/JMC.html Justin Morrill College], [http://www.msu.edu/ Michigan State University]. 5ea76aca7f5bbe9da83a89ded9fed5fd2a8b26dd 0 9 377 348 2009-12-29T12:30:56Z Vipul 2 wikitext text/x-wiki <noinclude>[[Status::Basic definition| ]][[Topic::Group theory| ]][[Primary wiki::Groupprops| ]]</noinclude> '''Normal subgroup''': A subgroup of a group that occurs as the kernel of a homomorphism, or equivalently, such that every left coset and right coset are equal. Related terms: Normality (the property of a subgroup being normal), [[Groupprops:Normal core|normal core]] (the largest normal subgroup contained in a given subgroup), [[Groupprops:Normal closure|normal closure]] (the smallest normal subgroup containing a given subgroup), [[Groupprops:Normalizer|normalizer]] (the largest subgroup containing a given subgroup, in which it is normal), [[Groupprops:Normal automorphism|normal automorphism]] (an automorphism that restricts to an automorphism on every normal subgroup). Term variations: [[Groupprops:Subnormal subgroup|subnormal subgroup]], [[Groupprops:Abnormal subgroup|abnormal subgroup]], [[Groupprops:Quasinormal subgroup|quasinormal subgroup]], and others. See [[Groupprops:Category:Variations of normality]], [[Groupprops:Category:Opposites of normality]], and [[Groupprops:Category:Analogues of normality]]. Primary subject wiki entry: [[Groupprops:Normal subgroup]]. See also [[Groupprops:Questions:Normal subgroup]]. Other subject wiki entries: [[Diffgeom:Normal subgroup]] Coverage in guided tours: Groupprops guided tour for beginners; section not yet prepared. Also located at: [[Wikipedia:Normal subgroup]], [[Planetmath:NormalSubgroup]], [[Mathworld:NormalSubgroup]], [[Sor:N/n067690|Springer Online Reference Works]], [[Citizendium:Normal subgroup]] a2b0cc62dc4bf64d923d27fa14201804d3548672 106 107 378 265 2009-12-30T11:58:07Z Vipul 2 wikitext text/x-wiki In all subject wikis, articles form the fundamental ''unit of knowledge''. Each article is treated as a separate, independent unit of knowledge. Articles are strongly linked to each other but all inter-article dependencies are loose. ==Goal of an article== The goal of each article is to: * The ''content goal'': Answer certain specific questions clearly and comprehensively. * The ''link goal'': Provide a vantage point for exploration of related ideas and for answers to related questions. The two goals together mean that each article should be very clear about what questions it provides comprehensive answers to, and what questions it provides relevant links for. ==Types of articles== ===Definition articles=== {{further|[[Subwiki:Definition article]], [[Subwiki:Definition]]}} Most subject wikis have articles defining terms. These articles are called definition articles or terminology articles. A definition article must contain the definition, and, depending on the type of term and the organizational paradigms of the subject wiki, it may contain other sections giving further context to the term. Definition articles follow a brief and terse style. Their main purpose is to serve as vantage points for exploring a term and understanding related terms and facts. Definition articles do ''not'' contain proofs, explanatory arguments, exploration and motivation. Rather, they give short statements with clear explanations, linking to more complete proofs and explanation. ===Fact articles=== {{further|[[Subwiki:Fact article]], [[Subwiki:Fact]]}} Fact articles aim to state a fact clearly, and provide a proof or explanation of the fact. Where the proof is too long, involved, or unilluminating, it is outlined and separate components are linked to. Fact articles follow a brief and terse style. Their main purpose is to serve as vantage points for exploring the fact, its underlying ideas, and related terms and facts. Fact articles contain definitions of important terms used, and state facts used, but do ''not'' attempt to prove other facts being used in their proof. ===Specific information articles=== {{further|[[Subwiki:Specific information article]]}} Specific information articles on a subject wiki provide specific information of a certain type about a definition or fact. The style here is reasonably terse and to-the-point, though there may be a little more leeway than for definition and fact articles. Depending on the type of information, however, the specific information article on a particular topic may have to follow a certain template. For instance, [[Groupprops:Subgroup structure of dihedral group:D8|subgroup structure of dihedral group:D8]] article on Groupprops provides specific information about the dihedral group of order eight, namely, it describes the subgroup structure of this group. ===Survey articles=== {{further|[[Subwiki:Survey article]]}} Survey articles explore an idea. The idea could be based on a definition or fact or it could be based on the relation between multiple definitions and facts. Survey articles do not follow rigid guidelines like definition and fact articles, and the value they offer is considerably more variable. This is because a survey article is not usually intended to answer a ''specific question'', but rather, it is intended to address general curiosities. Unlike definition and fact articles, survey articles are not obliged to provide complete and comprehensive statements of definitions and facts used. They can provide short summaries of the definitions and facts, linking to the full article for further information. The distinction between specific information articles and survey articles is not completely clear. The general idea is that for survey articles, it is usually not that easy to say, looking at the title of the article, what content should definitely go in to the article, since there is greater subjectivity. 8866d9bb3d1118339985c8409ede9ccdd93b6858 106 95 379 204 2009-12-30T11:59:10Z Vipul 2 wikitext text/x-wiki Each subject wiki has certain kinds of articles called ''definition articles'' or ''terminology articles''. Each such article is about a term being defined. Definition articles form an extremely important part of every subject wiki. This article discusses the overall design and organization of definition articles on subject wikis. We'll use the word ''definiendum'' for the term being defined. ==Goals of the definition article== ===Content goals=== The definition article should provide: * A clear definition * A representative range of examples * An idea of the different ways in which the definiendum is understood and used ===Link goals=== The definition article should provide brief information with further links to: * Articles detailing the history of the definiendum * Survey articles exploring how to understand the definition * Related terms, either related by an implication (one being stronger or weaker than the other), or by other means * Proofs of properties, conditions and results satisfied by the definiendum ==Article-tagging templates== There are boxes at the top of many definition articles on subject wikis. These boxes give general information about the definiendum, such as the ''type'' of term it is. Each such box is generated by an ''article-tagging template''. {{further|[[Subwiki:Definition article-tagging template]]}} ==Sections of the definition article== ===History section=== {{further|[[Subwiki:History section in definition articles]]}} This is not a mandatory section, but when it exists, it is usually at the beginning of the article, ''before'' the definition section. The history section provides a ''brief'' overview of the origin of the term and of the notion, along with the name of the person who came up with the term. This section ''might'' also contain further information about the evolution of the term or concept since it was introduced. ''Links from this section'': If more detailed information on the history of the concept or term is available on a separate page, a link to that page is provided. ===Definition section=== {{further|[[Subwiki:Definition]]}} This section is mandatory for all definition articles. The definition section, titled '''Definition''', contains ''at least one'' definition of the definiendum. It may contain multiple equivalent definitions, and may also contain the same definition stated in different ways (for instance, in words and in symbols). Some definition sections have a ''Quick phrases'' box on top with quick, easy-to-remember phrases that capture the meaning of the term. ''Links from this section'': The section links to pages explaining the equivalence of multiple definitions, or pages proving facts implicit in the formulation of the definition. It also links to survey articles that parse and study the definition carefully and to survey articles that consider further alternative definitions. ===Examples section=== {{further|[[Subwiki:Examples]]}} This section is recommended for all definition articles, though it is currently not present in too many of them. The section may be divided into subsections containing different kinds of examples. The aim is for examples to provide a representative range, cover the extremes, and provide a litmus test for definition understanding. Non-examples may also be included to contrast with examples. ===Formalisms section=== {{further|[[Subwiki:Formalisms]]}} ===Relation section=== {{further|[[Subwiki:Relations in definition article]]}} ===Facts section or more refined section choices=== {{fillin}} 18500a6f8f004a7435502f281ae4d049525980fc 380 379 2009-12-30T11:59:44Z Vipul 2 /* Link goals */ wikitext text/x-wiki Each subject wiki has certain kinds of articles called ''definition articles'' or ''terminology articles''. Each such article is about a term being defined. Definition articles form an extremely important part of every subject wiki. This article discusses the overall design and organization of definition articles on subject wikis. We'll use the word ''definiendum'' for the term being defined. ==Goals of the definition article== ===Content goals=== The definition article should provide: * A clear definition * A representative range of examples * An idea of the different ways in which the definiendum is understood and used ===Link goals=== The definition article should provide brief information with further links to: * Articles detailing the history of the definiendum * Specific information articles about the definiendum * Survey articles exploring how to understand the definition * Related terms, either related by an implication (one being stronger or weaker than the other), or by other means * Proofs of properties, conditions and results satisfied by the definiendum ==Article-tagging templates== There are boxes at the top of many definition articles on subject wikis. These boxes give general information about the definiendum, such as the ''type'' of term it is. Each such box is generated by an ''article-tagging template''. {{further|[[Subwiki:Definition article-tagging template]]}} ==Sections of the definition article== ===History section=== {{further|[[Subwiki:History section in definition articles]]}} This is not a mandatory section, but when it exists, it is usually at the beginning of the article, ''before'' the definition section. The history section provides a ''brief'' overview of the origin of the term and of the notion, along with the name of the person who came up with the term. This section ''might'' also contain further information about the evolution of the term or concept since it was introduced. ''Links from this section'': If more detailed information on the history of the concept or term is available on a separate page, a link to that page is provided. ===Definition section=== {{further|[[Subwiki:Definition]]}} This section is mandatory for all definition articles. The definition section, titled '''Definition''', contains ''at least one'' definition of the definiendum. It may contain multiple equivalent definitions, and may also contain the same definition stated in different ways (for instance, in words and in symbols). Some definition sections have a ''Quick phrases'' box on top with quick, easy-to-remember phrases that capture the meaning of the term. ''Links from this section'': The section links to pages explaining the equivalence of multiple definitions, or pages proving facts implicit in the formulation of the definition. It also links to survey articles that parse and study the definition carefully and to survey articles that consider further alternative definitions. ===Examples section=== {{further|[[Subwiki:Examples]]}} This section is recommended for all definition articles, though it is currently not present in too many of them. The section may be divided into subsections containing different kinds of examples. The aim is for examples to provide a representative range, cover the extremes, and provide a litmus test for definition understanding. Non-examples may also be included to contrast with examples. ===Formalisms section=== {{further|[[Subwiki:Formalisms]]}} ===Relation section=== {{further|[[Subwiki:Relations in definition article]]}} ===Facts section or more refined section choices=== {{fillin}} 1b46778e4fe1292e2808a48bed400d4a0033a330 381 380 2009-12-30T12:01:04Z Vipul 2 /* Definition section */ wikitext text/x-wiki Each subject wiki has certain kinds of articles called ''definition articles'' or ''terminology articles''. Each such article is about a term being defined. Definition articles form an extremely important part of every subject wiki. This article discusses the overall design and organization of definition articles on subject wikis. We'll use the word ''definiendum'' for the term being defined. ==Goals of the definition article== ===Content goals=== The definition article should provide: * A clear definition * A representative range of examples * An idea of the different ways in which the definiendum is understood and used ===Link goals=== The definition article should provide brief information with further links to: * Articles detailing the history of the definiendum * Specific information articles about the definiendum * Survey articles exploring how to understand the definition * Related terms, either related by an implication (one being stronger or weaker than the other), or by other means * Proofs of properties, conditions and results satisfied by the definiendum ==Article-tagging templates== There are boxes at the top of many definition articles on subject wikis. These boxes give general information about the definiendum, such as the ''type'' of term it is. Each such box is generated by an ''article-tagging template''. {{further|[[Subwiki:Definition article-tagging template]]}} ==Sections of the definition article== ===History section=== {{further|[[Subwiki:History section in definition articles]]}} This is not a mandatory section, but when it exists, it is usually at the beginning of the article, ''before'' the definition section. The history section provides a ''brief'' overview of the origin of the term and of the notion, along with the name of the person who came up with the term. This section ''might'' also contain further information about the evolution of the term or concept since it was introduced. ''Links from this section'': If more detailed information on the history of the concept or term is available on a separate page, a link to that page is provided. ===Definition section=== {{further|[[Subwiki:Definition]]}} This section is mandatory for all definition articles. The definition section, titled '''Definition''', contains ''at least one'' definition of the definiendum. It may contain multiple equivalent definitions, and may also contain the same definition stated in different ways (for instance, in words and in symbols). Some definition sections have a ''Quick phrases'' box on top with quick, easy-to-remember phrases that capture the meaning of the term. ''Links from this section'': The section links to terms used to express the definition, pages explaining the equivalence of multiple definitions, or pages proving facts implicit in the formulation of the definition. It also links to survey articles that parse and study the definition carefully and to survey articles that consider further alternative definitions. ===Examples section=== {{further|[[Subwiki:Examples]]}} This section is recommended for all definition articles, though it is currently not present in too many of them. The section may be divided into subsections containing different kinds of examples. The aim is for examples to provide a representative range, cover the extremes, and provide a litmus test for definition understanding. Non-examples may also be included to contrast with examples. ===Formalisms section=== {{further|[[Subwiki:Formalisms]]}} ===Relation section=== {{further|[[Subwiki:Relations in definition article]]}} ===Facts section or more refined section choices=== {{fillin}} e6d86f5d848c486d2cf678e64a3fe2c9dd08ed88 382 381 2009-12-30T12:02:15Z Vipul 2 /* Examples section */ wikitext text/x-wiki Each subject wiki has certain kinds of articles called ''definition articles'' or ''terminology articles''. Each such article is about a term being defined. Definition articles form an extremely important part of every subject wiki. This article discusses the overall design and organization of definition articles on subject wikis. We'll use the word ''definiendum'' for the term being defined. ==Goals of the definition article== ===Content goals=== The definition article should provide: * A clear definition * A representative range of examples * An idea of the different ways in which the definiendum is understood and used ===Link goals=== The definition article should provide brief information with further links to: * Articles detailing the history of the definiendum * Specific information articles about the definiendum * Survey articles exploring how to understand the definition * Related terms, either related by an implication (one being stronger or weaker than the other), or by other means * Proofs of properties, conditions and results satisfied by the definiendum ==Article-tagging templates== There are boxes at the top of many definition articles on subject wikis. These boxes give general information about the definiendum, such as the ''type'' of term it is. Each such box is generated by an ''article-tagging template''. {{further|[[Subwiki:Definition article-tagging template]]}} ==Sections of the definition article== ===History section=== {{further|[[Subwiki:History section in definition articles]]}} This is not a mandatory section, but when it exists, it is usually at the beginning of the article, ''before'' the definition section. The history section provides a ''brief'' overview of the origin of the term and of the notion, along with the name of the person who came up with the term. This section ''might'' also contain further information about the evolution of the term or concept since it was introduced. ''Links from this section'': If more detailed information on the history of the concept or term is available on a separate page, a link to that page is provided. ===Definition section=== {{further|[[Subwiki:Definition]]}} This section is mandatory for all definition articles. The definition section, titled '''Definition''', contains ''at least one'' definition of the definiendum. It may contain multiple equivalent definitions, and may also contain the same definition stated in different ways (for instance, in words and in symbols). Some definition sections have a ''Quick phrases'' box on top with quick, easy-to-remember phrases that capture the meaning of the term. ''Links from this section'': The section links to terms used to express the definition, pages explaining the equivalence of multiple definitions, or pages proving facts implicit in the formulation of the definition. It also links to survey articles that parse and study the definition carefully and to survey articles that consider further alternative definitions. ===Examples section=== {{further|[[Subwiki:Examples]]}} This section is recommended for all definition articles, though it is currently not present in too many of them. The section may be divided into subsections containing different kinds of examples. The aim is for examples to provide a representative range, cover the extremes, and provide a litmus test for definition understanding. Non-examples may also be included to contrast with examples. ''Links from this section'': Links usually go to pages that discuss the examples, either providing background information of the examples or explaining why those are indeed examples. ===Formalisms section=== {{further|[[Subwiki:Formalisms]]}} ===Relation section=== {{further|[[Subwiki:Relations in definition article]]}} ===Facts section or more refined section choices=== {{fillin}} 28749d288273234a7282be1ceba241a854cedbdc 106 109 383 272 2009-12-30T12:06:37Z Vipul 2 /* Stronger and weaker */ wikitext text/x-wiki [[Subwiki:Definition article|Definition articles]], or article defining new terms, have ''relation'' sections. These relation sections describe how the term being defined (henceforth, called the ''definiendum'') is related to other terms and concepts. ==Relation with similarly typed terms== The ''type'' of a term is the specific role that term plays. For instance, some terms may be of the ''property'' type: ''prime number'' is a property of natural numbers. Some terms may be of the ''species'' type, so ''mangifera indica'' is a species of plant. (Learn more about typing at [[Subwiki:Type-based organization]]). The relation with similarly typed terms describes how the definiendum is related to terms of the same ''type''. ===Stronger and weaker=== For properties over some context space, one can talk of ''stronger properties'' and ''weaker properties'', to describe the relation with other properties over the same context space. Thus, for instance, the property of natural numbers of being a perfect fourth power is ''stronger than'' the property of being a perfect square, while the property of being a perfect power is ''weaker than'' the property of being a perfect square. Something similar is true for extra structures placed. A ''stronger structure'' in this case is a structure that carries more information (or more constraints) than a weaker structure, and from which the weaker structure can be recovered. For instance, the structure of a Riemannian manifold is ''stronger than'' the structure of a topological manifold, and the structure of a group is ''stronger than'' the structure of a magma (a set with a binary operation). We typically say that the stronger property (resp., structure) ''implies'' (resp., gives rise to) the weaker property (resp., structure). Here's how this is presented in the case of properties: * In a section titled '''Relation with other properties''', there are subsections titled '''Stronger properties''' and '''Weaker properties'''. * In the section titled '''Stronger properties''', there are bullet points listing the stronger properties. When available or desired, each property is accompanied by a short description either of the property or of why it is stronger; there is usually also a link to a more complete proof, both of the implication, and of why the converse implication fails in general. There may also be a link to a survey article comparing the properties. ''Note: On some of the subject wikis, we have gradually been moving from bullet points to tables, that allow for a more compact and easy-to-read explanation of the two properties, including: the name and meaning of the other property, proof of the implication, proof of failure of the reverse implication (plus all examples), and intermediate notions.'' * In the section titled '''Weaker properties''', there are bullet points listing the weaker properties. The format is similar to that for '''Stronger properties'''. * There may also be a section titled '''Related properties''' or '''Incomparable properties''' that lists properties that are neither stronger nor weaker but are still closely related. A similar format is followed for stronger/weaker structure and for stronger notions and weaker notions in other senses. ===Conjunctions and disjunctions=== Sometimes new terms are obtained by taking the definiendum and '''AND'''ing it with something else. For instance, we can take the '''AND''' of the property of being a prime number and the property of being an odd number, and get the property of being an odd prime. We can take the '''OR''' of being a number that is 1 mod 3 and a number tha tis 2 mod 3, to get the property of being a number that is not a multiple of 3. For property articles, conjunctions and disjunctions are typically handled in separate subsections under the '''Relation with other properties''' section. The section with conjunctions is titled '''Conjunction with other properties''' while the section on disjunctions is titled '''Disjunction with other properties'''. ===Semantic information for stronger and weaker=== If <math>B</math> is stronger than <math>A</math>, and is mentioned in the article on <math>A</math>, the semantic property [[Property:Weaker than]] is used. This stores the information that <math>A</math> is weaker than <math>B</math>. Conversely, if <math>A</math> is stronger than <math>B</math>, the property [[Property:Stronger than]] is used. These can be used in semantic search to locate particular properties or structures that are stronger than or weaker than given ones. Note that conjunctions are ''stronger than'' the components, and disjunctions are ''weaker than'' the components, and this information is stored, ''even though'' conjunctions and disjunctions are usually listed in separate subsections. ==Relation with terms of related but different types== Apart from being related with terms of the same type, the definiendum may also be related with terms of different types. These relations may be somewhat more complicated. ===A broad banner of related properties=== Suppose a subgroup property has some related group properties. Then, these group properties are listed in a subsection titled '''Related group properties''' under the section '''Relation with other properties'''. ==Examples== ===Groupprops=== Here are some examples of pages with relations sections: * [[Groupprops:Normal subgroup]]: Here is its [[Groupprops:Normal subgroup#Relation with other properties|relation with other properties section]]. This has many subsections, including stronger properties, weaker properties, conjunction with other properties. For some of them, there are links to proofs of the implications and of the reverse implications being false, and for some, there are links to survey articles explaining the differences between the properties. * [[Groupprops:Pronormal subgroup]]: Here is its [[Groupprops:Pronormal subgroup#Relation with other properties|relation with other properties section]]. This has a list of stronger properties, a list of weaker properties, and a separate subsection titled '''Conjunction with other properties'''. * [[Groupprops:Complete group]]: Here is its [[Groupprops:Complete group#Relation with other properties|relation with other properties section]]. This has a list of stronger properties and a list of weaker properties. For some of these, there are links to proofs of the implications. * [[Groupprops:Monoid]]: Here is its [[Groupprops:Monoid#Relation with other structures|relation with other structures section]]. This has a list of stronger structures and a list of weaker structures. * [[Groupprops:Conjugate subgroups]]: Here is its [[Groupprops:Conjugate subgroups#Relation with other equivalence relations|relation with other equivalence relations section]]. This has a list of stronger equivalence relations and weaker equivalence relations. * [[Groupprops:Transitive subgroup property]]: Here is its [[Groupprops:Transitive subgroup property#Relation with other metaproperties|relation with other metaproperties section]]. This has a list of stronger metaproperties and a list of weaker metaproperties. These relations between properties, structures and equivalence relations of various sorts can be used to locate properties. Here are some examples of semantic queries (that can be executed by typing the query term in [[Groupprops:Special:Ask|the Special:Ask page]]: * For a list of subgroup properties that are ''stronger than'' subnormality and ''weaker than'' normality: <pre>[[Stronger than::Subnormal subgroup]][[Weaker than::Normal subgroup]]</pre> * For a list of group properties that are ''stronger than'' solvable group and ''weaker than'' Abelian group: <pre>[[Stronger than::Solvable group]][[Weaker than::Abelian group]]</pre> * We can restrict attention to ''pivotal'' subgroup properties that are stronger than normality: <pre>[[Stronger than::Normal subgroup]][[Category:Pivotal subgroup properties]]</pre> * For a list of the subgroup properties that are formed as '''AND''' of the property of normality with something else: <pre>[[Conjunction involving::Normal subgroup]][[Category:Subgroup properties]]</pre> * For a list of the ''transitive'' subgroup properties that are stronger than the property of normality: <pre>[[Stronger than::Normal subgroup]][[Category:Transitive subgroup properties]]</pre> * For a list of the subgroup properties that are variations of normality, and are stronger than the property of being a subnormal subgroup: <pre>[[Stronger than::Subnormal subgroup]][[Category:Variations of normality]]</pre> ===Topospaces=== Here are some examples of pages with relations sections: * [[Topospaces:Normal space]]: Here is its [[Topospaces:Normal space#Relation with other properties|relation with other properties section]]. This has two subsections: stronger properties and weaker properties. For some of these, links to proof of the implication are given. * [[Topospaces:Closed subset]]: Here is its [[Topospaces:Closed subset#Relation with other properties}relation with other properties section]]. This has two subsections: stronger properties and weaker properties. These relations between properties, structures and equivalence relations of various sorts can be used to locate properties. Here are some examples of semantic queries (that can be executed by typing the query term in [[Groupprops:Special:Ask|the Special:Ask page]]: * For a list of properties that are stronger than Hausdorffness but weaker than normality: <pre>[[Weaker than::Normal space]][[Stronger than::Hausdorff space]]</pre> * For a list of properties that are stronger than Hausdorffness and are ''variations'' of Hausdorffness: <pre>[[Stronger than::Hausdorff space]][[Category:Variations of Hausdorffness]]</pre> * For a list of retract-hereditary properties of topological spaces that are weaker than contractibility: <pre>[[Weaker than::Contractible space]][[Category:Retract-hereditary properties of topological spaces]]</pre> ===Commalg=== Here are some examples of pages with a relations section: * [[Commalg:Principal ideal domain]]: This has a [[Commalg:Principal ideal domain#Relation with other properties|relation with other properties section]]. The section includes subsections titled '''stronger properties''', '''weaker properties''', as well as a subsection giving pairs of properties whose '''AND''' gives the property of being a principal ideal domain. * [[Commalg:Noetherian ring]]: This has a [[Commalg:Noetherian ring#Relation with other properties|relation with other properties section]]. This section includes subsections titled '''stronger properties''', '''weaker properties'''', as well as '''conjunction with other properties'''. * [[Commalg:Intersection of maximal ideals]]: This has a [[Commalg:Intersection of maximal ideals#Relation with other properties|relation with other properties section]], listing stronger properties as well as weaker properties. These relations between properties, structures and equivalence relations of various sorts can be used to locate properties. Here are some examples of semantic queries (that can be executed by typing the query term in [[Commalg:Special:Ask|the Special:Ask page]]: * For a list of properties of commutative unital rings that are stronger than being a Noetherian domain but weaker than being a Euclidean domain, try: <pre>[[Weaker than::Euclidean domain]][[Stronger than::Noetherian domain]]</pre> 13938e6cae95326ed339d61daae4dc1c4f037585 0 12 390 23 2010-01-18T15:43:59Z Vipul 2 wikitext text/x-wiki <noinclude>[[Status::Basic definition| ]] [[Topic::Classical mechanics| ]]</noinclude> '''Normal force''': The component of contact force between two bodies, that acts in the direction perpendicular to the plane of contact. It manifests a tendency of the bodies to avoid moving into each other, and acts outwards on both bodies. Primary subject wiki entry: [[Mech:Normal force]] Also located at: [[Wikipedia:Normal force]], [[Citizendium:Normal force]] 56bad67561cdc7327d544ebded13db89dc39c080 10 162 391 314 2010-05-18T14:41:45Z Vipul 2 wikitext text/x-wiki {{quotation|Learn more about {{{word}}}<br> Dictionary definitions: [[oed:{{{oednumber}}}|Oxford English Dictionary]], [[merriamwebster:{{{word}}}|Merriam Webster]], [[Wiktionary:{{{word}}}|Wiktionary]], [[thefreedictionary:{{{word}}}|The Free Dictionary]]<br> Visual Thesaurus: [[visualthesaurus:{{{word}}}|Visual Thesaurus]]<br> Wolfram Alpha: [http://www.wolframalpha.com/input/?i=word+{{PAGENAME}} Wolfram Alpha]}} d24d77100c0ffbf2a8670bc695b1771d0b142c14 392 391 2010-05-18T14:42:39Z Vipul 2 wikitext text/x-wiki {{quotation|Learn more about {{{word}}}<br> Dictionary definitions: [[oed:{{{oednumber}}}|Oxford English Dictionary]], [[merriamwebster:{{{word}}}|Merriam Webster]], [[Wiktionary:{{{word}}}|Wiktionary]], [[thefreedictionary:{{{word}}}|The Free Dictionary]]<br> Visual Thesaurus: [[visualthesaurus:{{{word}}}|Visual Thesaurus]]<br> Wolfram Alpha: [http://www.wolframalpha.com/input/?i=word+{{{word}}} Wolfram Alpha]}} 252042baddc9de14d49b2601b1b31e24d4ef377c 393 392 2010-05-18T14:42:58Z Vipul 2 wikitext text/x-wiki {{quotation|Learn more about {{{word}}}<br> Dictionary definitions: [[oed:{{{oednumber}}}|Oxford English Dictionary]], [[merriamwebster:{{{word}}}|Merriam Webster]], [[Wiktionary:{{{word}}}|Wiktionary]], [[thefreedictionary:{{{word}}}|The Free Dictionary]]<br> Visual Thesaurus: [[visualthesaurus:{{{word}}}|Visual Thesaurus]]}} ad0f02fbd74f433341b165d03f3b38def7b6fc12 0 180 395 2010-06-09T18:09:14Z Jon Awbrey 3 + project page wikitext text/x-wiki <font size="3">&#9758;</font> This page serves as a '''focal node''' for a collection of related resources. ==Participants== Interested parties may add their names on [[Inquiry Live/Participants|this page]]. ==Rudiments of organization== ===Focal nodes=== '''Focal nodes''' serve as hubs for collections of related resources, in particular, activity sites and article contents. {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== '''Peer nodes''' are roughly parallel pages on different sites that are not necessarily identical in content &mdash; especially as they develop in time across different environments through interaction with diverse populations &mdash; but they should preserve enough information to reproduce each other, more or less, in case of damage or loss. For example, the peer nodes of the present page are listed here: {{col-begin}} {{col-break}} * [http://mywikibiz.com/Inquiry_Live Inquiry Live @ MyWikiBiz] * [http://mathweb.org/wiki/Inquiry_Live Inquiry Live @ MathWeb Wiki] * [http://www.netknowledge.org/wiki/Inquiry_Live Inquiry Live @ NetKnowledge] {{col-break}} * [http://wiki.oercommons.org/mediawiki/index.php/Inquiry_Live Inquiry Live @ OER Commons] * [http://p2pfoundation.net/Inquiry_Live Inquiry Live @ P2P Foundation] * [http://semanticweb.org/wiki/Inquiry_Live Inquiry Live @ SemanticWeb] {{col-end}} [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] bbf52ddf639604f4add30b5ddae52faf8e36a61d 400 395 2010-06-09T18:36:05Z Jon Awbrey 3 /* Peer nodes */ + 2 wikitext text/x-wiki <font size="3">&#9758;</font> This page serves as a '''focal node''' for a collection of related resources. ==Participants== Interested parties may add their names on [[Inquiry Live/Participants|this page]]. ==Rudiments of organization== ===Focal nodes=== '''Focal nodes''' serve as hubs for collections of related resources, in particular, activity sites and article contents. {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== '''Peer nodes''' are roughly parallel pages on different sites that are not necessarily identical in content &mdash; especially as they develop in time across different environments through interaction with diverse populations &mdash; but they should preserve enough information to reproduce each other, more or less, in case of damage or loss. For example, the peer nodes of the present page are listed here: {{col-begin}} {{col-break}} * [http://mywikibiz.com/Inquiry_Live Inquiry Live @ MyWikiBiz] * [http://mathweb.org/wiki/Inquiry_Live Inquiry Live @ MathWeb Wiki] * [http://www.netknowledge.org/wiki/Inquiry_Live Inquiry Live @ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Inquiry_Live Inquiry Live @ OER Commons] {{col-break}} * [http://p2pfoundation.net/Inquiry_Live Inquiry Live @ P2P Foundation] * [http://semanticweb.org/wiki/Inquiry_Live Inquiry Live @ SemanticWeb] * [http://ref.subwiki.org/wiki/Inquiry_Live Inquiry Live @ Subject Wikis] * [http://beta.wikiversity.org/wiki/Inquiry_Live Inquiry Live @ Wikiversity Beta] {{col-end}} [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] 1fcad7291b1a6a64ba628f3af32500721a1a6d43 0 181 396 2010-06-09T18:10:59Z Jon Awbrey 3 + focal node wikitext text/x-wiki <font size="3">&#9758;</font> This page serves as a '''focal node''' for a collection of related resources. ==Participants== * Interested parties may add their names on [[Logic Live/Participants|this page]]. ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://mywikibiz.com/Logic_Live Logic Live @ MyWikiBiz] * [http://mathweb.org/wiki/Logic_Live Logic Live @ MathWeb Wiki] * [http://netknowledge.org/wiki/Logic_Live Logic Live @ NetKnowledge] {{col-break}} * [http://wiki.oercommons.org/mediawiki/index.php/Logic_Live Logic Live @ OER Commons] * [http://p2pfoundation.net/Logic_Live Logic Live @ P2P Foundation] * [http://semanticweb.org/wiki/Logic_Live Logic Live @ SemanticWeb] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;] * [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;] [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] b8947c302d53abcac4131d6438f3753a2a362d72 10 182 397 2010-06-09T18:21:32Z Jon Awbrey 3 + col-begin template wikitext text/x-wiki {| cellpadding="0" cellspacing="0" style="background:{{{bgColor|transparent}}}; width:{{{width|100%}}};" <noinclude>|}</noinclude> c4e67ca623657064231f9574154af7782f728ca2 10 183 398 2010-06-09T18:23:20Z Jon Awbrey 3 + co-break template wikitext text/x-wiki <p></p> | align="{{{align|left}}}" valign="{{{valign|top}}}" width="{{{width|}}}" | 61ac8525ef0e9366ff7276b28a1c2a4ff891decb 10 184 399 2010-06-09T18:24:11Z Jon Awbrey 3 + col-end template wikitext text/x-wiki <p></p> |} 5b917baad25f31b188af1a668118b18578e1a732 0 181 401 396 2010-06-09T19:26:37Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page serves as a '''focal node''' for a collection of related resources. ==Participants== * Interested parties may add their names on [[Logic Live/Participants|this page]]. ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://mywikibiz.com/Logic_Live Logic Live @ MyWikiBiz] * [http://mathweb.org/wiki/Logic_Live Logic Live @ MathWeb Wiki] * [http://netknowledge.org/wiki/Logic_Live Logic Live @ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Logic_Live Logic Live @ OER Commons] {{col-break}} * [http://p2pfoundation.net/Logic_Live Logic Live @ P2P Foundation] * [http://semanticweb.org/wiki/Logic_Live Logic Live @ SemanticWeb] * [http://ref.subwiki.org/wiki/Logic_Live Logic Live @ Subject Wikis] * [http://beta.wikiversity.org/wiki/Logic_Live Logic Live @ Wikiversity Beta] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;] * [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;] [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] 5ce7416d5fb13f557ddfa09dd05dcdfdd4c11fa7 0 185 402 2010-06-09T19:44:07Z Jon Awbrey 3 + participant page wikitext text/x-wiki ==Leaders== * [[User:Jon Awbrey|Jon Awbrey]] ==Participants== * &hellip; {add name above} [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] b5751bfe1b6f2692af1db7c61c36c25a95b9be3c 0 186 403 2010-06-09T19:45:26Z Jon Awbrey 3 + participant page wikitext text/x-wiki ==Leaders== * [[User:Jon Awbrey|Jon Awbrey]] ==Participants== * &hellip; {add name above} [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] b5751bfe1b6f2692af1db7c61c36c25a95b9be3c 107 165 404 350 2010-06-09T21:10:08Z Jon Awbrey 3 /* Back in Town */ wikitext text/x-wiki This discussion page is intended for a discussion on using property theory as a means of organizing material in subject wikis, as well as the underlying issues of mathematics, logic, computation and philosophy. [[User:Vipul|Vipul]] 20:20, 25 February 2009 (UTC) ==Prospects for a Logic Wiki== ===Initial Discussion=== Previously on Buffer &hellip; * [http://blog.subwiki.org/?p=13 The History of Subject Wikis] * [http://blog.subwiki.org/?p=13&cpage=1#comment-2 Comments] <blockquote> <p>Vipul,</p> <p>From the initial sample of your writing distribution that I’ve read so far, your approach to mathematical objects by way of their properties is essentially a logical angle.</p> <p>If one elects to think in functional, computational terms from the very beginning, then it is convenient to think of a property <math>p\!</math> as being a mapping from a space <math>X,\!</math> the universe of discourse, to a space of 2 elements, say, <math>\mathbb{B} = \{ 0, 1 \}.</math> So a property is something of the form <math>p : X \to \mathbb{B}.</math> That level of consideration amounts to propositional calculus. Trivial as it may seem, it’s been my experience that getting efficient computational support at this level is key to many of the things we’d like to do at higher levels in the way of mathematical knowledge management.</p> <p>Well, enough for now, as I’m not even sure the above formatting will work here. Let’s look for a wiki discussion page where it will be easier to talk. Either at MyWikiBiz, one of your Subject Wikis, or if you like email archiving there is the Inquiry List that an e-friend set up for me.</p> <p>[http://blog.subwiki.org/?p=13&cpage=1#comment-6 Jon Awbrey, 24 Feb 2009, 20:42 UTC]</p> </blockquote> VN: Jon, I agree with your basic ideas. There's of course the issue that the space <math>X</math> under consideration is often ''not'' a ''set'' (it's too large to be one -- for instance, a [[Groupprops:group property|group property]] is defined as a ''function'' on the collection of all groups), so we cnanot simply treat it as a set map. Still, it behaves largely like a set map. And as you point out, properties behave a lot like propositions with the ''parameter'' being from the space <math>X</math>. At some point in time, I was interested in the mathematical aspects of properties. Properties form a Boolean algebra, inheriting conjunction and disjunction etc. (in set-theoretic jargon, this would just be the power set of <math>X</math>, except that <math>X</math> isn't a set). Then, we can use various partial binary operations on <math>X</math> to define corresponding binary operations on the property space. I believe this, too, has been explored by some people, who came up with structures such as quantales and gaggles. The broad idea, as far as I understand, is that the binary operation must distribute over disjunctions (ORs). Many of the operators that I discuss in the group theory wiki, such as the [[groupprops:composition operator for subgroup properties|composition operator for subgroup properties]] and the [[groupprops:join operator|join operator]], are quantalic operators. Some of the terminology that I use for some property-theoretic notation is borrowed from terminology used for such operators, for instance, the [[groupprops:left residual operator for composition|left residual]] and [[groupprops:right residual operator for composition|right residual]]. Some of the basic structural theorems and approaches are also taken from corresponding ideas that already existed in logic. (I arrived at some of them before becoming aware of the corresponding structures explored in logic, and later found that much of this had been done before). As far as I know, though, there has been no systematic application of the ideas developed in different parts of logic to this ''real world'' (?) application to understanding properties in different mathematical domains. This puzzled me since I think that using the ideas of logic and logical structure in this way can help achieve a better understanding of many disciplines, particularly those in mathematics and computer science. Perhaps it is because there are no deep theorems here? My own use of property theory as an organizational principle seems to rely very minimally on the logical ideas beneath and rests more on the practical results that are important within the specific subject. But perhaps there are deeper theoretical logic issues that should be explored. I'd be glad to hear about your perspective, logical, philosophical, or any other. I'd also like to hear more about your views on what you mean by supporting such a model computationally. On the group theory wiki, I'm using MediaWiki's basic tools as well as Semantic MediaWiki to help in property-theoretic exploration. This isn't ''smart'' in any sense on the machine's part, but it meets some of the practical needs. Nonetheless, if you have ideas for something more sophisticated, I'd be glad to hear. [[User:Vipul|Vipul]] 22:13, 25 February 2009 (UTC) JA: Sure, there is all that. My main interest at present is focused on using logic as a tool for practical knowledge management in mathematical domains. That means starting small, breaking off this or that chunk of suitably interesting or useful domains and representing them in such a way that you can not only get computers to help you reason about them but in such a way that human beings can grok what's going on. JA: Uh-oh, I'm being called to dinner, I will have to continue later tonight. [[User:Jon Awbrey|Jon Awbrey]] 23:08, 25 February 2009 (UTC) VN: Well, that's interesting. I'm myself interested in the practical aspects, too, though perhaps from a different angle. I'd like to hear more. I'll also go through some of your past writings to understand your perspective. If you have comments on the way things have been done on the subject wikis specifically, I'd like to hear those as well. [[User:Vipul|Vipul]] 00:00, 26 February 2009 (UTC) JA: Maybe it will be quickest just to list a few links. Most of the longer pieces at MyWikiBiz are unpublished papers and project workups of mine that I'm still in the process TeXing and Wikifying and redoing the graphics that got mushed up moving between different platforms over the years, so most of them have a point where I last left off reformatting or rewriting. At any rate, here are a few serving suggestions on the basic logic side of things: :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Propositional_Calculus Differential Propositional Calculus] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Functional_Logic_:_Quantification_Theory Functional Logic : Quantification Theory] JA: [[User:Jon Awbrey|Jon Awbrey]] 03:26, 26 February 2009 (UTC) VN: Jon, I've started looking at your work. Your language and approach are new to me, so it'll take me some time to go through it. Other work has also come up. I hope to get back to you in a couple of weeks. [[User:Vipul|Vipul]] 18:05, 1 March 2009 (UTC) JA: No particular rush. I went looking for some old e-lectures of mine where I worked up much more concrete examples, but they need to be converted from ASCII to HTML-TeX-Wiki formats to be readable at all, and that will take me a while. I will probably keep outlining the occasional notes here in the interval. Later, [[User:Jon Awbrey|Jon Awbrey]] 22:40, 1 March 2009 (UTC) ===Scope and Purpose of a Logic Wiki=== ; Objectives : Maintain a focus of active methods that is parallel to the focus on static content. :: Algorithms :: Computational tools :: Efficient calculi :: Heuristics :: Proofs : Here's a project that looks interesting in this connection &mdash; :: [http://www.mat.univie.ac.at/~neum/FMathL.html Arnold Neumaier : Automated Mathematical Research Assistant] VN: Jon, that certainly looks interesting. Thanks for the link. [[User:Vipul|Vipul]] 15:57, 7 March 2009 (UTC) VN: I think the discussion has two parallel tracks here: what a logic wiki should contain and what logical paradigms can be used efficiently to organize content in all subject wikis (may be you're talking of only one of these right now). VN: Regarding the organization of content across subject wikis. I had a quick look at the FMathL link. From what I can figure out, this is interesting, but still preliminary. What I'm interested in right now is something actionable that I can use to make the definitions and proofs easier and quicker to understand, easier to link to, etc. I'm sure projects such as FMathL can give some useful ideas in that connection, so I'd be glad to hear what ideas you've extracted from there. I [http://blog.subwiki.org/?p=25 wrote a recent blog post] describing some of the issues that I'm thinking about, or rather, the place I'm starting from. VN: Regarding a specific logic wiki, it seems to me that logic is an extremely big subject. Your comments and past writings give me the impression that you're interested more in the proof theory/proof calculus part of the subject. I'm assuming you're not interested so much in the model theory side of things. Okay, so how about calling it a proof theory wiki or something like that? Of course, I might be misunderstanding the scope, so more clarification from you is welcome. I'm also not sure how Neumaier's automated research assistant connects up with the content of the proposed wiki. Are you sugesting that FMathL's ideas be used to organize the wiki, or that the wiki contain information about FMathL? [[User:Vipul|Vipul]] 23:54, 11 March 2009 (UTC) JA: First time I've stopped by this way in a while. It will take me a while longer to work up the concrete example that I had in mind, which is currently in progress here: :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Analytic_Turing_Automata Differential Analytic Turing Automata (DATA)]. JA: I will have to leave the whole of logic to others. The most I can hope to do with my time here is maybe to grease the wheels of our practical vehicles at critical points. JA: Lunch''!'' Back later &hellip; [[User:Jon Awbrey|Jon Awbrey]] 16:02, 16 March 2009 (UTC) JA: Just passing through. Currently reviving some old work I did on the Propositions As Types Analogy, located here: :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositions_As_Types Propositions As Types] ===Narrative Memoir=== JA: I thought it might help to switch from didactic to personal narrative. Recent discussions on various lists reminded of why I first started work on "automatic theorem proving" (ATM) or "mechanized mathematical reasoning", as they called it back then. I had long been interested in the interplay between algebra, geometry, and logic &mdash; what one of my professors later called "the lambda point" &mdash; and when I started grad school in math in the late 70's this interest congealed into a study of the linkages between group theory, number theory, graph theory, and logic. On the graph theory face of it I got bitten by the Reconstruction Conjecture bug and eventually worked out a group-theoretic (orbit counting) way of looking at the problem. But it was all getting to be a bit too much for my unaided brain, and so one of my office-mates suggested that I try bringing the mainframe to bear on it. I was more than hesitant at first, but had been taking courses in Lisp, so I eventually said <math>y_0?\!</math> and went to work on trying to write a theorem prover. [[User:Jon Awbrey|Jon Awbrey]] 20:16, 16 March 2009 (UTC) JA: To be continued &hellip; ===Back in Town=== I was thinking I might include the Subject Ref Wiki in a trans-site project related to Logic that I've been developing. See the "focal nodes" [[Inquiry Live]] and [[Logic Live]] for an indication of what I'm trying to do. If you ever get around to creating a subject wiki for logic then the material could be moved there. [[User:Jon Awbrey|Jon Awbrey]] 14:10, 9 June 2010 (PDT) ad4f4b3c7b51e2bd597b1f612210d2212b766602 405 404 2010-06-10T01:24:23Z Jon Awbrey 3 /* Back in Town */ not sure why its giving me pacific daylight time ... wikitext text/x-wiki This discussion page is intended for a discussion on using property theory as a means of organizing material in subject wikis, as well as the underlying issues of mathematics, logic, computation and philosophy. [[User:Vipul|Vipul]] 20:20, 25 February 2009 (UTC) ==Prospects for a Logic Wiki== ===Initial Discussion=== Previously on Buffer &hellip; * [http://blog.subwiki.org/?p=13 The History of Subject Wikis] * [http://blog.subwiki.org/?p=13&cpage=1#comment-2 Comments] <blockquote> <p>Vipul,</p> <p>From the initial sample of your writing distribution that I’ve read so far, your approach to mathematical objects by way of their properties is essentially a logical angle.</p> <p>If one elects to think in functional, computational terms from the very beginning, then it is convenient to think of a property <math>p\!</math> as being a mapping from a space <math>X,\!</math> the universe of discourse, to a space of 2 elements, say, <math>\mathbb{B} = \{ 0, 1 \}.</math> So a property is something of the form <math>p : X \to \mathbb{B}.</math> That level of consideration amounts to propositional calculus. Trivial as it may seem, it’s been my experience that getting efficient computational support at this level is key to many of the things we’d like to do at higher levels in the way of mathematical knowledge management.</p> <p>Well, enough for now, as I’m not even sure the above formatting will work here. Let’s look for a wiki discussion page where it will be easier to talk. Either at MyWikiBiz, one of your Subject Wikis, or if you like email archiving there is the Inquiry List that an e-friend set up for me.</p> <p>[http://blog.subwiki.org/?p=13&cpage=1#comment-6 Jon Awbrey, 24 Feb 2009, 20:42 UTC]</p> </blockquote> VN: Jon, I agree with your basic ideas. There's of course the issue that the space <math>X</math> under consideration is often ''not'' a ''set'' (it's too large to be one -- for instance, a [[Groupprops:group property|group property]] is defined as a ''function'' on the collection of all groups), so we cnanot simply treat it as a set map. Still, it behaves largely like a set map. And as you point out, properties behave a lot like propositions with the ''parameter'' being from the space <math>X</math>. At some point in time, I was interested in the mathematical aspects of properties. Properties form a Boolean algebra, inheriting conjunction and disjunction etc. (in set-theoretic jargon, this would just be the power set of <math>X</math>, except that <math>X</math> isn't a set). Then, we can use various partial binary operations on <math>X</math> to define corresponding binary operations on the property space. I believe this, too, has been explored by some people, who came up with structures such as quantales and gaggles. The broad idea, as far as I understand, is that the binary operation must distribute over disjunctions (ORs). Many of the operators that I discuss in the group theory wiki, such as the [[groupprops:composition operator for subgroup properties|composition operator for subgroup properties]] and the [[groupprops:join operator|join operator]], are quantalic operators. Some of the terminology that I use for some property-theoretic notation is borrowed from terminology used for such operators, for instance, the [[groupprops:left residual operator for composition|left residual]] and [[groupprops:right residual operator for composition|right residual]]. Some of the basic structural theorems and approaches are also taken from corresponding ideas that already existed in logic. (I arrived at some of them before becoming aware of the corresponding structures explored in logic, and later found that much of this had been done before). As far as I know, though, there has been no systematic application of the ideas developed in different parts of logic to this ''real world'' (?) application to understanding properties in different mathematical domains. This puzzled me since I think that using the ideas of logic and logical structure in this way can help achieve a better understanding of many disciplines, particularly those in mathematics and computer science. Perhaps it is because there are no deep theorems here? My own use of property theory as an organizational principle seems to rely very minimally on the logical ideas beneath and rests more on the practical results that are important within the specific subject. But perhaps there are deeper theoretical logic issues that should be explored. I'd be glad to hear about your perspective, logical, philosophical, or any other. I'd also like to hear more about your views on what you mean by supporting such a model computationally. On the group theory wiki, I'm using MediaWiki's basic tools as well as Semantic MediaWiki to help in property-theoretic exploration. This isn't ''smart'' in any sense on the machine's part, but it meets some of the practical needs. Nonetheless, if you have ideas for something more sophisticated, I'd be glad to hear. [[User:Vipul|Vipul]] 22:13, 25 February 2009 (UTC) JA: Sure, there is all that. My main interest at present is focused on using logic as a tool for practical knowledge management in mathematical domains. That means starting small, breaking off this or that chunk of suitably interesting or useful domains and representing them in such a way that you can not only get computers to help you reason about them but in such a way that human beings can grok what's going on. JA: Uh-oh, I'm being called to dinner, I will have to continue later tonight. [[User:Jon Awbrey|Jon Awbrey]] 23:08, 25 February 2009 (UTC) VN: Well, that's interesting. I'm myself interested in the practical aspects, too, though perhaps from a different angle. I'd like to hear more. I'll also go through some of your past writings to understand your perspective. If you have comments on the way things have been done on the subject wikis specifically, I'd like to hear those as well. [[User:Vipul|Vipul]] 00:00, 26 February 2009 (UTC) JA: Maybe it will be quickest just to list a few links. Most of the longer pieces at MyWikiBiz are unpublished papers and project workups of mine that I'm still in the process TeXing and Wikifying and redoing the graphics that got mushed up moving between different platforms over the years, so most of them have a point where I last left off reformatting or rewriting. At any rate, here are a few serving suggestions on the basic logic side of things: :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Propositional_Calculus Differential Propositional Calculus] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Functional_Logic_:_Quantification_Theory Functional Logic : Quantification Theory] JA: [[User:Jon Awbrey|Jon Awbrey]] 03:26, 26 February 2009 (UTC) VN: Jon, I've started looking at your work. Your language and approach are new to me, so it'll take me some time to go through it. Other work has also come up. I hope to get back to you in a couple of weeks. [[User:Vipul|Vipul]] 18:05, 1 March 2009 (UTC) JA: No particular rush. I went looking for some old e-lectures of mine where I worked up much more concrete examples, but they need to be converted from ASCII to HTML-TeX-Wiki formats to be readable at all, and that will take me a while. I will probably keep outlining the occasional notes here in the interval. Later, [[User:Jon Awbrey|Jon Awbrey]] 22:40, 1 March 2009 (UTC) ===Scope and Purpose of a Logic Wiki=== ; Objectives : Maintain a focus of active methods that is parallel to the focus on static content. :: Algorithms :: Computational tools :: Efficient calculi :: Heuristics :: Proofs : Here's a project that looks interesting in this connection &mdash; :: [http://www.mat.univie.ac.at/~neum/FMathL.html Arnold Neumaier : Automated Mathematical Research Assistant] VN: Jon, that certainly looks interesting. Thanks for the link. [[User:Vipul|Vipul]] 15:57, 7 March 2009 (UTC) VN: I think the discussion has two parallel tracks here: what a logic wiki should contain and what logical paradigms can be used efficiently to organize content in all subject wikis (may be you're talking of only one of these right now). VN: Regarding the organization of content across subject wikis. I had a quick look at the FMathL link. From what I can figure out, this is interesting, but still preliminary. What I'm interested in right now is something actionable that I can use to make the definitions and proofs easier and quicker to understand, easier to link to, etc. I'm sure projects such as FMathL can give some useful ideas in that connection, so I'd be glad to hear what ideas you've extracted from there. I [http://blog.subwiki.org/?p=25 wrote a recent blog post] describing some of the issues that I'm thinking about, or rather, the place I'm starting from. VN: Regarding a specific logic wiki, it seems to me that logic is an extremely big subject. Your comments and past writings give me the impression that you're interested more in the proof theory/proof calculus part of the subject. I'm assuming you're not interested so much in the model theory side of things. Okay, so how about calling it a proof theory wiki or something like that? Of course, I might be misunderstanding the scope, so more clarification from you is welcome. I'm also not sure how Neumaier's automated research assistant connects up with the content of the proposed wiki. Are you sugesting that FMathL's ideas be used to organize the wiki, or that the wiki contain information about FMathL? [[User:Vipul|Vipul]] 23:54, 11 March 2009 (UTC) JA: First time I've stopped by this way in a while. It will take me a while longer to work up the concrete example that I had in mind, which is currently in progress here: :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Analytic_Turing_Automata Differential Analytic Turing Automata (DATA)]. JA: I will have to leave the whole of logic to others. The most I can hope to do with my time here is maybe to grease the wheels of our practical vehicles at critical points. JA: Lunch''!'' Back later &hellip; [[User:Jon Awbrey|Jon Awbrey]] 16:02, 16 March 2009 (UTC) JA: Just passing through. Currently reviving some old work I did on the Propositions As Types Analogy, located here: :* [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositions_As_Types Propositions As Types] ===Narrative Memoir=== JA: I thought it might help to switch from didactic to personal narrative. Recent discussions on various lists reminded of why I first started work on "automatic theorem proving" (ATM) or "mechanized mathematical reasoning", as they called it back then. I had long been interested in the interplay between algebra, geometry, and logic &mdash; what one of my professors later called "the lambda point" &mdash; and when I started grad school in math in the late 70's this interest congealed into a study of the linkages between group theory, number theory, graph theory, and logic. On the graph theory face of it I got bitten by the Reconstruction Conjecture bug and eventually worked out a group-theoretic (orbit counting) way of looking at the problem. But it was all getting to be a bit too much for my unaided brain, and so one of my office-mates suggested that I try bringing the mainframe to bear on it. I was more than hesitant at first, but had been taking courses in Lisp, so I eventually said <math>y_0?\!</math> and went to work on trying to write a theorem prover. [[User:Jon Awbrey|Jon Awbrey]] 20:16, 16 March 2009 (UTC) JA: To be continued &hellip; ===Back In Town=== I was thinking I might include the Subject Ref Wiki in a trans-site project related to Logic that I've been developing. See the "focal nodes" [[Inquiry Live]] and [[Logic Live]] for an indication of what I'm trying to do. If you ever get around to creating a subject wiki for logic then the material could be moved there. [[User:Jon Awbrey|Jon Awbrey]] 18:24, 9 June 2010 (PDT) 017935412e3c9e7213c596a4cd824874d4704f47 0 187 406 2010-06-10T01:28:31Z Jon Awbrey 3 + article wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''logical graph''' is a [[graph theory|graph-theoretic]] structure in one of the systems of graphical [[syntax]] that [http://mywikibiz.com/Charles_Sanders_Peirce Charles Sanders Peirce] developed for [[logic]]. In his papers on ''[[qualitative logic]]'', ''[[entitative graph]]s'', and ''[[existential graph]]s'', Peirce developed several versions of a graphical formalism, or a graph-theoretic formal language, designed to be interpreted for logic. In the century since Peirce initiated this line of development, a variety of formal systems have branched out from what is abstractly the same formal base of graph-theoretic structures. This article examines the common basis of these formal systems from a bird's eye view, focusing on those aspects of form that are shared by the entire family of algebras, calculi, or languages, however they happen to be viewed in a given application. ==Abstract point of view== {| width="100%" cellpadding="2" cellspacing="0" | width="60%" | &nbsp; | width="40%" | ''Wollust ward dem Wurm gegeben &hellip;'' |- | &nbsp; | align="right" | &mdash; Friedrich Schiller, ''An die Freude'' |} The bird's eye view in question is more formally known as the perspective of formal equivalence, from which remove one cannot see many distinctions that appear momentous from lower levels of abstraction. In particular, expressions of different formalisms whose syntactic structures are [[isomorphic]] from the standpoint of [[algebra]] or [[topology]] are not recognized as being different from each other in any significant sense. Though we may note in passing such historical details as the circumstance that Charles Sanders Peirce used a ''streamer-cross symbol'' where [[George Spencer Brown]] used a ''carpenter's square marker'', the theme of principal interest at the abstract level of form is neutral with regard to variations of that order. ==In lieu of a beginning== Consider the formal equations indicated in Figures&nbsp;1 and 2. {| align="center" border="0" cellpadding="10" cellspacing="0" | [[Image:Logical_Graph_Figure_1_Visible_Frame.jpg|500px]] || (1) |- | [[Image:Logical_Graph_Figure_2_Visible_Frame.jpg|500px]] || (2) |} For the time being these two forms of transformation may be referred to as ''[[axioms]]'' or ''initial equations''. ==Duality : logical and topological== There are two types of duality that have to be kept separately mind in the use of logical graphs &mdash; logical duality and topological duality. There is a standard way that graphs of the order that Peirce considered, those embedded in a continuous [[manifold]] like that commonly represented by a plane sheet of paper &mdash; with or without the paper bridges that Peirce used to augment its topological genus &mdash; can be represented in linear text as what are called ''parse strings'' or ''traversal strings'' and parsed into ''pointer structures'' in computer memory. A blank sheet of paper can be represented in linear text as a blank space, but that way of doing it tends to be confusing unless the logical expression under consideration is set off in a separate display. For example, consider the axiom or initial equation that is shown below: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_3_Visible_Frame.jpg|500px]] || (3) |} This can be written inline as <math>{}^{\backprime\backprime} \texttt{(~(~)~)} = \quad {}^{\prime\prime}</math> or set off in a text display as follows: {| align="center" cellpadding="10" | width="33%" | <math>\texttt{(~(~)~)}</math> | width="34%" | <math>=\!</math> | width="33%" | &nbsp; |} When we turn to representing the corresponding expressions in computer memory, where they can be manipulated with utmost facility, we begin by transforming the planar graphs into their topological duals. The planar regions of the original graph correspond to nodes (or points) of the [[dual graph]], and the boundaries between planar regions in the original graph correspond to edges (or lines) between the nodes of the dual graph. For example, overlaying the corresponding dual graphs on the plane-embedded graphs shown above, we get the following composite picture: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_4_Visible_Frame.jpg|500px]] || (4) |} Though it's not really there in the most abstract topology of the matter, for all sorts of pragmatic reasons we find ourselves compelled to single out the outermost region of the plane in a distinctive way and to mark it as the ''[[root node]]'' of the corresponding dual graph. In the present style of Figure the root nodes are marked by horizontal strike-throughs. Extracting the dual graphs from their composite matrices, we get this picture: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_5_Visible_Frame.jpg|500px]] || (5) |} It is easy to see the relationship between the parenthetical expressions of Peirce's logical graphs, that somewhat clippedly picture the ordered containments of their formal contents, and the associated dual graphs, that constitute the species of [[rooted tree]]s here to be described. In the case of our last example, a moment's contemplation of the following picture will lead us to see that we can get the corresponding parenthesis string by starting at the root of the tree, climbing up the left side of the tree until we reach the top, then climbing back down the right side of the tree until we return to the root, all the while reading off the symbols, in this case either <math>{}^{\backprime\backprime} \texttt{(} {}^{\prime\prime}</math> or <math>{}^{\backprime\backprime} \texttt{)} {}^{\prime\prime},</math> that we happen to encounter in our travels. {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_6_Visible_Frame.jpg|500px]] || (6) |} This ritual is called ''[[traversing]]'' the tree, and the string read off is called the ''[[traversal string]]'' of the tree. The reverse ritual, that passes from the string to the tree, is called ''[[parsing]]'' the string, and the tree constructed is called the ''[[parse graph]]'' of the string. The speakers thereof tend to be a bit loose in this language, often using ''[[parse string]]'' to mean the string that gets parsed into the associated graph. We have treated in some detail various forms of the initial equation or logical axiom that is formulated in string form as <math>{}^{\backprime\backprime} \texttt{(~(~)~)} = \quad {}^{\prime\prime}.</math> For the sake of comparison, let's record the plane-embedded and topological dual forms of the axiom that is formulated in string form as <math>{}^{\backprime\backprime} \texttt{(~)(~)} = \texttt{(~)} {}^{\prime\prime}.</math> First the plane-embedded maps: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_7_Visible_Frame.jpg|500px]] || (7) |} Next the plane-embedded maps and their dual trees superimposed: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_8_Visible_Frame.jpg|500px]] || (8) |} Finally the dual trees by themselves: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_9_Visible_Frame.jpg|500px]] || (9) |} And here are the parse trees with their traversal strings indicated: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_10_Visible_Frame.jpg|500px]] || (10) |} We have at this point enough material to begin thinking about the forms of [[analogy]], [[iconicity]], [[metaphor]], [[morphism]], whatever you want to call them, that are pertinent to the use of logical graphs in their various logical interpretations, for instance, those that Peirce described as ''[[entitative graph]]s'' and ''[[existential graph]]s''. ==Computational representation== The parse graphs that we've been looking at so far bring us one step closer to the pointer graphs that it takes to make these maps and trees live in computer memory, but they are still a couple of steps too abstract to detail the concrete species of dynamic data structures that we need. The time has come to flesh out the skeletons that we've drawn up to this point. Nodes in a graph represent ''records'' in computer memory. A record is a collection of data that can be conceived to reside at a specific ''address''. The address of a record is analogous to a demonstrative pronoun, on which account programmers commonly describe it as a ''pointer'' and semioticians recognize it as a type of sign called an ''index''. At the next level of concreteness, a pointer-record structure is represented as follows: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_11_Visible_Frame.jpg|500px]] || (11) |} This portrays the pointer <math>\mathit{index}_0\!</math> as the address of a record that contains the following data: {| align="center" cellpadding="10" | <math>\mathit{datum}_1, \mathit{datum}_2, \mathit{datum}_3, \ldots,\!</math> and so on. |} What makes it possible to represent graph-theoretical structures as data structures in computer memory is the fact that an address is just another datum, and so we may have a state of affairs like the following: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_12_Visible_Frame.jpg|500px]] || (12) |} Returning to the abstract level, it takes three nodes to represent the three data records illustrated above: one root node connected to a couple of adjacent nodes. The items of data that do not point any further up the tree are then treated as labels on the record-nodes where they reside, as shown below: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_13_Visible_Frame.jpg|500px]] || (13) |} Notice that drawing the arrows is optional with rooted trees like these, since singling out a unique node as the root induces a unique orientation on all the edges of the tree, with ''up'' being the same direction as ''away from the root''. ==Quick tour of the neighborhood== This much preparation allows us to take up the founding axioms or initial equations that determine the entire system of logical graphs. ===Primary arithmetic as semiotic system=== Though it may not seem too exciting, logically speaking, there are many reasons to make oneself at home with the system of forms that is represented indifferently, topologically speaking, by rooted trees, balanced strings of parentheses, or finite sets of non-intersecting simple closed curves in the plane. :* One reason is that it gives us a respectable example of a sign domain on which to cut our semiotic teeth, non-trivial in the sense that it contains a [[countable]] [[infinity]] of signs. :* Another reason is that it allows us to study a simple form of [[computation]] that is recognizable as a species of ''[[semiosis]]'', or sign-transforming process. This space of forms, along with the two axioms that induce its [[partition of a set|partition]] into exactly two [[equivalence class]]es, is what [[George Spencer Brown]] called the ''primary arithmetic''. The axioms of the primary arithmetic are shown below, as they appear in both graph and string forms, along with pairs of names that come in handy for referring to the two opposing directions of applying the axioms. {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_14_Banner.jpg|500px]] || (14) |- | [[Image:Logical_Graph_Figure_15_Banner.jpg|500px]] || (15) |} Let <math>S\!</math> be the set of rooted trees and let <math>S_0\!</math> be the 2-element subset of <math>S\!</math> that consists of a rooted node and a rooted edge. {| align="center" cellpadding="10" style="text-align:center" | <math>S\!</math> | <math>=\!</math> | <math>\{ \text{rooted trees} \}\!</math> |- | <math>S_0\!</math> | <math>=\!</math> | <math>\{ \ominus, \vert \} = \{</math>[[Image:Rooted Node.jpg|16px]], [[Image:Rooted Edge.jpg|12px]]<math>\}\!</math> |} Simple intuition, or a simple inductive proof, assures us that any rooted tree can be reduced by way of the arithmetic initials either to a root node [[Image:Rooted Node.jpg|16px]] or else to a rooted edge [[Image:Rooted Edge.jpg|12px]]&nbsp;. For example, consider the reduction that proceeds as follows: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_16.jpg|500px]] || (16) |} Regarded as a semiotic process, this amounts to a sequence of signs, every one after the first serving as the ''[[interpretant]]'' of its predecessor, ending in a final sign that may be taken as the canonical sign for their common object, in the upshot being the result of the computation process. Simple as it is, this exhibits the main features of any computation, namely, a semiotic process that proceeds from an obscure sign to a clear sign of the same object, being in its aim and effect an action on behalf of clarification. ===Primary algebra as pattern calculus=== Experience teaches that complex objects are best approached in a gradual, laminar, [[module|modular]] fashion, one step, one layer, one piece at a time, and it's just as much the case when the complexity of the object is irreducible, that is, when the articulations of the representation are necessarily at joints that are cloven disjointedly from nature, with some assembly required in the synthetic integrity of the intuition. That's one good reason for spending so much time on the first half of [[zeroth order logic]], represented here by the primary arithmetic, a level of formal structure that C.S. Peirce verged on intuiting at numerous points and times in his work on logical graphs, and that Spencer Brown named and brought more completely to life. There is one other reason for lingering a while longer in these primitive forests, and this is that an acquaintance with "bare trees", those as yet unadorned with literal or numerical labels, will provide a firm basis for understanding what's really at issue in such problems as the "ontological status of variables". It is probably best to illustrate this theme in the setting of a concrete case, which we can do by reviewing the previous example of reductive evaluation shown in Figure&nbsp;16. The observation of several ''semioses'', or sign-transformations, of roughly this shape will most likely lead an observer with any observational facility whatever to notice that it doesn't really matter what sorts of branches happen to sprout from the side of the root aside from the lone edge that also grows there &mdash; the end result will always be the same. Our observer might think to summarize the results of many such observations by introducing a label or variable to signify any shape of branch whatever, writing something like the following: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_17.jpg|500px]] || (17) |} Observations like that, made about an arithmetic of any variety, germinated by their summarizations, are the root of all algebra. Speaking of algebra, and having encountered already one example of an algebraic law, we might as well introduce the axioms of the ''primary algebra'', once again deriving their substance and their name from the works of [[Charles Sanders Peirce]] and [[George Spencer Brown]], respectively. {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_18.jpg|500px]] || (18) |- | [[Image:Logical_Graph_Figure_19.jpg|500px]] || (19) |} The choice of axioms for any formal system is to some degree a matter of aesthetics, as it is commonly the case that many different selections of formal rules will serve as axioms to derive all the rest as theorems. As it happens, the example of an algebraic law that we noticed first, <math>a(~) = (~),</math> as simple as it appears, proves to be provable as a theorem on the grounds of the foregoing axioms. We might also notice at this point a subtle difference between the primary arithmetic and the primary algebra with respect to the grounds of justification that we have naturally if tacitly adopted for their respective sets of axioms. The arithmetic axioms were introduced by fiat, in a quasi-[[apriori]] fashion, though of course it is only long prior experience with the practical uses of comparably developed generations of formal systems that would actually induce us to such a quasi-primal move. The algebraic axioms, in contrast, can be seen to derive their motive and their justice from the observation and summarization of patterns that are visible in the arithmetic spectrum. ==Formal development== What precedes this point is intended as an informal introduction to the axioms of the primary arithmetic and primary algebra, and hopefully provides the reader with an intuitive sense of their motivation and rationale. The next order of business is to give the exact forms of the axioms that are used in the following more formal development, devolving from Peirce's various systems of logical graphs via Spencer-Brown's ''Laws of Form'' (LOF). In formal proofs, a variation of the annotation scheme from LOF will be used to mark each step of the proof according to which axiom, or ''initial'', is being invoked to justify the corresponding step of syntactic transformation, whether it applies to graphs or to strings. ===Axioms=== The axioms are just four in number, divided into the ''arithmetic initials'', <math>I_1\!</math> and <math>I_2,\!</math> and the ''algebraic initials'', <math>J_1\!</math> and <math>J_2.\!</math> {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_20.jpg|500px]] || (20) |- | [[Image:Logical_Graph_Figure_21.jpg|500px]] || (21) |- | [[Image:Logical_Graph_Figure_22.jpg|500px]] || (22) |- | [[Image:Logical_Graph_Figure_23.jpg|500px]] || (23) |} One way of assigning logical meaning to the initial equations is known as the ''entitative interpretation'' (EN). Under EN, the axioms read as follows: {| align="center" cellpadding="10" | <math>\begin{array}{ccccc} I_1 & : & \operatorname{true}\ \operatorname{or}\ \operatorname{true} & = & \operatorname{true} \\ I_2 & : & \operatorname{not}\ \operatorname{true}\ & = & \operatorname{false} \\ J_1 & : & a\ \operatorname{or}\ \operatorname{not}\ a & = & \operatorname{true} \\ J_2 & : & (a\ \operatorname{or}\ b)\ \operatorname{and}\ (a\ \operatorname{or}\ c) & = & a\ \operatorname{or}\ (b\ \operatorname{and}\ c) \\ \end{array}</math> |} Another way of assigning logical meaning to the initial equations is known as the ''existential interpretation'' (EX). Under EX, the axioms read as follows: {| align="center" cellpadding="10" | <math>\begin{array}{ccccc} I_1 & : & \operatorname{false}\ \operatorname{and}\ \operatorname{false} & = & \operatorname{false} \\ I_2 & : & \operatorname{not}\ \operatorname{false} & = & \operatorname{true} \\ J_1 & : & a\ \operatorname{and}\ \operatorname{not}\ a & = & \operatorname{false} \\ J_2 & : & (a\ \operatorname{and}\ b)\ \operatorname{or}\ (a\ \operatorname{and}\ c) & = & a\ \operatorname{and}\ (b\ \operatorname{or}\ c) \\ \end{array}</math> |} All of the axioms in this set have the form of equations. This means that all of the inference steps that they allow are reversible. The proof annotation scheme employed below makes use of a double bar <math>\overline{\underline{~~~~~~}}</math> to mark this fact, although it will often be left to the reader to decide which of the two possible directions is the one required for applying the indicated axiom. ===Frequently used theorems=== The actual business of proof is a far more strategic affair than the simple cranking of inference rules might suggest. Part of the reason for this lies in the circumstance that the usual brands of inference rules combine the moving forward of a state of inquiry with the losing of information along the way that doesn't appear to be immediately relevant, at least, not as viewed in the local focus and the short run of the moment to moment proceedings of the proof in question. Over the long haul, this has the pernicious side-effect that one is forever strategically required to reconstruct much of the information that one had strategically thought to forget in earlier stages of the proof, if "before the proof started" can be counted as an earlier stage of the proof in view. For this reason, among others, it is very instructive to study equational inference rules of the sort that our axioms have just provided. Although equational forms of reasoning are paramount in mathematics, they are less familiar to the student of conventional logic textbooks, who may find a few surprises here. By way of gaining a minimal experience with how equational proofs look in the present forms of syntax, let us examine the proofs of a few essential theorems in the primary algebra. ====C₁. Double negation==== The first theorem goes under the names of ''Consequence&nbsp;1'' <math>(C_1)\!</math>, the ''double negation theorem'' (DNT), or ''Reflection''. {| align="center" cellpadding="10" | [[Image:Double Negation 1.0 Splash Page.png|500px]] || (24) |} The proof that follows is adapted from the one that was given by [[George Spencer Brown]] in his book ''Laws of Form'' (LOF) and credited to two of his students, John Dawes and D.A. Utting. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Double Negation 1.0 Marquee Title.png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 1.png|500px]] |- | [[Image:Equational Inference I2 Elicit (( )).png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 2.png|500px]] |- | [[Image:Equational Inference J1 Insert (a).png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 3.png|500px]] |- | [[Image:Equational Inference J2 Distribute ((a)).png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 4.png|500px]] |- | [[Image:Equational Inference J1 Delete (a).png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 5.png|500px]] |- | [[Image:Equational Inference J1 Insert a.png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 6.png|500px]] |- | [[Image:Equational Inference J2 Collect a.png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 7.png|500px]] |- | [[Image:Equational Inference J1 Delete ((a)).png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 8.png|500px]] |- | [[Image:Equational Inference I2 Cancel (( )).png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 9.png|500px]] |- | [[Image:Equational Inference Marquee QED.png|500px]] |} | (25) |} The steps of this proof are replayed in the following animation. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Double Negation 2.0 Animation.gif]] |} | (26) |} ====C₂. Generation theorem==== One theorem of frequent use goes under the nickname of the ''weed and seed theorem'' (WAST). The proof is just an exercise in mathematical induction, once a suitable basis is laid down, and it will be left as an exercise for the reader. What the WAST says is that a label can be freely distributed or freely erased anywhere in a subtree whose root is labeled with that label. The second in our list of frequently used theorems is in fact the base case of this weed and seed theorem. In LOF, it goes by the names of ''Consequence&nbsp;2'' <math>(C_2)\!</math> or ''Generation''. {| align="center" cellpadding="10" | [[Image:Generation Theorem 1.0 Splash Page.png|500px]] || (27) |} Here is a proof of the Generation Theorem. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Generation Theorem 1.0 Marquee Title.png|500px]] |- | [[Image:Generation Theorem 1.0 Storyboard 1.png|500px]] |- | [[Image:Equational Inference C1 Reflect a(b).png|500px]] |- | [[Image:Generation Theorem 1.0 Storyboard 2.png|500px]] |- | [[Image:Equational Inference I2 Elicit (( )).png|500px]] |- | [[Image:Generation Theorem 1.0 Storyboard 3.png|500px]] |- | [[Image:Equational Inference J1 Insert a.png|500px]] |- | [[Image:Generation Theorem 1.0 Storyboard 4.png|500px]] |- | [[Image:Equational Inference J2 Collect a.png|500px]] |- | [[Image:Generation Theorem 1.0 Storyboard 5.png|500px]] |- | [[Image:Equational Inference C1 Reflect a, b.png|500px]] |- | [[Image:Generation Theorem 1.0 Storyboard 6.png|500px]] |- | [[Image:Equational Inference Marquee QED.png|500px]] |} | (28) |} The steps of this proof are replayed in the following animation. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Generation Theorem 2.0 Animation.gif]] |} | (29) |} ====C₃. Dominant form theorem==== The third of the frequently used theorems of service to this survey is one that Spencer-Brown annotates as ''Consequence&nbsp;3'' <math>(C_3)\!</math> or ''Integration''. A better mnemonic might be ''dominance and recession theorem'' (DART), but perhaps the brevity of ''dominant form theorem'' (DFT) is sufficient reminder of its double-edged role in proofs. {| align="center" cellpadding="10" | [[Image:Dominant Form 1.0 Splash Page.png|500px]] || (30) |} Here is a proof of the Dominant Form Theorem. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Dominant Form 1.0 Marquee Title.png|500px]] |- | [[Image:Dominant Form 1.0 Storyboard 1.png|500px]] |- | [[Image:Equational Inference C2 Regenerate a.png|500px]] |- | [[Image:Dominant Form 1.0 Storyboard 2.png|500px]] |- | [[Image:Equational Inference J1 Delete a.png|500px]] |- | [[Image:Dominant Form 1.0 Storyboard 3.png|500px]] |- | [[Image:Equational Inference Marquee QED.png|500px]] |} | (31) |} The following animation provides an instant re*play. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Dominant Form 2.0 Animation.gif]] |} | (32) |} ===Exemplary proofs=== Based on the axioms given at the outest, and aided by the theorems recorded so far, it is possible to prove a multitude of much more complex theorems. A couple of all-time favorites are given next. ====Peirce's law==== : ''Main article'' : [[Peirce's law]] Peirce's law is commonly written in the following form: {| align="center" cellpadding="10" | <math>((p \Rightarrow q) \Rightarrow p) \Rightarrow p</math> |} The existential graph representation of Peirce's law is shown in Figure&nbsp;33. {| align="center" cellpadding="10" | [[Image:Peirce's Law 1.0 Splash Page.png|500px]] || (33) |} A graphical proof of Peirce's law is shown in Figure&nbsp;34. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Peirce's Law 1.0 Marquee Title.png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 1.png|500px]] |- | [[Image:Equational Inference Band Collect p.png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 2.png|500px]] |- | [[Image:Equational Inference Band Quit ((q)).png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 3.png|500px]] |- | [[Image:Equational Inference Band Cancel (( )).png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 4.png|500px]] |- | [[Image:Equational Inference Band Delete p.png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 5.png|500px]] |- | [[Image:Equational Inference Band Cancel (( )).png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 6.png|500px]] |- | [[Image:Equational Inference Marquee QED.png|500px]] |} | (34) |} The following animation replays the steps of the proof. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Peirce's Law 2.0 Animation.gif]] |} | (35) |} ====Praeclarum theorema==== An illustrious example of a propositional theorem is the ''praeclarum theorema'', the ''admirable'', ''shining'', or ''splendid'' theorem of [[Leibniz]]. {| align="center" cellspacing="6" width="90%" <!--QUOTE--> | <p>If ''a'' is ''b'' and ''d'' is ''c'', then ''ad'' will be ''bc''.</p> <p>This is a fine theorem, which is proved in this way:</p> <p>''a'' is ''b'', therefore ''ad'' is ''bd'' (by what precedes),</p> <p>''d'' is ''c'', therefore ''bd'' is ''bc'' (again by what precedes),</p> <p>''ad'' is ''bd'', and ''bd'' is ''bc'', therefore ''ad'' is ''bc''. Q.E.D.</p> <p>(Leibniz, ''Logical Papers'', p. 41).</p> |} Under the existential interpretation, the praeclarum theorema is represented by means of the following logical graph. {| align="center" cellpadding="10" | [[Image:Praeclarum Theorema 1.0 Splash Page.png|500px]] || (36) |} And here's a neat proof of that nice theorem. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Praeclarum Theorema 1.0 Marquee Title.png|500px]] |- | [[Image:Praeclarum Theorema 1.0 Storyboard 1.png|500px]] |- | [[Image:Equational Inference Rule Reflect ad(bc).png|500px]] |- | [[Image:Praeclarum Theorema 1.0 Storyboard 2.png|500px]] |- | [[Image:Equational Inference Rule Weed a, d.png|500px]] |- | [[Image:Praeclarum Theorema 1.0 Storyboard 3.png|500px]] |- | [[Image:Equational Inference Rule Reflect b, c.png|500px]] |- | [[Image:Praeclarum Theorema 1.0 Storyboard 4.png|500px]] |- | [[Image:Equational Inference Rule Weed bc.png|500px]] |- | [[Image:Praeclarum Theorema 1.0 Storyboard 5.png|500px]] |- | [[Image:Equational Inference Rule Quit abcd.png|500px]] |- | [[Image:Praeclarum Theorema 1.0 Storyboard 6.png|500px]] |- | [[Image:Equational Inference Rule Cancel (( )).png|500px]] |- | [[Image:Praeclarum Theorema 1.0 Storyboard 7.png|500px]] |- | [[Image:Equational Inference Marquee QED.png|500px]] |} | (37) |} The steps of the proof are replayed in the following animation. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Praeclarum Theorema 2.0 Animation.gif]] |} | (38) |} ====Two-thirds majority function==== Consider the following equation in boolean algebra, posted as a [http://mathoverflow.net/questions/9292/newbie-boolean-algebra-question problem for proof] at [http://mathoverflow.net/ MathOverFlow]. {| align="center" cellpadding="20" | <math>\begin{matrix} a b \bar{c} + a \bar{b} c + \bar{a} b c + a b c \\[6pt] \iff \\[6pt] a b + a c + b c \end{matrix}</math> | &nbsp;&nbsp;&nbsp;&nbsp; |} The required equation can be proven in the medium of logical graphs as shown in the following Figure. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Two-Thirds Majority Eq 1 Pf 1 Banner Title.png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 1.png|500px]] |- | [[Image:Equational Inference Bar Reflect ab, ac, bc.png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 2.png|500px]] |- | [[Image:Equational Inference Bar Distribute (abc).png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 3.png|500px]] |- | [[Image:Equational Inference Bar Collect ab, ac, bc.png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 4.png|500px]] |- | [[Image:Equational Inference Bar Quit (a), (b), (c).png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 5.png|500px]] |- | [[Image:Equational Inference Bar Cancel (( )).png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 6.png|500px]] |- | [[Image:Equational Inference Bar Weed ab, ac, bc.png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 7.png|500px]] |- | [[Image:Equational Inference Bar Delete a, b, c.png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 8.png|500px]] |- | [[Image:Equational Inference Bar Cancel (( )).png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 9.png|500px]] |- | [[Image:Equational Inference Banner QED.png|500px]] |} | (39) |} Here's an animated recap of the graphical transformations that occur in the above proof: {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Two-Thirds Majority Function 500 x 250 Animation.gif]] |} | (40) |} ==Bibliography== * [[Gottfried Leibniz|Leibniz, G.W.]] (1679&ndash;1686 ?), "Addenda to the Specimen of the Universal Calculus", pp. 40&ndash;46 in G.H.R. Parkinson (ed. and trans., 1966), ''Leibniz : Logical Papers'', Oxford University Press, London, UK. * [[Charles Peirce (Bibliography)|Peirce, C.S., Bibliography]]. * [[Charles Peirce|Peirce, C.S.]] (1931&ndash;1935, 1958), ''Collected Papers of Charles Sanders Peirce'', vols. 1&ndash;6, [[Charles Hartshorne]] and [[Paul Weiss]] (eds.), vols. 7&ndash;8, [[Arthur W. Burks]] (ed.), Harvard University Press, Cambridge, MA. Cited as (CP&nbsp;volume.paragraph). * Peirce, C.S. (1981&ndash;), ''Writings of Charles S. Peirce : A Chronological Edition'', [[Peirce Edition Project]] (eds.), Indiana University Press, Bloomington and Indianopolis, IN. Cited as (CE&nbsp;volume, page). * Peirce, C.S. (1885), "On the Algebra of Logic : A Contribution to the Philosophy of Notation", ''American Journal of Mathematics'' 7 (1885), 180&ndash;202. Reprinted as CP&nbsp;3.359&ndash;403 and CE&nbsp;5, 162&ndash;190. * Peirce, C.S. (''c.'' 1886), "Qualitative Logic", MS 736. Published as pp. 101&ndash;115 in Carolyn Eisele (ed., 1976), ''The New Elements of Mathematics by Charles S. Peirce, Volume 4, Mathematical Philosophy'', Mouton, The Hague. * Peirce, C.S. (1886 a), "Qualitative Logic", MS 582. Published as pp. 323&ndash;371 in ''Writings of Charles S. Peirce : A Chronological Edition, Volume 5, 1884&ndash;1886'', Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1993. * Peirce, C.S. (1886 b), "The Logic of Relatives : Qualitative and Quantitative", MS 584. Published as pp. 372&ndash;378 in ''Writings of Charles S. Peirce : A Chronological Edition, Volume 5, 1884&ndash;1886'', Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1993. * [[George Spencer Brown|Spencer Brown, George]] (1969), ''[[Laws of Form]]'', George Allen and Unwin, London, UK. ==Resources== * [http://planetmath.org/ PlanetMath] ** [http://planetmath.org/encyclopedia/LogicalGraph.html Logical Graph : Introduction] ** [http://planetmath.org/encyclopedia/LogicalGraphFormalDevelopment.html Logical Graph : Formal Development] * Bergman and Paavola (eds.), [http://www.helsinki.fi/science/commens/dictionary.html Commens Dictionary of Peirce's Terms] ** [http://www.helsinki.fi/science/commens/terms/graphexis.html Existential Graph] ** [http://www.helsinki.fi/science/commens/terms/graphlogi.html Logical Graph] * [http://dr-dau.net/index.shtml Dau, Frithjof] ** [http://dr-dau.net/eg_readings.shtml Peirce's Existential Graphs : Readings and Links] ** [http://dr-dau.net/pc.shtml Existential Graphs as Moving Pictures of Thought] &mdash; Computer Animated Proof of Leibniz's Praeclarum Theorema * [http://www.math.uic.edu/~kauffman/ Kauffman, Louis H.] ** [http://www.math.uic.edu/~kauffman/Arithmetic.htm Box Algebra, Boundary Mathematics, Logic, and Laws of Form] * [http://mathworld.wolfram.com/ MathWorld : A Wolfram Web Resource] ** [http://mathworld.wolfram.com/about/author.html Weisstein, Eric W.], [http://mathworld.wolfram.com/Spencer-BrownForm.html Spencer-Brown Form] * Shoup, Richard (ed.), [http://www.lawsofform.org/ Laws of Form Web Site] ** Spencer-Brown, George (1973), [http://www.lawsofform.org/aum/session1.html Transcript Session One], [http://www.lawsofform.org/aum/ AUM Conference], Esalen, CA. ==Translations== * [http://pt.wikipedia.org/wiki/Grafo_l%C3%B3gico Grafo lógico], [http://pt.wikipedia.org/ Portuguese Wikipedia]. ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://mywikibiz.com/Logical_graph Logical Graph @ MyWikiBiz] * [http://mathweb.org/wiki/Logical_graph Logical Graph @ MathWeb Wiki] * [http://netknowledge.org/wiki/Logical_graph Logical Graph @ NetKnowledge] {{col-break}} * [http://wiki.oercommons.org/mediawiki/index.php/Logical_graph Logical Graph @ OER Commons] * [http://p2pfoundation.net/Logical_Graph Logical Graph @ P2P Foundation] * [http://semanticweb.org/wiki/Logical_graph Logical Graph @ SemanticWeb] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;] * [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;] ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Logical_graph Logical Graph], [http://mywikibiz.com/ MyWikiBiz] * [http://mathweb.org/wiki/Logical_graph Logical Graph], [http://mathweb.org/wiki/ MathWeb Wiki] * [http://netknowledge.org/wiki/Logical_graph Logical Graph], [http://netknowledge.org/ NetKnowledge] * [http://p2pfoundation.net/Logical_Graph Logical Graph], [http://p2pfoundation.net/ P2P Foundation] * [http://semanticweb.org/wiki/Logical_graph Logical Graph], [http://semanticweb.org/ Semantic Web] {{col-break}} * [http://proofwiki.org/wiki/Definition:Logical_Graph Logical Graph], [http://proofwiki.org/ ProofWiki] * [http://planetmath.org/encyclopedia/LogicalGraph.html Logical Graph : 1], [http://planetmath.org/ PlanetMath] * [http://planetmath.org/encyclopedia/LogicalGraphFormalDevelopment.html Logical Graph : 2], [http://planetmath.org/ PlanetMath] * [http://knol.google.com/k/logical-graphs-1 Logical Graph : 1], [http://knol.google.com/ Google Knol] * [http://knol.google.com/k/logical-graphs-2 Logical Graph : 2], [http://knol.google.com/ Google Knol] {{col-break}} * [http://beta.wikiversity.org/wiki/Logical_graph Logical Graph], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://getwiki.net/-Logical_Graph Logical Graph], [http://getwiki.net/ GetWiki] * [http://wikinfo.org/index.php/Logical_graph Logical Graph], [http://wikinfo.org/ Wikinfo] * [http://textop.org/wiki/index.php?title=Logical_graph Logical Graph], [http://textop.org/wiki/ Textop Wiki] * [http://en.wikipedia.org/wiki/Logical_graph Logical Graph], [http://en.wikipedia.org/ Wikipedia] {{col-end}} 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da39a3ee5e6b4b0d3255bfef95601890afd80709 6 229 448 2010-06-10T03:04:46Z Jon Awbrey 3 wikitext text/x-wiki da39a3ee5e6b4b0d3255bfef95601890afd80709 6 230 449 2010-06-10T03:05:27Z Jon Awbrey 3 wikitext text/x-wiki da39a3ee5e6b4b0d3255bfef95601890afd80709 6 231 450 2010-06-10T03:06:09Z Jon Awbrey 3 wikitext text/x-wiki da39a3ee5e6b4b0d3255bfef95601890afd80709 6 232 451 2010-06-10T03:08:13Z Jon Awbrey 3 wikitext text/x-wiki da39a3ee5e6b4b0d3255bfef95601890afd80709 6 233 452 2010-06-10T03:10:11Z Jon Awbrey 3 wikitext text/x-wiki da39a3ee5e6b4b0d3255bfef95601890afd80709 0 187 453 406 2010-06-10T05:22:21Z Jon Awbrey 3 /* Peer nodes */ + 2 wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''logical graph''' is a [[graph theory|graph-theoretic]] structure in one of the systems of graphical [[syntax]] that [http://mywikibiz.com/Charles_Sanders_Peirce Charles Sanders Peirce] developed for [[logic]]. In his papers on ''[[qualitative logic]]'', ''[[entitative graph]]s'', and ''[[existential graph]]s'', Peirce developed several versions of a graphical formalism, or a graph-theoretic formal language, designed to be interpreted for logic. In the century since Peirce initiated this line of development, a variety of formal systems have branched out from what is abstractly the same formal base of graph-theoretic structures. This article examines the common basis of these formal systems from a bird's eye view, focusing on those aspects of form that are shared by the entire family of algebras, calculi, or languages, however they happen to be viewed in a given application. ==Abstract point of view== {| width="100%" cellpadding="2" cellspacing="0" | width="60%" | &nbsp; | width="40%" | ''Wollust ward dem Wurm gegeben &hellip;'' |- | &nbsp; | align="right" | &mdash; Friedrich Schiller, ''An die Freude'' |} The bird's eye view in question is more formally known as the perspective of formal equivalence, from which remove one cannot see many distinctions that appear momentous from lower levels of abstraction. In particular, expressions of different formalisms whose syntactic structures are [[isomorphic]] from the standpoint of [[algebra]] or [[topology]] are not recognized as being different from each other in any significant sense. Though we may note in passing such historical details as the circumstance that Charles Sanders Peirce used a ''streamer-cross symbol'' where [[George Spencer Brown]] used a ''carpenter's square marker'', the theme of principal interest at the abstract level of form is neutral with regard to variations of that order. ==In lieu of a beginning== Consider the formal equations indicated in Figures&nbsp;1 and 2. {| align="center" border="0" cellpadding="10" cellspacing="0" | [[Image:Logical_Graph_Figure_1_Visible_Frame.jpg|500px]] || (1) |- | [[Image:Logical_Graph_Figure_2_Visible_Frame.jpg|500px]] || (2) |} For the time being these two forms of transformation may be referred to as ''[[axioms]]'' or ''initial equations''. ==Duality : logical and topological== There are two types of duality that have to be kept separately mind in the use of logical graphs &mdash; logical duality and topological duality. There is a standard way that graphs of the order that Peirce considered, those embedded in a continuous [[manifold]] like that commonly represented by a plane sheet of paper &mdash; with or without the paper bridges that Peirce used to augment its topological genus &mdash; can be represented in linear text as what are called ''parse strings'' or ''traversal strings'' and parsed into ''pointer structures'' in computer memory. A blank sheet of paper can be represented in linear text as a blank space, but that way of doing it tends to be confusing unless the logical expression under consideration is set off in a separate display. For example, consider the axiom or initial equation that is shown below: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_3_Visible_Frame.jpg|500px]] || (3) |} This can be written inline as <math>{}^{\backprime\backprime} \texttt{(~(~)~)} = \quad {}^{\prime\prime}</math> or set off in a text display as follows: {| align="center" cellpadding="10" | width="33%" | <math>\texttt{(~(~)~)}</math> | width="34%" | <math>=\!</math> | width="33%" | &nbsp; |} When we turn to representing the corresponding expressions in computer memory, where they can be manipulated with utmost facility, we begin by transforming the planar graphs into their topological duals. The planar regions of the original graph correspond to nodes (or points) of the [[dual graph]], and the boundaries between planar regions in the original graph correspond to edges (or lines) between the nodes of the dual graph. For example, overlaying the corresponding dual graphs on the plane-embedded graphs shown above, we get the following composite picture: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_4_Visible_Frame.jpg|500px]] || (4) |} Though it's not really there in the most abstract topology of the matter, for all sorts of pragmatic reasons we find ourselves compelled to single out the outermost region of the plane in a distinctive way and to mark it as the ''[[root node]]'' of the corresponding dual graph. In the present style of Figure the root nodes are marked by horizontal strike-throughs. Extracting the dual graphs from their composite matrices, we get this picture: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_5_Visible_Frame.jpg|500px]] || (5) |} It is easy to see the relationship between the parenthetical expressions of Peirce's logical graphs, that somewhat clippedly picture the ordered containments of their formal contents, and the associated dual graphs, that constitute the species of [[rooted tree]]s here to be described. In the case of our last example, a moment's contemplation of the following picture will lead us to see that we can get the corresponding parenthesis string by starting at the root of the tree, climbing up the left side of the tree until we reach the top, then climbing back down the right side of the tree until we return to the root, all the while reading off the symbols, in this case either <math>{}^{\backprime\backprime} \texttt{(} {}^{\prime\prime}</math> or <math>{}^{\backprime\backprime} \texttt{)} {}^{\prime\prime},</math> that we happen to encounter in our travels. {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_6_Visible_Frame.jpg|500px]] || (6) |} This ritual is called ''[[traversing]]'' the tree, and the string read off is called the ''[[traversal string]]'' of the tree. The reverse ritual, that passes from the string to the tree, is called ''[[parsing]]'' the string, and the tree constructed is called the ''[[parse graph]]'' of the string. The speakers thereof tend to be a bit loose in this language, often using ''[[parse string]]'' to mean the string that gets parsed into the associated graph. We have treated in some detail various forms of the initial equation or logical axiom that is formulated in string form as <math>{}^{\backprime\backprime} \texttt{(~(~)~)} = \quad {}^{\prime\prime}.</math> For the sake of comparison, let's record the plane-embedded and topological dual forms of the axiom that is formulated in string form as <math>{}^{\backprime\backprime} \texttt{(~)(~)} = \texttt{(~)} {}^{\prime\prime}.</math> First the plane-embedded maps: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_7_Visible_Frame.jpg|500px]] || (7) |} Next the plane-embedded maps and their dual trees superimposed: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_8_Visible_Frame.jpg|500px]] || (8) |} Finally the dual trees by themselves: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_9_Visible_Frame.jpg|500px]] || (9) |} And here are the parse trees with their traversal strings indicated: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_10_Visible_Frame.jpg|500px]] || (10) |} We have at this point enough material to begin thinking about the forms of [[analogy]], [[iconicity]], [[metaphor]], [[morphism]], whatever you want to call them, that are pertinent to the use of logical graphs in their various logical interpretations, for instance, those that Peirce described as ''[[entitative graph]]s'' and ''[[existential graph]]s''. ==Computational representation== The parse graphs that we've been looking at so far bring us one step closer to the pointer graphs that it takes to make these maps and trees live in computer memory, but they are still a couple of steps too abstract to detail the concrete species of dynamic data structures that we need. The time has come to flesh out the skeletons that we've drawn up to this point. Nodes in a graph represent ''records'' in computer memory. A record is a collection of data that can be conceived to reside at a specific ''address''. The address of a record is analogous to a demonstrative pronoun, on which account programmers commonly describe it as a ''pointer'' and semioticians recognize it as a type of sign called an ''index''. At the next level of concreteness, a pointer-record structure is represented as follows: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_11_Visible_Frame.jpg|500px]] || (11) |} This portrays the pointer <math>\mathit{index}_0\!</math> as the address of a record that contains the following data: {| align="center" cellpadding="10" | <math>\mathit{datum}_1, \mathit{datum}_2, \mathit{datum}_3, \ldots,\!</math> and so on. |} What makes it possible to represent graph-theoretical structures as data structures in computer memory is the fact that an address is just another datum, and so we may have a state of affairs like the following: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_12_Visible_Frame.jpg|500px]] || (12) |} Returning to the abstract level, it takes three nodes to represent the three data records illustrated above: one root node connected to a couple of adjacent nodes. The items of data that do not point any further up the tree are then treated as labels on the record-nodes where they reside, as shown below: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_13_Visible_Frame.jpg|500px]] || (13) |} Notice that drawing the arrows is optional with rooted trees like these, since singling out a unique node as the root induces a unique orientation on all the edges of the tree, with ''up'' being the same direction as ''away from the root''. ==Quick tour of the neighborhood== This much preparation allows us to take up the founding axioms or initial equations that determine the entire system of logical graphs. ===Primary arithmetic as semiotic system=== Though it may not seem too exciting, logically speaking, there are many reasons to make oneself at home with the system of forms that is represented indifferently, topologically speaking, by rooted trees, balanced strings of parentheses, or finite sets of non-intersecting simple closed curves in the plane. :* One reason is that it gives us a respectable example of a sign domain on which to cut our semiotic teeth, non-trivial in the sense that it contains a [[countable]] [[infinity]] of signs. :* Another reason is that it allows us to study a simple form of [[computation]] that is recognizable as a species of ''[[semiosis]]'', or sign-transforming process. This space of forms, along with the two axioms that induce its [[partition of a set|partition]] into exactly two [[equivalence class]]es, is what [[George Spencer Brown]] called the ''primary arithmetic''. The axioms of the primary arithmetic are shown below, as they appear in both graph and string forms, along with pairs of names that come in handy for referring to the two opposing directions of applying the axioms. {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_14_Banner.jpg|500px]] || (14) |- | [[Image:Logical_Graph_Figure_15_Banner.jpg|500px]] || (15) |} Let <math>S\!</math> be the set of rooted trees and let <math>S_0\!</math> be the 2-element subset of <math>S\!</math> that consists of a rooted node and a rooted edge. {| align="center" cellpadding="10" style="text-align:center" | <math>S\!</math> | <math>=\!</math> | <math>\{ \text{rooted trees} \}\!</math> |- | <math>S_0\!</math> | <math>=\!</math> | <math>\{ \ominus, \vert \} = \{</math>[[Image:Rooted Node.jpg|16px]], [[Image:Rooted Edge.jpg|12px]]<math>\}\!</math> |} Simple intuition, or a simple inductive proof, assures us that any rooted tree can be reduced by way of the arithmetic initials either to a root node [[Image:Rooted Node.jpg|16px]] or else to a rooted edge [[Image:Rooted Edge.jpg|12px]]&nbsp;. For example, consider the reduction that proceeds as follows: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_16.jpg|500px]] || (16) |} Regarded as a semiotic process, this amounts to a sequence of signs, every one after the first serving as the ''[[interpretant]]'' of its predecessor, ending in a final sign that may be taken as the canonical sign for their common object, in the upshot being the result of the computation process. Simple as it is, this exhibits the main features of any computation, namely, a semiotic process that proceeds from an obscure sign to a clear sign of the same object, being in its aim and effect an action on behalf of clarification. ===Primary algebra as pattern calculus=== Experience teaches that complex objects are best approached in a gradual, laminar, [[module|modular]] fashion, one step, one layer, one piece at a time, and it's just as much the case when the complexity of the object is irreducible, that is, when the articulations of the representation are necessarily at joints that are cloven disjointedly from nature, with some assembly required in the synthetic integrity of the intuition. That's one good reason for spending so much time on the first half of [[zeroth order logic]], represented here by the primary arithmetic, a level of formal structure that C.S. Peirce verged on intuiting at numerous points and times in his work on logical graphs, and that Spencer Brown named and brought more completely to life. There is one other reason for lingering a while longer in these primitive forests, and this is that an acquaintance with "bare trees", those as yet unadorned with literal or numerical labels, will provide a firm basis for understanding what's really at issue in such problems as the "ontological status of variables". It is probably best to illustrate this theme in the setting of a concrete case, which we can do by reviewing the previous example of reductive evaluation shown in Figure&nbsp;16. The observation of several ''semioses'', or sign-transformations, of roughly this shape will most likely lead an observer with any observational facility whatever to notice that it doesn't really matter what sorts of branches happen to sprout from the side of the root aside from the lone edge that also grows there &mdash; the end result will always be the same. Our observer might think to summarize the results of many such observations by introducing a label or variable to signify any shape of branch whatever, writing something like the following: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_17.jpg|500px]] || (17) |} Observations like that, made about an arithmetic of any variety, germinated by their summarizations, are the root of all algebra. Speaking of algebra, and having encountered already one example of an algebraic law, we might as well introduce the axioms of the ''primary algebra'', once again deriving their substance and their name from the works of [[Charles Sanders Peirce]] and [[George Spencer Brown]], respectively. {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_18.jpg|500px]] || (18) |- | [[Image:Logical_Graph_Figure_19.jpg|500px]] || (19) |} The choice of axioms for any formal system is to some degree a matter of aesthetics, as it is commonly the case that many different selections of formal rules will serve as axioms to derive all the rest as theorems. As it happens, the example of an algebraic law that we noticed first, <math>a(~) = (~),</math> as simple as it appears, proves to be provable as a theorem on the grounds of the foregoing axioms. We might also notice at this point a subtle difference between the primary arithmetic and the primary algebra with respect to the grounds of justification that we have naturally if tacitly adopted for their respective sets of axioms. The arithmetic axioms were introduced by fiat, in a quasi-[[apriori]] fashion, though of course it is only long prior experience with the practical uses of comparably developed generations of formal systems that would actually induce us to such a quasi-primal move. The algebraic axioms, in contrast, can be seen to derive their motive and their justice from the observation and summarization of patterns that are visible in the arithmetic spectrum. ==Formal development== What precedes this point is intended as an informal introduction to the axioms of the primary arithmetic and primary algebra, and hopefully provides the reader with an intuitive sense of their motivation and rationale. The next order of business is to give the exact forms of the axioms that are used in the following more formal development, devolving from Peirce's various systems of logical graphs via Spencer-Brown's ''Laws of Form'' (LOF). In formal proofs, a variation of the annotation scheme from LOF will be used to mark each step of the proof according to which axiom, or ''initial'', is being invoked to justify the corresponding step of syntactic transformation, whether it applies to graphs or to strings. ===Axioms=== The axioms are just four in number, divided into the ''arithmetic initials'', <math>I_1\!</math> and <math>I_2,\!</math> and the ''algebraic initials'', <math>J_1\!</math> and <math>J_2.\!</math> {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_20.jpg|500px]] || (20) |- | [[Image:Logical_Graph_Figure_21.jpg|500px]] || (21) |- | [[Image:Logical_Graph_Figure_22.jpg|500px]] || (22) |- | [[Image:Logical_Graph_Figure_23.jpg|500px]] || (23) |} One way of assigning logical meaning to the initial equations is known as the ''entitative interpretation'' (EN). Under EN, the axioms read as follows: {| align="center" cellpadding="10" | <math>\begin{array}{ccccc} I_1 & : & \operatorname{true}\ \operatorname{or}\ \operatorname{true} & = & \operatorname{true} \\ I_2 & : & \operatorname{not}\ \operatorname{true}\ & = & \operatorname{false} \\ J_1 & : & a\ \operatorname{or}\ \operatorname{not}\ a & = & \operatorname{true} \\ J_2 & : & (a\ \operatorname{or}\ b)\ \operatorname{and}\ (a\ \operatorname{or}\ c) & = & a\ \operatorname{or}\ (b\ \operatorname{and}\ c) \\ \end{array}</math> |} Another way of assigning logical meaning to the initial equations is known as the ''existential interpretation'' (EX). Under EX, the axioms read as follows: {| align="center" cellpadding="10" | <math>\begin{array}{ccccc} I_1 & : & \operatorname{false}\ \operatorname{and}\ \operatorname{false} & = & \operatorname{false} \\ I_2 & : & \operatorname{not}\ \operatorname{false} & = & \operatorname{true} \\ J_1 & : & a\ \operatorname{and}\ \operatorname{not}\ a & = & \operatorname{false} \\ J_2 & : & (a\ \operatorname{and}\ b)\ \operatorname{or}\ (a\ \operatorname{and}\ c) & = & a\ \operatorname{and}\ (b\ \operatorname{or}\ c) \\ \end{array}</math> |} All of the axioms in this set have the form of equations. This means that all of the inference steps that they allow are reversible. The proof annotation scheme employed below makes use of a double bar <math>\overline{\underline{~~~~~~}}</math> to mark this fact, although it will often be left to the reader to decide which of the two possible directions is the one required for applying the indicated axiom. ===Frequently used theorems=== The actual business of proof is a far more strategic affair than the simple cranking of inference rules might suggest. Part of the reason for this lies in the circumstance that the usual brands of inference rules combine the moving forward of a state of inquiry with the losing of information along the way that doesn't appear to be immediately relevant, at least, not as viewed in the local focus and the short run of the moment to moment proceedings of the proof in question. Over the long haul, this has the pernicious side-effect that one is forever strategically required to reconstruct much of the information that one had strategically thought to forget in earlier stages of the proof, if "before the proof started" can be counted as an earlier stage of the proof in view. For this reason, among others, it is very instructive to study equational inference rules of the sort that our axioms have just provided. Although equational forms of reasoning are paramount in mathematics, they are less familiar to the student of conventional logic textbooks, who may find a few surprises here. By way of gaining a minimal experience with how equational proofs look in the present forms of syntax, let us examine the proofs of a few essential theorems in the primary algebra. ====C₁. Double negation==== The first theorem goes under the names of ''Consequence&nbsp;1'' <math>(C_1)\!</math>, the ''double negation theorem'' (DNT), or ''Reflection''. {| align="center" cellpadding="10" | [[Image:Double Negation 1.0 Splash Page.png|500px]] || (24) |} The proof that follows is adapted from the one that was given by [[George Spencer Brown]] in his book ''Laws of Form'' (LOF) and credited to two of his students, John Dawes and D.A. Utting. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Double Negation 1.0 Marquee Title.png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 1.png|500px]] |- | [[Image:Equational Inference I2 Elicit (( )).png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 2.png|500px]] |- | [[Image:Equational Inference J1 Insert (a).png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 3.png|500px]] |- | [[Image:Equational Inference J2 Distribute ((a)).png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 4.png|500px]] |- | [[Image:Equational Inference J1 Delete (a).png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 5.png|500px]] |- | [[Image:Equational Inference J1 Insert a.png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 6.png|500px]] |- | [[Image:Equational Inference J2 Collect a.png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 7.png|500px]] |- | [[Image:Equational Inference J1 Delete ((a)).png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 8.png|500px]] |- | [[Image:Equational Inference I2 Cancel (( )).png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 9.png|500px]] |- | [[Image:Equational Inference Marquee QED.png|500px]] |} | (25) |} The steps of this proof are replayed in the following animation. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Double Negation 2.0 Animation.gif]] |} | (26) |} ====C₂. Generation theorem==== One theorem of frequent use goes under the nickname of the ''weed and seed theorem'' (WAST). The proof is just an exercise in mathematical induction, once a suitable basis is laid down, and it will be left as an exercise for the reader. What the WAST says is that a label can be freely distributed or freely erased anywhere in a subtree whose root is labeled with that label. The second in our list of frequently used theorems is in fact the base case of this weed and seed theorem. In LOF, it goes by the names of ''Consequence&nbsp;2'' <math>(C_2)\!</math> or ''Generation''. {| align="center" cellpadding="10" | [[Image:Generation Theorem 1.0 Splash Page.png|500px]] || (27) |} Here is a proof of the Generation Theorem. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Generation Theorem 1.0 Marquee Title.png|500px]] |- | [[Image:Generation Theorem 1.0 Storyboard 1.png|500px]] |- | [[Image:Equational Inference C1 Reflect a(b).png|500px]] |- | [[Image:Generation Theorem 1.0 Storyboard 2.png|500px]] |- | [[Image:Equational Inference I2 Elicit (( )).png|500px]] |- | [[Image:Generation Theorem 1.0 Storyboard 3.png|500px]] |- | [[Image:Equational Inference J1 Insert a.png|500px]] |- | [[Image:Generation Theorem 1.0 Storyboard 4.png|500px]] |- | [[Image:Equational Inference J2 Collect a.png|500px]] |- | [[Image:Generation Theorem 1.0 Storyboard 5.png|500px]] |- | [[Image:Equational Inference C1 Reflect a, b.png|500px]] |- | [[Image:Generation Theorem 1.0 Storyboard 6.png|500px]] |- | [[Image:Equational Inference Marquee QED.png|500px]] |} | (28) |} The steps of this proof are replayed in the following animation. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Generation Theorem 2.0 Animation.gif]] |} | (29) |} ====C₃. Dominant form theorem==== The third of the frequently used theorems of service to this survey is one that Spencer-Brown annotates as ''Consequence&nbsp;3'' <math>(C_3)\!</math> or ''Integration''. A better mnemonic might be ''dominance and recession theorem'' (DART), but perhaps the brevity of ''dominant form theorem'' (DFT) is sufficient reminder of its double-edged role in proofs. {| align="center" cellpadding="10" | [[Image:Dominant Form 1.0 Splash Page.png|500px]] || (30) |} Here is a proof of the Dominant Form Theorem. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Dominant Form 1.0 Marquee Title.png|500px]] |- | [[Image:Dominant Form 1.0 Storyboard 1.png|500px]] |- | [[Image:Equational Inference C2 Regenerate a.png|500px]] |- | [[Image:Dominant Form 1.0 Storyboard 2.png|500px]] |- | [[Image:Equational Inference J1 Delete a.png|500px]] |- | [[Image:Dominant Form 1.0 Storyboard 3.png|500px]] |- | [[Image:Equational Inference Marquee QED.png|500px]] |} | (31) |} The following animation provides an instant re*play. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Dominant Form 2.0 Animation.gif]] |} | (32) |} ===Exemplary proofs=== Based on the axioms given at the outest, and aided by the theorems recorded so far, it is possible to prove a multitude of much more complex theorems. A couple of all-time favorites are given next. ====Peirce's law==== : ''Main article'' : [[Peirce's law]] Peirce's law is commonly written in the following form: {| align="center" cellpadding="10" | <math>((p \Rightarrow q) \Rightarrow p) \Rightarrow p</math> |} The existential graph representation of Peirce's law is shown in Figure&nbsp;33. {| align="center" cellpadding="10" | [[Image:Peirce's Law 1.0 Splash Page.png|500px]] || (33) |} A graphical proof of Peirce's law is shown in Figure&nbsp;34. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Peirce's Law 1.0 Marquee Title.png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 1.png|500px]] |- | [[Image:Equational Inference Band Collect p.png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 2.png|500px]] |- | [[Image:Equational Inference Band Quit ((q)).png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 3.png|500px]] |- | [[Image:Equational Inference Band Cancel (( )).png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 4.png|500px]] |- | [[Image:Equational Inference Band Delete p.png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 5.png|500px]] |- | [[Image:Equational Inference Band Cancel (( )).png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 6.png|500px]] |- | [[Image:Equational Inference Marquee QED.png|500px]] |} | (34) |} The following animation replays the steps of the proof. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Peirce's Law 2.0 Animation.gif]] |} | (35) |} ====Praeclarum theorema==== An illustrious example of a propositional theorem is the ''praeclarum theorema'', the ''admirable'', ''shining'', or ''splendid'' theorem of [[Leibniz]]. {| align="center" cellspacing="6" width="90%" <!--QUOTE--> | <p>If ''a'' is ''b'' and ''d'' is ''c'', then ''ad'' will be ''bc''.</p> <p>This is a fine theorem, which is proved in this way:</p> <p>''a'' is ''b'', therefore ''ad'' is ''bd'' (by what precedes),</p> <p>''d'' is ''c'', therefore ''bd'' is ''bc'' (again by what precedes),</p> <p>''ad'' is ''bd'', and ''bd'' is ''bc'', therefore ''ad'' is ''bc''. Q.E.D.</p> <p>(Leibniz, ''Logical Papers'', p. 41).</p> |} Under the existential interpretation, the praeclarum theorema is represented by means of the following logical graph. {| align="center" cellpadding="10" | [[Image:Praeclarum Theorema 1.0 Splash Page.png|500px]] || (36) |} And here's a neat proof of that nice theorem. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Praeclarum Theorema 1.0 Marquee Title.png|500px]] |- | [[Image:Praeclarum Theorema 1.0 Storyboard 1.png|500px]] |- | [[Image:Equational Inference Rule Reflect ad(bc).png|500px]] |- | [[Image:Praeclarum Theorema 1.0 Storyboard 2.png|500px]] |- | [[Image:Equational Inference Rule Weed a, d.png|500px]] |- | [[Image:Praeclarum Theorema 1.0 Storyboard 3.png|500px]] |- | [[Image:Equational Inference Rule Reflect b, c.png|500px]] |- | [[Image:Praeclarum Theorema 1.0 Storyboard 4.png|500px]] |- | [[Image:Equational Inference Rule Weed bc.png|500px]] |- | [[Image:Praeclarum Theorema 1.0 Storyboard 5.png|500px]] |- | [[Image:Equational Inference Rule Quit abcd.png|500px]] |- | [[Image:Praeclarum Theorema 1.0 Storyboard 6.png|500px]] |- | [[Image:Equational Inference Rule Cancel (( )).png|500px]] |- | [[Image:Praeclarum Theorema 1.0 Storyboard 7.png|500px]] |- | [[Image:Equational Inference Marquee QED.png|500px]] |} | (37) |} The steps of the proof are replayed in the following animation. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Praeclarum Theorema 2.0 Animation.gif]] |} | (38) |} ====Two-thirds majority function==== Consider the following equation in boolean algebra, posted as a [http://mathoverflow.net/questions/9292/newbie-boolean-algebra-question problem for proof] at [http://mathoverflow.net/ MathOverFlow]. {| align="center" cellpadding="20" | <math>\begin{matrix} a b \bar{c} + a \bar{b} c + \bar{a} b c + a b c \\[6pt] \iff \\[6pt] a b + a c + b c \end{matrix}</math> | &nbsp;&nbsp;&nbsp;&nbsp; |} The required equation can be proven in the medium of logical graphs as shown in the following Figure. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Two-Thirds Majority Eq 1 Pf 1 Banner Title.png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 1.png|500px]] |- | [[Image:Equational Inference Bar Reflect ab, ac, bc.png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 2.png|500px]] |- | [[Image:Equational Inference Bar Distribute (abc).png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 3.png|500px]] |- | [[Image:Equational Inference Bar Collect ab, ac, bc.png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 4.png|500px]] |- | [[Image:Equational Inference Bar Quit (a), (b), (c).png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 5.png|500px]] |- | [[Image:Equational Inference Bar Cancel (( )).png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 6.png|500px]] |- | [[Image:Equational Inference Bar Weed ab, ac, bc.png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 7.png|500px]] |- | [[Image:Equational Inference Bar Delete a, b, c.png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 8.png|500px]] |- | [[Image:Equational Inference Bar Cancel (( )).png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 9.png|500px]] |- | [[Image:Equational Inference Banner QED.png|500px]] |} | (39) |} Here's an animated recap of the graphical transformations that occur in the above proof: {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Two-Thirds Majority Function 500 x 250 Animation.gif]] |} | (40) |} ==Bibliography== * [[Gottfried Leibniz|Leibniz, G.W.]] (1679&ndash;1686 ?), "Addenda to the Specimen of the Universal Calculus", pp. 40&ndash;46 in G.H.R. Parkinson (ed. and trans., 1966), ''Leibniz : Logical Papers'', Oxford University Press, London, UK. * [[Charles Peirce (Bibliography)|Peirce, C.S., Bibliography]]. * [[Charles Peirce|Peirce, C.S.]] (1931&ndash;1935, 1958), ''Collected Papers of Charles Sanders Peirce'', vols. 1&ndash;6, [[Charles Hartshorne]] and [[Paul Weiss]] (eds.), vols. 7&ndash;8, [[Arthur W. Burks]] (ed.), Harvard University Press, Cambridge, MA. Cited as (CP&nbsp;volume.paragraph). * Peirce, C.S. (1981&ndash;), ''Writings of Charles S. Peirce : A Chronological Edition'', [[Peirce Edition Project]] (eds.), Indiana University Press, Bloomington and Indianopolis, IN. Cited as (CE&nbsp;volume, page). * Peirce, C.S. (1885), "On the Algebra of Logic : A Contribution to the Philosophy of Notation", ''American Journal of Mathematics'' 7 (1885), 180&ndash;202. Reprinted as CP&nbsp;3.359&ndash;403 and CE&nbsp;5, 162&ndash;190. * Peirce, C.S. (''c.'' 1886), "Qualitative Logic", MS 736. Published as pp. 101&ndash;115 in Carolyn Eisele (ed., 1976), ''The New Elements of Mathematics by Charles S. Peirce, Volume 4, Mathematical Philosophy'', Mouton, The Hague. * Peirce, C.S. (1886 a), "Qualitative Logic", MS 582. Published as pp. 323&ndash;371 in ''Writings of Charles S. Peirce : A Chronological Edition, Volume 5, 1884&ndash;1886'', Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1993. * Peirce, C.S. (1886 b), "The Logic of Relatives : Qualitative and Quantitative", MS 584. Published as pp. 372&ndash;378 in ''Writings of Charles S. Peirce : A Chronological Edition, Volume 5, 1884&ndash;1886'', Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1993. * [[George Spencer Brown|Spencer Brown, George]] (1969), ''[[Laws of Form]]'', George Allen and Unwin, London, UK. ==Resources== * [http://planetmath.org/ PlanetMath] ** [http://planetmath.org/encyclopedia/LogicalGraph.html Logical Graph : Introduction] ** [http://planetmath.org/encyclopedia/LogicalGraphFormalDevelopment.html Logical Graph : Formal Development] * Bergman and Paavola (eds.), [http://www.helsinki.fi/science/commens/dictionary.html Commens Dictionary of Peirce's Terms] ** [http://www.helsinki.fi/science/commens/terms/graphexis.html Existential Graph] ** [http://www.helsinki.fi/science/commens/terms/graphlogi.html Logical Graph] * [http://dr-dau.net/index.shtml Dau, Frithjof] ** [http://dr-dau.net/eg_readings.shtml Peirce's Existential Graphs : Readings and Links] ** [http://dr-dau.net/pc.shtml Existential Graphs as Moving Pictures of Thought] &mdash; Computer Animated Proof of Leibniz's Praeclarum Theorema * [http://www.math.uic.edu/~kauffman/ Kauffman, Louis H.] ** [http://www.math.uic.edu/~kauffman/Arithmetic.htm Box Algebra, Boundary Mathematics, Logic, and Laws of Form] * [http://mathworld.wolfram.com/ MathWorld : A Wolfram Web Resource] ** [http://mathworld.wolfram.com/about/author.html Weisstein, Eric W.], [http://mathworld.wolfram.com/Spencer-BrownForm.html Spencer-Brown Form] * Shoup, Richard (ed.), [http://www.lawsofform.org/ Laws of Form Web Site] ** Spencer-Brown, George (1973), [http://www.lawsofform.org/aum/session1.html Transcript Session One], [http://www.lawsofform.org/aum/ AUM Conference], Esalen, CA. ==Translations== * [http://pt.wikipedia.org/wiki/Grafo_l%C3%B3gico Grafo lógico], [http://pt.wikipedia.org/ Portuguese Wikipedia]. ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://mywikibiz.com/Logical_graph Logical Graph @ MyWikiBiz] * [http://mathweb.org/wiki/Logical_graph Logical Graph @ MathWeb Wiki] * [http://netknowledge.org/wiki/Logical_graph Logical Graph @ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Logical_graph Logical Graph @ OER Commons] {{col-break}} * [http://p2pfoundation.net/Logical_Graph Logical Graph @ P2P Foundation] * [http://semanticweb.org/wiki/Logical_graph Logical Graph @ SemanticWeb] * [http://ref.subwiki.org/wiki/Logical_graph Logical Graph @ Subject Wikis] * [http://beta.wikiversity.org/wiki/Logical_graph Logical Graph @ Wikiversity Beta] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;] * [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;] ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Logical_graph Logical Graph], [http://mywikibiz.com/ MyWikiBiz] * [http://mathweb.org/wiki/Logical_graph Logical Graph], [http://mathweb.org/wiki/ MathWeb Wiki] * [http://netknowledge.org/wiki/Logical_graph Logical Graph], [http://netknowledge.org/ NetKnowledge] * [http://p2pfoundation.net/Logical_Graph Logical Graph], [http://p2pfoundation.net/ P2P Foundation] * [http://semanticweb.org/wiki/Logical_graph Logical Graph], [http://semanticweb.org/ Semantic Web] {{col-break}} * [http://proofwiki.org/wiki/Definition:Logical_Graph Logical Graph], [http://proofwiki.org/ ProofWiki] * [http://planetmath.org/encyclopedia/LogicalGraph.html Logical Graph : 1], [http://planetmath.org/ PlanetMath] * [http://planetmath.org/encyclopedia/LogicalGraphFormalDevelopment.html Logical Graph : 2], [http://planetmath.org/ PlanetMath] * [http://knol.google.com/k/logical-graphs-1 Logical Graph : 1], [http://knol.google.com/ Google Knol] * [http://knol.google.com/k/logical-graphs-2 Logical Graph : 2], [http://knol.google.com/ Google Knol] {{col-break}} * [http://beta.wikiversity.org/wiki/Logical_graph Logical Graph], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://getwiki.net/-Logical_Graph Logical Graph], [http://getwiki.net/ GetWiki] * [http://wikinfo.org/index.php/Logical_graph Logical Graph], [http://wikinfo.org/ Wikinfo] * [http://textop.org/wiki/index.php?title=Logical_graph Logical Graph], [http://textop.org/wiki/ Textop Wiki] * [http://en.wikipedia.org/wiki/Logical_graph Logical Graph], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Artificial Intelligence]] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Cybernetics]] [[Category:Equational Reasoning]] [[Category:Formal Languages]] [[Category:Formal Systems]] [[Category:Graph Theory]] [[Category:History of Logic]] [[Category:History of Mathematics]] [[Category:Knowledge Representation]] [[Category:Logic]] [[Category:Logical Graphs]] [[Category:Mathematics]] [[Category:Philosophy]] [[Category:Semiotics]] [[Category:Visualization]] f39d3412d9a318a1996abccae4907d882600090e 6 234 454 2010-06-10T12:01:49Z Jon Awbrey 3 wikitext text/x-wiki da39a3ee5e6b4b0d3255bfef95601890afd80709 6 235 455 2010-06-10T12:02:53Z Jon Awbrey 3 wikitext text/x-wiki da39a3ee5e6b4b0d3255bfef95601890afd80709 6 236 456 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article wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. '''Peirce's law''' is a formula in [[propositional calculus]] that is commonly expressed in the following form: {| align="center" cellpadding="10" | <math>((p \Rightarrow q) \Rightarrow p) \Rightarrow p</math> |} Peirce's law holds in classical propositional calculus, but not in intuitionistic propositional calculus. The precise axiom system that one chooses for classical propositional calculus determines whether Peirce's law is taken as an axiom or proven as a theorem. ==History== Here is Peirce's own statement and proof of the law: {| align="center" cellpadding="8" width="90%" | <p>A ''fifth icon'' is required for the principle of excluded middle and other propositions connected with it. One of the simplest formulae of this kind is:</p> <center> <p><math>\{ (x \,-\!\!\!< y) \,-\!\!\!< x \} \,-\!\!\!< x.</math></p> </center> <p>This is hardly axiomatical. That it is true appears as follows. It can only be false by the final consequent <math>x\!</math> being false while its antecedent <math>(x \,-\!\!\!< y) \,-\!\!\!< x</math> is true. If this is true, either its consequent, <math>x,\!</math> is true, when the whole formula would be true, or its antecedent <math>x \,-\!\!\!< y</math> is false. But in the last case the antecedent of <math>x \,-\!\!\!< y,</math> that is <math>x,\!</math> must be true. (Peirce, CP&nbsp;3.384).</p> |} Peirce goes on to point out an immediate application of the law: {| align="center" cellpadding="8" width="90%" | <p>From the formula just given, we at once get:</p> <center> <p><math>\{ (x \,-\!\!\!< y) \,-\!\!\!< a \} \,-\!\!\!< x,</math></p> </center> <p>where the <math>a\!</math> is used in such a sense that <math>(x \,-\!\!\!< y) \,-\!\!\!< a</math> means that from <math>(x \,-\!\!\!< y)</math> every proposition follows. With that understanding, the formula states the principle of excluded middle, that from the falsity of the denial of <math>x\!</math> follows the truth of <math>x.\!</math> (Peirce, CP&nbsp;3.384).</p> |} '''Note.''' Peirce uses the ''sign of illation'' “<math>-\!\!\!<</math>” for implication. In one place he explains “<math>-\!\!\!<</math>” as a variant of the sign “<math>\le</math>” for ''less than or equal to''; in another place he suggests that <math>A \,-\!\!\!< B</math> is an iconic way of representing a state of affairs where <math>A,\!</math> in every way that it can be, is <math>B.\!</math> ==Graphical proof== Under the existential interpretation of Peirce's [[logical graph]]s, Peirce's law is represented by means of the following formal equivalence or logical equation. {| align="center" border="0" cellpadding="10" cellspacing="0" | [[Image:Peirce's Law 1.0 Splash Page.png|500px]] || (1) |} '''Proof.''' Using the axiom set given in the entry for [[logical graphs]], Peirce's law may be proved in the following manner. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Peirce's Law 1.0 Marquee Title.png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 1.png|500px]] |- | [[Image:Equational Inference Band Collect p.png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 2.png|500px]] |- | [[Image:Equational Inference Band Quit ((q)).png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 3.png|500px]] |- | [[Image:Equational Inference Band Cancel (( )).png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 4.png|500px]] |- | [[Image:Equational Inference Band Delete p.png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 5.png|500px]] |- | [[Image:Equational Inference Band Cancel (( )).png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 6.png|500px]] |- | [[Image:Equational Inference Marquee QED.png|500px]] |} | (2) |} The following animation replays the steps of the proof. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Peirce's Law 2.0 Animation.gif]] |} | (3) |} ==Equational form== A stronger form of Peirce's law also holds, in which the final implication is observed to be reversible: {| align="center" cellpadding="10" | <math>((p \Rightarrow q) \Rightarrow p) \Leftrightarrow p</math> |} ===Proof 1=== Given what precedes, it remains to show that: {| align="center" cellpadding="10" | <math>p \Rightarrow ((p \Rightarrow q) \Rightarrow p)</math> |} But this is immediate, since <math>p \Rightarrow (r \Rightarrow p)</math> for any proposition <math>r.\!</math> ===Proof 2=== Representing propositions as logical graphs under the existential interpretation, the strong form of Peirce's law is expressed by the following equation: {| align="center" border="0" cellpadding="10" cellspacing="0" | [[Image:Peirce's Law Strong Form 1.0 Splash Page.png|500px]] || (4) |} Using the axioms and theorems listed in the article on [[logical graphs]], the equational form of Peirce's law may be proved in the following manner: {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Peirce's Law Strong Form 1.0 Marquee Title.png|500px]] |- | [[Image:Peirce's Law Strong Form 1.0 Storyboard 1.png|500px]] |- | [[Image:Equational Inference Rule Collect p.png|500px]] |- | [[Image:Peirce's Law Strong Form 1.0 Storyboard 2.png|500px]] |- | [[Image:Equational Inference Rule Quit ((q)).png|500px]] |- | [[Image:Peirce's Law Strong Form 1.0 Storyboard 3.png|500px]] |- | [[Image:Equational Inference Rule Cancel (( )).png|500px]] |- | [[Image:Peirce's Law Strong Form 1.0 Storyboard 4.png|500px]] |- | [[Image:Equational Inference Marquee QED.png|500px]] |} | (5) |} The following animation replays the steps of the proof. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Peirce's Law Strong Form 2.0 Animation.gif]] |} | (6) |} ==Bibliography== * [[Charles Sanders Peirce|Peirce, Charles Sanders]] (1885), "On the Algebra of Logic : A Contribution to the Philosophy of Notation", ''American Journal of Mathematics'' 7 (1885), 180&ndash;202. Reprinted (CP&nbsp;3.359&ndash;403), (CE&nbsp;5, 162&ndash;190). * Peirce, Charles Sanders (1931&ndash;1935, 1958), ''Collected Papers of Charles Sanders Peirce'', vols. 1&ndash;6, Charles Hartshorne and Paul Weiss (eds.), vols. 7&ndash;8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA. Cited as (CP&nbsp;volume.paragraph). * Peirce, Charles Sanders (1981&ndash;), ''Writings of Charles S. Peirce : A Chronological Edition'', Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN. Cited as (CE&nbsp;volume,&nbsp;page). ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://mywikibiz.com/Peirce's_law Peirce's Law @ MyWikiBiz] * [http://mathweb.org/wiki/Peirce's_law Peirce's Law @ MathWeb Wiki] * [http://netknowledge.org/wiki/Peirce's_law Peirce's Law @ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Peirce's_law Peirce's Law @ OER Commons] {{col-break}} * [http://p2pfoundation.net/Peirce's_Law Peirce's Law @ P2P Foundation] * [http://semanticweb.org/wiki/Peirce's_law Peirce's Law @ SemanticWeb] * [http://ref.subwiki.org/wiki/Peirce's_law Peirce's Law @ Subject Wikis] * [http://beta.wikiversity.org/wiki/Peirce's_law Peirce's Law @ Wikiversity Beta] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;] * [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;] ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Peirce's_law Peirce's Law], [http://mywikibiz.com/ MyWikiBiz] * [http://mathweb.org/wiki/Peirce's_law Peirce's Law], [http://mathweb.org/ MathWeb Wiki] * [http://www.netknowledge.org/wiki/Peirce's_law Peirce's Law], [http://www.netknowledge.org/ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Peirce's_law Peirce's Law], [http://wiki.oercommons.org/ OER Commons] * [http://planetmath.org/encyclopedia/PeircesLaw.html Peirce's Law], [http://planetmath.org/ PlanetMath] * [http://www.proofwiki.org/wiki/Peirce's_Law Peirce's Law], [http://www.proofwiki.org/ ProofWiki] {{col-break}} * [http://beta.wikiversity.org/wiki/Peirce's_law Peirce's Law], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://knol.google.com/k/jon-awbrey/peirce-s-law/3fkwvf69kridz/10 Peirce's Law], [http://knol.google.com/ Google Knol] * [http://www.getwiki.net/-Peirce%92s_Law Peirce's Law], [http://www.getwiki.net/ GetWiki] * [http://www.wikinfo.org/index.php?title=Peirce%92s_law&oldid=29173 Peirce's Law], [http://www.wikinfo.org/ Wikinfo] * [http://www.textop.org/wiki/index.php?title=Peirce's_law Peirce's Law], [http://www.textop.org/wiki/ Textop Wiki] * [http://en.wikipedia.org/w/index.php?title=Peirce%27s_law&oldid=60606482 Peirce's Law], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Artificial Intelligence]] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Cybernetics]] [[Category:Equational Reasoning]] [[Category:Formal Languages]] [[Category:Formal Systems]] [[Category:Graph Theory]] [[Category:History of Logic]] [[Category:History of Mathematics]] [[Category:Knowledge Representation]] [[Category:Logic]] [[Category:Logical Graphs]] [[Category:Mathematics]] [[Category:Philosophy]] [[Category:Semiotics]] [[Category:Visualization]] bf5344fc65c9933c2bb31a3155d0936dffce5cd5 532 521 2010-06-14T19:32:25Z Jon Awbrey 3 plural wiki links wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. '''Peirce's law''' is a formula in [[propositional calculus]] that is commonly expressed in the following form: {| align="center" cellpadding="10" | <math>((p \Rightarrow q) \Rightarrow p) \Rightarrow p</math> |} Peirce's law holds in classical propositional calculus, but not in intuitionistic propositional calculus. The precise axiom system that one chooses for classical propositional calculus determines whether Peirce's law is taken as an axiom or proven as a theorem. ==History== Here is Peirce's own statement and proof of the law: {| align="center" cellpadding="8" width="90%" | <p>A ''fifth icon'' is required for the principle of excluded middle and other propositions connected with it. One of the simplest formulae of this kind is:</p> <center> <p><math>\{ (x \,-\!\!\!< y) \,-\!\!\!< x \} \,-\!\!\!< x.</math></p> </center> <p>This is hardly axiomatical. That it is true appears as follows. It can only be false by the final consequent <math>x\!</math> being false while its antecedent <math>(x \,-\!\!\!< y) \,-\!\!\!< x</math> is true. If this is true, either its consequent, <math>x,\!</math> is true, when the whole formula would be true, or its antecedent <math>x \,-\!\!\!< y</math> is false. But in the last case the antecedent of <math>x \,-\!\!\!< y,</math> that is <math>x,\!</math> must be true. (Peirce, CP&nbsp;3.384).</p> |} Peirce goes on to point out an immediate application of the law: {| align="center" cellpadding="8" width="90%" | <p>From the formula just given, we at once get:</p> <center> <p><math>\{ (x \,-\!\!\!< y) \,-\!\!\!< a \} \,-\!\!\!< x,</math></p> </center> <p>where the <math>a\!</math> is used in such a sense that <math>(x \,-\!\!\!< y) \,-\!\!\!< a</math> means that from <math>(x \,-\!\!\!< y)</math> every proposition follows. With that understanding, the formula states the principle of excluded middle, that from the falsity of the denial of <math>x\!</math> follows the truth of <math>x.\!</math> (Peirce, CP&nbsp;3.384).</p> |} '''Note.''' Peirce uses the ''sign of illation'' “<math>-\!\!\!<</math>” for implication. In one place he explains “<math>-\!\!\!<</math>” as a variant of the sign “<math>\le</math>” for ''less than or equal to''; in another place he suggests that <math>A \,-\!\!\!< B</math> is an iconic way of representing a state of affairs where <math>A,\!</math> in every way that it can be, is <math>B.\!</math> ==Graphical proof== Under the existential interpretation of Peirce's [[logical graphs]], Peirce's law is represented by means of the following formal equivalence or logical equation. {| align="center" border="0" cellpadding="10" cellspacing="0" | [[Image:Peirce's Law 1.0 Splash Page.png|500px]] || (1) |} '''Proof.''' Using the axiom set given in the entry for [[logical graphs]], Peirce's law may be proved in the following manner. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Peirce's Law 1.0 Marquee Title.png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 1.png|500px]] |- | [[Image:Equational Inference Band Collect p.png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 2.png|500px]] |- | [[Image:Equational Inference Band Quit ((q)).png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 3.png|500px]] |- | [[Image:Equational Inference Band Cancel (( )).png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 4.png|500px]] |- | [[Image:Equational Inference Band Delete p.png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 5.png|500px]] |- | [[Image:Equational Inference Band Cancel (( )).png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 6.png|500px]] |- | [[Image:Equational Inference Marquee QED.png|500px]] |} | (2) |} The following animation replays the steps of the proof. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Peirce's Law 2.0 Animation.gif]] |} | (3) |} ==Equational form== A stronger form of Peirce's law also holds, in which the final implication is observed to be reversible: {| align="center" cellpadding="10" | <math>((p \Rightarrow q) \Rightarrow p) \Leftrightarrow p</math> |} ===Proof 1=== Given what precedes, it remains to show that: {| align="center" cellpadding="10" | <math>p \Rightarrow ((p \Rightarrow q) \Rightarrow p)</math> |} But this is immediate, since <math>p \Rightarrow (r \Rightarrow p)</math> for any proposition <math>r.\!</math> ===Proof 2=== Representing propositions as logical graphs under the existential interpretation, the strong form of Peirce's law is expressed by the following equation: {| align="center" border="0" cellpadding="10" cellspacing="0" | [[Image:Peirce's Law Strong Form 1.0 Splash Page.png|500px]] || (4) |} Using the axioms and theorems listed in the article on [[logical graphs]], the equational form of Peirce's law may be proved in the following manner: {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Peirce's Law Strong Form 1.0 Marquee Title.png|500px]] |- | [[Image:Peirce's Law Strong Form 1.0 Storyboard 1.png|500px]] |- | [[Image:Equational Inference Rule Collect p.png|500px]] |- | [[Image:Peirce's Law Strong Form 1.0 Storyboard 2.png|500px]] |- | [[Image:Equational Inference Rule Quit ((q)).png|500px]] |- | [[Image:Peirce's Law Strong Form 1.0 Storyboard 3.png|500px]] |- | [[Image:Equational Inference Rule Cancel (( )).png|500px]] |- | [[Image:Peirce's Law Strong Form 1.0 Storyboard 4.png|500px]] |- | [[Image:Equational Inference Marquee QED.png|500px]] |} | (5) |} The following animation replays the steps of the proof. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Peirce's Law Strong Form 2.0 Animation.gif]] |} | (6) |} ==Bibliography== * [[Charles Sanders Peirce|Peirce, Charles Sanders]] (1885), "On the Algebra of Logic : A Contribution to the Philosophy of Notation", ''American Journal of Mathematics'' 7 (1885), 180&ndash;202. Reprinted (CP&nbsp;3.359&ndash;403), (CE&nbsp;5, 162&ndash;190). * Peirce, Charles Sanders (1931&ndash;1935, 1958), ''Collected Papers of Charles Sanders Peirce'', vols. 1&ndash;6, Charles Hartshorne and Paul Weiss (eds.), vols. 7&ndash;8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA. Cited as (CP&nbsp;volume.paragraph). * Peirce, Charles Sanders (1981&ndash;), ''Writings of Charles S. Peirce : A Chronological Edition'', Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN. Cited as (CE&nbsp;volume,&nbsp;page). ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://mywikibiz.com/Peirce's_law Peirce's Law @ MyWikiBiz] * [http://mathweb.org/wiki/Peirce's_law Peirce's Law @ MathWeb Wiki] * [http://netknowledge.org/wiki/Peirce's_law Peirce's Law @ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Peirce's_law Peirce's Law @ OER Commons] {{col-break}} * [http://p2pfoundation.net/Peirce's_Law Peirce's Law @ P2P Foundation] * [http://semanticweb.org/wiki/Peirce's_law Peirce's Law @ SemanticWeb] * [http://ref.subwiki.org/wiki/Peirce's_law Peirce's Law @ Subject Wikis] * [http://beta.wikiversity.org/wiki/Peirce's_law Peirce's Law @ Wikiversity Beta] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;] * [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;] ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Peirce's_law Peirce's Law], [http://mywikibiz.com/ MyWikiBiz] * [http://mathweb.org/wiki/Peirce's_law Peirce's Law], [http://mathweb.org/ MathWeb Wiki] * [http://www.netknowledge.org/wiki/Peirce's_law Peirce's Law], [http://www.netknowledge.org/ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Peirce's_law Peirce's Law], [http://wiki.oercommons.org/ OER Commons] * [http://planetmath.org/encyclopedia/PeircesLaw.html Peirce's Law], [http://planetmath.org/ PlanetMath] * [http://www.proofwiki.org/wiki/Peirce's_Law Peirce's Law], [http://www.proofwiki.org/ ProofWiki] {{col-break}} * [http://beta.wikiversity.org/wiki/Peirce's_law Peirce's Law], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://knol.google.com/k/jon-awbrey/peirce-s-law/3fkwvf69kridz/10 Peirce's Law], [http://knol.google.com/ Google Knol] * [http://www.getwiki.net/-Peirce%92s_Law Peirce's Law], [http://www.getwiki.net/ GetWiki] * [http://www.wikinfo.org/index.php?title=Peirce%92s_law&oldid=29173 Peirce's Law], [http://www.wikinfo.org/ Wikinfo] * [http://www.textop.org/wiki/index.php?title=Peirce's_law Peirce's Law], [http://www.textop.org/wiki/ Textop Wiki] * [http://en.wikipedia.org/w/index.php?title=Peirce%27s_law&oldid=60606482 Peirce's Law], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Artificial Intelligence]] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Cybernetics]] [[Category:Equational Reasoning]] [[Category:Formal Languages]] [[Category:Formal Systems]] [[Category:Graph Theory]] [[Category:History of Logic]] [[Category:History of Mathematics]] [[Category:Knowledge Representation]] [[Category:Logic]] [[Category:Logical Graphs]] [[Category:Mathematics]] [[Category:Philosophy]] [[Category:Semiotics]] [[Category:Visualization]] 79aba0f232f5e13dbbb164a3c6024f4ff38a7104 0 302 522 2010-06-14T16:20:13Z Jon Awbrey 3 #REDIRECT [[Logical graph]] wikitext text/x-wiki #REDIRECT [[Logical graph]] add5e5f00fb4e2fa1d40009ce232532446881cc3 6 303 523 2010-06-14T19:20:18Z Jon Awbrey 3 wikitext text/x-wiki da39a3ee5e6b4b0d3255bfef95601890afd80709 6 304 524 2010-06-14T19:20:33Z Jon Awbrey 3 wikitext 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'''Relation theory''', or the '''theory of relations''', treats the subject matter of [[relations]] in its combinatorial aspect, as distinguished from, though related to, its more properly logical study on one side and its more generally mathematical study on another. A '''relation''', as conceived in the combinatorial theory of relations, is a mathematical object that in general can have a very complex type, the complexity of which is best approached in several stages, as indicated next. In order to approach the combinatorial definition of a relation, it helps to introduce a few preliminary notions that can serve as stepping stones to the general idea. A relation in mathematics is defined as an object that has its existence as such within a definite context or setting. It is literally the case that to change this setting is to change the relation that is being defined. The particular type of context that is needed here is formalized as a collection of elements from which are chosen the elements of the relation in question. This larger collection of ''[[elementary relation]]s'' or ''[[tuple]]s'' is constructed by means of the set-theoretic product commonly known as the ''[[cartesian product]]''. ==Preliminaries== A '''relation''' <math>L\!</math> is defined by specifying two mathematical objects as its constituent parts: {| align="center" cellspacing="6" width="90%" | <p>The first part is called the ''figure'' of <math>L,\!</math> notated as <math>\operatorname{figure}(L).</math></p> |- | <p>The second part is called the ''ground'' of <math>L,\!</math> notated as <math>\operatorname{ground}(L).</math></p> |} In the special case of a ''finitary relation'', for concreteness a ''<math>k\!</math>-place relation'', the concepts of figure and ground are defined as follows: {| align="center" cellspacing="6" width="90%" | <p>The ''ground'' of <math>L\!</math> is a sequence of <math>k\!</math> nonempty sets, <math>X_1, \ldots, X_k,</math> called the ''domains'' of the relation <math>L.\!</math></p> |- | <p>The ''figure'' of <math>L\!</math> is a subset of the cartesian product taken over the domains of <math>L.\!</math></p> |- | <p>In sum we have: <math>\operatorname{figure}(L) ~\subseteq~ \operatorname{ground}(L) ~=~ X_1 \times \ldots, \times X_k.</math></p> |} Strictly speaking, then, the relation <math>L\!</math> is an ordered pair of mathematical objects, <math>L = (\operatorname{figure}(L), \operatorname{ground}(L)),</math> but it is customary in loose speech to use the single name <math>L\!</math> in a systematically equivocal fashion, taking it to denote either the pair <math>L = (\operatorname{figure}(L), \operatorname{ground}(L))</math> or the figure <math>\operatorname{figure}(L).</math> There is usually no confusion about this so long as the ground of the relation can be gathered from context. ==Definition== The formal definition of a '''finite arity relation''', specifically, a '''k-ary relation''' can now be stated. * '''Definition.''' A '''k-ary relation''' ''L'' over the nonempty sets ''X''₁,&nbsp;…,&nbsp;''X''_''k'' is a (1+''k'')-tuple ''L'' = (''F''(''L''),&nbsp;''X''₁,&nbsp;…,&nbsp;''X''_''k'') where ''F''(''L'') is a subset of the cartesian product ''X''₁&nbsp;&times;&nbsp;…&nbsp;&times;&nbsp;''X''_''k''. If all of the ''X''_''j'' for ''j''&nbsp;=&nbsp;1&nbsp;to&nbsp;''k'' are the same set ''X'', then ''L'' is more simply called a '''''k''-ary relation over ''X'''''. The set ''F''(''L'') is called the ''figure'' of ''L'' and, provided that the sequence of sets ''X''₁,&nbsp;…,&nbsp;''X''_''k'' is fixed throughout a given discussion or is otherwise determinate in context, one may regard the relation ''L'' as being determined by its figure ''F''(''L''). The formal definition simply repeats more concisely what was said above, merely unwrapping the conceptual packaging of the relation's ground to define the relation in 1&nbsp;+&nbsp;''k'' parts, as ''L'' = (''F''(''X''),&nbsp;''X''₁,&nbsp;…,&nbsp;''X''_''k''), rather than just the two, as ''L'' = (''F''(''L''),&nbsp;''G''(''L'')). A ''k''-ary '''predicate''' is a ''[[boolean-valued function]]'' on ''k'' variables. ==Local incidence properties== A '''local incidence property''' (LIP) of a relation ''L'' is a property that depends in turn on the properties of special subsets of ''L'' that are known as its ''local flags''. The local flags of a relation are defined in the following way: Let ''L'' be a ''k''-place relation ''L'' &sube; ''X''₁ × … × ''X''_''k''. Select a relational domain ''X''_''j'' and one of its elements ''x''. Then ''L''_''x''.''j'' is a subset of ''L'' that is referred to as the ''flag'' of ''L'' with ''x'' at ''j'', or the ''x''.''j''-flag of ''L'', an object which has the following definition: :* ''L''_''x''.''j'' = { (''x''₁, …, ''x''_''j'', …, ''x''_''k'') &isin; ''L'' : ''x''_''j'' = ''x'' }. Any property ''C'' of the local flag ''L''_''x''.''j'' &sube; ''L'' is said to be a ''local incidence property'' of ''L'' with respect to the locus ''x'' at ''j''. A ''k''-adic relation ''L'' &sube; ''X''₁ &times; … &times; ''X''_''k'' is said to be ''C''-regular at ''j'' if and only if every flag of ''L'' with ''x'' at ''j'' has the property ''C'', where ''x'' is taken to vary over the ''theme'' of the fixed domain ''X''_''j''. Expressed in symbols, ''L'' is ''C''-regular at ''j'' if and only if ''C''(''L''_''x''.''j'') is true for all ''x'' in ''X''_''j''. ==Numerical incidence properties== A '''numerical incidence property''' (NIP) of a relation is a local incidence property that depends on the [[cardinality|cardinalities]] of its local flags. For example, ''L'' is said to be ''c''-regular at ''j'' if and only if the cardinality of the local flag ''L''_''x''.''j'' is ''c'' for all ''x'' in ''X''_''j'', or, to write it in symbols, if and only if |''L''_''x''.''j''| = ''c'' for all ''x'' in ''X''_''j''. In a similar fashion, one can define the NIPs (< ''c'')-regular at ''j'', (> ''c'')-regular at ''j'', and so on. For ease of reference, a few of these definitions are recorded here: {| cellpadding="4" | &nbsp; | <math>L</math> | is | align="center" | ''c''-regular | at ''j'' | if and only if | <math>|L_{x.j}|</math> | = | ''c'' | for all ''x'' in ''X''_j. |- | &nbsp; | <math>L</math> | is | align="center" | (< ''c'')-regular | at ''j'' | if and only if | <math>|L_{x.j}|</math> | < | ''c'' | for all ''x'' in ''X''_j. |- | &nbsp; | <math>L</math> | is | align="center" | (> ''c'')-regular | at ''j'' | if and only if | <math>|L_{x.j}|</math> | > | ''c'' | for all ''x'' in ''X''_j. |} The definition of a local flag can be broadened from a point ''x'' in ''X''_''j'' to a subset ''M'' of ''X''_''j'', arriving at the definition of a ''regional flag'' in the following way: Suppose that ''L'' &sube; ''X''₁ &times; … &times; ''X''_''k'', and choose a subset ''M'' &sube; ''X''_''j''. Then ''L''_''M''.''j'' is a subset of ''L'' that is said to be the ''flag'' of ''L'' with ''M'' at ''j'', or the ''M''.''j''-flag of ''L'', an object which has the following definition: {| cellpadding="4" | &nbsp; | ''L''_''M''.''j'' | = | { (''x''₁, …, ''x''_''j'', …, ''x''_''k'') &isin; ''L'' : ''x''_''j'' &isin; ''M'' }. |} Returning to 2-adic relations, it is useful to describe some familiar classes of objects in terms of their local and their numerical incidence properties. Let ''L'' &sube; ''S'' &times; ''T'' be an arbitrary 2-adic relation. The following properties of ''L'' can be defined: {| cellpadding="4" | &nbsp; | <math>L</math> | is | align="center" | ''total'' | at ''S'' | if and only if | <math>L</math> | is | (&#8805;1)-regular | at ''S''. |- | &nbsp; | <math>L</math> | is | align="center" | ''total'' | at ''T'' | if and only if | <math>L</math> | is | (&#8805;1)-regular | at ''T''. |- | &nbsp; | <math>L</math> | is | align="center" | ''tubular'' | at ''S'' | if and only if | <math>L</math> | is | (&#8804;1)-regular | at ''S''. |- | &nbsp; | <math>L</math> | is | align="center" | ''tubular'' | at ''T'' | if and only if | <math>L</math> | is | (&#8804;1)-regular | at ''T''. |} If ''L'' &sube; ''S'' × ''T'' is tubular at ''S'', then ''L'' is called a ''partial function'' or a ''prefunction'' from ''S'' to ''T'', sometimes indicated by giving ''L'' an alternate name, say, "''p''", and writing ''L'' = ''p'' : ''S'' <math>\rightharpoonup</math> ''T''. Just by way of formalizing the definition: {| cellpadding="4" | &nbsp; | ''L'' | = | ''p'' : ''S'' <math>\rightharpoonup</math> ''T'' | if and only if | ''L'' | is | tubular | at ''S''. |} If ''L'' is a prefunction ''p'' : ''S'' <math>\rightharpoonup</math> ''T'' that happens to be total at ''S'', then ''L'' is called a ''function'' from ''S'' to ''T'', indicated by writing ''L'' = ''f'' : ''S'' &rarr; ''T''. To say that a relation ''L'' &sube; ''S'' &times; ''T'' is ''totally tubular'' at ''S'' is to say that it is 1-regular at ''S''. Thus, we may formalize the following definition: {| cellpadding="4" | &nbsp; | ''L'' | = | ''f'' : ''S'' &rarr; ''T'' | if and only if | ''L'' | is | 1-regular | at ''S''. |} In the case of a function ''f'' : ''S'' &rarr; ''T'', one has the following additional definitions: {| cellpadding="4" | &nbsp; | ''f'' | is | align="center" | ''surjective'' | if and only if | ''f'' | is | align="center" | total | at ''T''. |- | &nbsp; | ''f'' | is | align="center" | ''injective'' | if and only if | ''f'' | is | align="center" | tubular | at ''T''. |- | &nbsp; | ''f'' | is | align="center" | ''bijective'' | if and only if | ''f'' | is | align="center" | 1-regular | at ''T''. |} ==Variations== Because the concept of a relation has been developed quite literally from the beginnings of logic and mathematics, and because it has incorporated contributions from a diversity of thinkers from many different times and intellectual climates, there is a wide variety of terminology that the reader may run across in connection with the subject. One dimension of variation is reflected in the names that are given to k-place relations, with some writers using ''medadic'', ''monadic'', ''dyadic'', ''triadic'', ''k-adic'' where other writers use ''nullary'', ''unary'', ''binary'', ''ternary'', ''k-ary''. One finds a relation on a finite number of domains described as either a ''finitary'' relation or a ''polyadic'' relation. If the number of domains is finite, say equal to ''k'', then the [[parameter]] ''k'' may be referred to as the ''[[arity]]'', the ''[[adicity]]'', or the ''[[dimension]]'' of the relation. In these cases, the relation may be described as a ''k-ary'' relation, a ''k-adic'' relation, or a ''k-dimensional'' relation, respectively. A more conceptual than nominal variation depends on whether one uses terms like 'predicate', 'relation', and even 'term' to refer to the formal object proper or else to the allied [[syntax|syntactic]] items that are used to denote them. Compounded with this variation is still another, frequently associated with philosophical differences over the status in reality accorded formal objects. Among those who speak of numbers, functions, properties, relations, and sets as being real, that is to say, as having objective properties, there are divergences as to whether some things are more real than others, especially whether particulars or properties are equally real or else one derivative of the other. Historically speaking, just about every combination of modalities has been used by one school of thought or another, but it suffices here merely to indicate how the options are generated. ==Examples== : ''See the articles on [[relation (mathematics)|relations]], [[relation composition]], [[relation reduction]], [[sign relation]]s, and [[triadic relation]]s for concrete examples of relations.'' Many relations of the greatest interest in mathematics are triadic relations, but this fact is somewhat disguised by the circumstance that many of them are referred to as ''[[binary operation]]s'', and because the most familiar of these have very specific properties that are dictated by their axioms. This makes it practical to study these operations for quite some time by focusing on their dyadic aspects before being forced to consider their proper characters as triadic relations. ==References== * [[Charles Sanders Peirce|Peirce, C.S.]], "Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic", ''Memoirs of the American Academy of Arts and Sciences'', 9, 317-378, 1870. Reprinted, ''Collected Papers'' CP 3.45-149, ''Chronological Edition'' CE 2, 359-429. * [[Stanis&#322;aw Marcin Ulam|Ulam, S.M.]] and [[Al Bednarek|Bednarek, A.R.]], "On the Theory of Relational Structures and Schemata for Parallel Computation", pp. 477-508 in A.R. Bednarek and Françoise Ulam (eds.), ''Analogies Between Analogies : The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators'', University of California Press, Berkeley, CA, 1990. ==Bibliography== * [[Michael Barr|Barr, M.]] and [[Charles Wells|Wells, C.]], ''Category Theory for Computing Science'', [[Hemel Hempstead]], UK, 1990. * [[Nicolas Bourbaki|Bourbaki, N.]], ''Elements of the History of Mathematics'', John Meldrum (trans.), Springer-Verlag, Berlin, Germany, 1994. * [[Rudolf Carnap|Carnap, Rudolf]], ''Introduction to Symbolic Logic with Applications'', Dover Publications, New York, NY, 1958. * Chang, C.C., and Keisler, H.J., ''Model Theory'', North-Holland, Amsterdam, Netherlands, 1973. * [[Dirk van Dalen|van Dalen, D.]], ''Logic and Structure'', 2nd edition, Springer-Verlag, Berlin, Germany, 1980. * [[Keith J. Devlin|Devlin, K.J.]], ''The Joy of Sets: Fundamentals of Contemporary Set Theory'', 2nd edition, Springer-Verlag, New York, NY, 1993. * [[Paul Richard Halmos|Halmos, P.R.]], ''Naive Set Theory'', D. Van Nostrand Company, Princeton, NJ, 1960. * [[Jean van Heijenoort|van Heijenoort, J.]], ''From Frege to Gödel, A Source Book in Mathematical Logic, 1879?1931'', Harvard University Press, Cambridge, MA, 1967. Reprinted with corrections, 1977. * [[John L. Kelley|Kelley, J.L.]], ''General Topology'', Van Nostrand Reinhold, New York, NY, 1955. * [[William Kneale|Kneale, W.]] and [[Martha Kneale|Kneale, M.]], ''The Development of Logic'', Oxford University Press, Oxford, UK, 1962. Reprinted with corrections, 1975. * [[Francis William Lawvere|Lawvere, F.W.]], and [[Robert Rosebrugh|Rosebrugh, R.]], ''Sets for Mathematics'', Cambridge University Press, Cambridge, UK, 2003. * [[Francis William Lawvere|Lawvere, F.W.]], and [[Stephen H. Schanuel|Schanuel, S.H.]], ''Conceptual Mathematics, A First Introduction to Categories'', Cambridge University Press, Cambridge, UK, 1997. Reprinted with corrections, 2000. * Maddux, R.D., ''Relation Algebras'', Volume 150, 'Studies in Logic and the Foundations of Mathematics', Elsevier Science, 2006. * [[Yu. I. Manin|Manin, Yu. I.]], ''A Course in Mathematical Logic'', Neal Koblitz (trans.), Springer-Verlag, New York, NY, 1977. * [[Mathematical Society of Japan]], ''Encyclopedic Dictionary of Mathematics'', 2nd edition, 2 vols., Kiyosi Itô (ed.), MIT Press, Cambridge, MA, 1993. * Mili, A., Desharnais, J., Mili, F., with Frappier, M., ''Computer Program Construction'', Oxford University Press, New York, NY, 1994. (Introduction to Tarskian relation theory and its applications within the relational programming paradigm.) * [[John C. Mitchell|Mitchell, J.C.]], ''Foundations for Programming Languages'', MIT Press, Cambridge, MA, 1996. * Peirce, C.S., ''Collected Papers of Charles Sanders Peirce'', vols. 1&ndash;6, [[Charles Hartshorne]] and [[Paul Weiss]] (eds.), vols. 7&ndash;8, [[Arthur W. Burks]] (ed.), Harvard University Press, Cambridge, MA, 1931&ndash;1935, 1958. Cited as CP volume.paragraph. * Peirce, C.S., ''Writings of Charles S. Peirce : A Chronological Edition, Volume 2, 1867&ndash;1871'', Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1984. Cited as CE 2. * [[Bruno Poizat|Poizat, B.]], ''A Course in Model Theory: An Introduction to Contemporary Mathematical Logic'', Moses Klein (trans.), Springer-Verlag, New York, NY, 2000. * [[Willard Van Orman Quine|Quine, W.V.]], ''Mathematical Logic'', 1940. Revised edition, Harvard University Press, Cambridge, MA, 1951. New preface, 1981. * [[Josiah Royce|Royce, J.]], ''The Principles of Logic'', Philosophical Library, New York, NY, 1961. * [[Dagobert D. Runes|Runes, D.D.]] (ed.), ''Dictionary of Philosophy'', Littlefield, Adams, and Company, Totowa, NJ, 1962. * [[Joseph R. Shoenfield|Shoenfield, J.R.]], ''Mathematical Logic'', Addison-Wesley, Reading, MA, 1967. * Styazhkin, N.I., ''History of Mathematical Logic from Leibniz to Peano'', MIT Press, Cambridge, MA, 1969. * [[Patrick Suppes|Suppes, Patrick]], ''Introduction to Logic'', 1st published 1957. Reprinted, Dover Publications, New York, NY, 1999. * Suppes, Patrick, ''Axiomatic Set Theory'', 1st published 1960. Reprinted, Dover Publications, New York, NY, 1972. * [[Alfred Tarski|Tarski, A.]], ''Logic, Semantics, Metamathematics, Papers from 1923 to 1938'', J.H. Woodger (trans.), first edition, Oxford University Press, 1956, second edition, J. Corcoran (ed.), Hackett Publishing, Indianapolis, IN, 1983. * [[Stanis&#322;aw Marcin Ulam|Ulam, S.M.]], ''Analogies Between Analogies : The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators'', A.R. Bednarek and Françoise Ulam (eds.), University of California Press, Berkeley, CA, 1990. * [[Jeffrey D. Ullman|Ullman, J.D.]], ''Principles of Database Systems'', Computer Science Press, Rockville, MD, 1980. * [[Paulus Venetus|Venetus, P.]], ''Logica Parva, Translation of the 1472 Edition with Introduction and Notes'', Alan R. Perreiah (trans.), Philosophia Verlag, Munich, Germany, 1984. ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://mywikibiz.com/Relation_theory Relation Theory @ MyWikiBiz] * [http://mathweb.org/wiki/Relation_theory Relation Theory @ MathWeb Wiki] * [http://netknowledge.org/wiki/Relation_theory Relation Theory @ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Relation_theory Relation Theory @ OER Commons] {{col-break}} * [http://p2pfoundation.net/Relation_Theory Relation Theory @ P2P Foundation] * [http://semanticweb.org/wiki/Relation_theory Relation Theory @ SemanticWeb] * [http://ref.subwiki.org/wiki/Relation_theory Relation Theory @ Subject Wikis] * [http://beta.wikiversity.org/wiki/Relation_theory Relation Theory @ Wikiversity Beta] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical 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relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;] * [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;] ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Relation_theory Relation Theory], [http://mywikibiz.com/ MyWikiBiz] * [http://mathweb.org/wiki/Relation_theory Relation Theory], [http://mathweb.org/wiki/ MathWeb Wiki] * [http://netknowledge.org/wiki/Relation_theory Relation Theory], [http://netknowledge.org/ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Relation_theory Relation Theory], [http://wiki.oercommons.org/ OER Commons] {{col-break}} * [http://p2pfoundation.net/Relation_Theory Relation Theory], [http://p2pfoundation.net/ P2P Foundation] * [http://semanticweb.org/wiki/Relation_theory Relation Theory], [http://semanticweb.org/ Semantic Web] * [http://planetmath.org/encyclopedia/RelationTheory.html Relation Theory], [http://planetmath.org/ PlanetMath] * [http://planetphysics.org/encyclopedia/RelationTheory.html Relation Theory], [http://planetphysics.org/ PlanetPhysics] {{col-break}} * [http://getwiki.net/-Theory_of_Relations Relation Theory], [http://getwiki.net/ GetWiki] * [http://wikinfo.org/index.php/Theory_of_relations Relation Theory], [http://wikinfo.org/ Wikinfo] * [http://textop.org/wiki/index.php?title=Theory_of_relations Relation Theory], [http://textop.org/wiki/ Textop Wiki] * [http://en.wikipedia.org/w/index.php?title=Theory_of_relations&oldid=45042729 Relation Theory], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Database Theory]] [[Category:Discrete Mathematics]] [[Category:Formal Sciences]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Relation Theory]] [[Category:Set Theory]] 3aac7ba84fbea7adeb7acd55aeaaf22d0ca10ebc 0 313 534 2010-06-18T20:18:50Z Jon Awbrey 3 + article wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. In [[logic]], [[mathematics]], and [[semiotics]], a '''triadic relation''' is an important special case of a [[relation (mathematics)|polyadic or finitary relation]], one in which the number of places in the relation is three. In other language that is often used, a triadic relation is called a '''ternary relation'''. One may also see the adjectives ''3-adic'', ''3-ary'', ''3-dimensional'', or ''3-place'' being used to describe these relations. Mathematics is positively rife with examples of 3-adic relations, and a [[sign relation]], the arch-idea of the whole field of semiotics, is a special case of a 3-adic relation. Therefore it will be useful to consider a few concrete examples from each of these two realms. ==Examples from mathematics== For the sake of topics to be taken up later, it is useful to examine a pair of 3-adic relations in tandem, <math>L_0\!</math> and <math>L_1,\!</math> that can be described in the following manner. The first order of business is to define the space in which the relations <math>L_0\!</math> and <math>L_1\!</math> take up residence. This space is constructed as a 3-fold [[cartesian power]] in the following way. The ''[[boolean domain]]'' is the set <math>\mathbb{B} = \{ 0, 1 \}.</math> The ''plus sign'' <math>^{\backprime\backprime} + ^{\prime\prime},</math> used in the context of the boolean domain <math>\mathbb{B},</math> denotes addition modulo 2. Interpreted for logic, the plus sign can be used to indicate either the boolean operation of ''[[exclusive disjunction]]'', <math>\operatorname{XOR} : \mathbb{B} \times \mathbb{B} \to \mathbb{B},</math> or the boolean relation of ''logical inequality'', <math>\operatorname{NEQ} \subseteq \mathbb{B} \times \mathbb{B}.</math> The third cartesian power of <math>\mathbb{B}</math> is the set <math>\mathbb{B}^3 = \mathbb{B} \times \mathbb{B} \times \mathbb{B} = \{ (x_1, x_2, x_3) : x_j \in \mathbb{B} ~\text{for}~ j = 1, 2, 3 \}.</math> In what follows, the space <math>X \times Y \times Z</math> is isomorphic to <math>\mathbb{B} \times \mathbb{B} \times \mathbb{B} ~=~ \mathbb{B}^3.</math> The relation <math>L_0\!</math> is defined as follows: : <math>L_0 = \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 0 \}.</math> The relation <math>L_0\!</math> is the set of four triples enumerated here: : <math>L_0 = \{ (0, 0, 0), (0, 1, 1), (1, 0, 1), (1, 1, 0) \}.\!</math> The relation <math>L_1\!</math> is defined as follows: : <math>L_1 = \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 1 \}.</math> The relation <math>L_1\!</math> is the set of four triples enumerated here: : <math>L_1 = \{ (0, 0, 1), (0, 1, 0), (1, 0, 0), (1, 1, 1) \}.\!</math> The triples that make up the relations <math>L_0\!</math> and <math>L_1\!</math> are conveniently arranged in the form of ''[[relational database|relational data tables]]'', as follows: <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:60%" |+ <math>L_0 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 0 \}</math> |- style="background:#e6e6ff" ! <math>X\!</math> !! <math>Y\!</math> !! <math>Z\!</math> |- | <math>0\!</math> || <math>0\!</math> || <math>0\!</math> |- | <math>0\!</math> || <math>1\!</math> || <math>1\!</math> |- | <math>1\!</math> || <math>0\!</math> || <math>1\!</math> |- | <math>1\!</math> || <math>1\!</math> || <math>0\!</math> |} <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:60%" |+ <math>L_1 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 1 \}</math> |- style="background:#e6e6ff" ! <math>X\!</math> !! <math>Y\!</math> !! <math>Z\!</math> |- | <math>0\!</math> || <math>0\!</math> || <math>1\!</math> |- | <math>0\!</math> || <math>1\!</math> || <math>0\!</math> |- | <math>1\!</math> || <math>0\!</math> || <math>0\!</math> |- | <math>1\!</math> || <math>1\!</math> || <math>1\!</math> |} <br> ==Examples from semiotics== The study of signs &mdash; the full variety of significant forms of expression &mdash; in relation to the things that signs are significant ''of'', and in relation to the beings that signs are significant ''to'', is known as ''[[semiotics]]'' or the ''theory of signs''. As just described, semiotics treats of a 3-place relation among ''signs'', their ''objects'', and their ''interpreters''. The term ''[[semiosis]]'' refers to any activity or process that involves signs. Studies of semiosis that deal with its more abstract form are not concerned with every concrete detail of the entities that act as signs, as objects, or as agents of semiosis, but only with the most salient patterns of relationship among these three roles. In particular, the formal theory of signs does not consider all of the properties of the interpretive agent but only the more striking features of the impressions that signs make on a representative interpreter. In its formal aspects, that impact or influence may be treated as just another sign, called the ''interpretant sign'', or the ''interpretant'' for short. Such a 3-adic relation, among objects, signs, and interpretants, is called a ''[[sign relation]]''. For example, consider the aspects of sign use that concern two people &mdash; let us say <math>\operatorname{Ann}</math> and <math>\operatorname{Bob}\!</math> &mdash; in using their own proper names, <math>^{\backprime\backprime} \operatorname{Ann} ^{\prime\prime}</math> and <math>^{\backprime\backprime} \operatorname{Bob} ^{\prime\prime},</math> together with the pronouns, <math>^{\backprime\backprime} \operatorname{I} ^{\prime\prime}</math> and <math>^{\backprime\backprime} \operatorname{you} ^{\prime\prime}.</math> For brevity, these four signs may be abbreviated to the set <math>\{ \, ^{\backprime\backprime} \operatorname{A} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{B} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{i} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{u} ^{\prime\prime} \, \}.</math> The abstract consideration of how <math>\operatorname{A}</math> and <math>\operatorname{B}</math> use this set of signs to refer to themselves and each other leads to the contemplation of a pair of 3-adic relations, the sign relations <math>L_\operatorname{A}</math> and <math>L_\operatorname{B},</math> that reflect the differential use of these signs by <math>\operatorname{A}</math> and <math>\operatorname{B},</math> respectively. Each of the sign relations, <math>L_\operatorname{A}</math> and <math>L_\operatorname{B},</math> consists of eight triples of the form <math>(x, y, z),\!</math> where the ''object'' <math>x\!</math> is an element of the ''object domain'' <math>O = \{ \operatorname{A}, \operatorname{B} \},</math> where the ''sign'' <math>y\!</math> is an element of the ''sign domain'' <math>S\!,</math> where the ''interpretant sign'' <math>z\!</math> is an element of the interpretant domain <math>I,\!</math> and where it happens in this case that <math>S = I = \{ \, ^{\backprime\backprime} \operatorname{A} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{B} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{i} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{u} ^{\prime\prime} \, \}.</math> In general, it is convenient to refer to the union <math>S \cup I</math> as the ''syntactic domain'', but in this case <math>S ~=~ I ~=~ S \cup I.</math> The set-up so far is summarized as follows: {| align="center" cellpadding="8" width="90%" | <math>\begin{array}{ccc} L_\operatorname{A}, L_\operatorname{B} & \subseteq & O \times S \times I \\ \\ O & = & \{ \operatorname{A}, \operatorname{B} \} \\ \\ S & = & \{ \, ^{\backprime\backprime} \operatorname{A} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{B} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{i} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{u} ^{\prime\prime} \, \} \\ \\ I & = & \{ \, ^{\backprime\backprime} \operatorname{A} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{B} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{i} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{u} ^{\prime\prime} \, \} \\ \\ \end{array}</math> |} The relation <math>L_\operatorname{A}</math> is the set of eight triples enumerated here: {| align="center" cellpadding="8" width="90%" | <math>\begin{array}{cccccc} \{ & (\operatorname{A}, \, ^{\backprime\backprime} \operatorname{A} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{A} ^{\prime\prime}), & (\operatorname{A}, \, ^{\backprime\backprime} \operatorname{A} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{i} ^{\prime\prime}), & (\operatorname{A}, \, ^{\backprime\backprime} \operatorname{i} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{A} ^{\prime\prime}), & (\operatorname{A}, \, ^{\backprime\backprime} \operatorname{i} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{i} ^{\prime\prime}), & \\ & (\operatorname{B}, \, ^{\backprime\backprime} \operatorname{B} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{B} ^{\prime\prime}), & (\operatorname{B}, \, ^{\backprime\backprime} \operatorname{B} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{u} ^{\prime\prime}), & (\operatorname{B}, \, ^{\backprime\backprime} \operatorname{u} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{B} ^{\prime\prime}), & (\operatorname{B}, \, ^{\backprime\backprime} \operatorname{u} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{u} ^{\prime\prime}) & \}. \end{array}</math> |} The triples in <math>L_\operatorname{A}</math> represent the way that interpreter <math>\operatorname{A}</math> uses signs. For example, the listing of the triple <math>(\operatorname{B}, \, ^{\backprime\backprime} \operatorname{u} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{B} ^{\prime\prime})</math> in <math>L_\operatorname{A}</math> represents the fact that <math>\operatorname{A}</math> uses <math>^{\backprime\backprime} \operatorname{B} ^{\prime\prime}</math> to mean the same thing that <math>\operatorname{A}</math> uses <math>^{\backprime\backprime} \operatorname{u} ^{\prime\prime}</math> to mean, namely, <math>\operatorname{B}.</math> The relation <math>L_\operatorname{B}</math> is the set of eight triples enumerated here: {| align="center" cellpadding="8" width="90%" | <math>\begin{array}{cccccc} \{ & (\operatorname{A}, \, ^{\backprime\backprime} \operatorname{A} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{A} ^{\prime\prime}), & (\operatorname{A}, \, ^{\backprime\backprime} \operatorname{A} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{u} ^{\prime\prime}), & (\operatorname{A}, \, ^{\backprime\backprime} \operatorname{u} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{A} ^{\prime\prime}), & (\operatorname{A}, \, ^{\backprime\backprime} \operatorname{u} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{u} ^{\prime\prime}), & \\ & (\operatorname{B}, \, ^{\backprime\backprime} \operatorname{B} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{B} ^{\prime\prime}), & (\operatorname{B}, \, ^{\backprime\backprime} \operatorname{B} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{i} ^{\prime\prime}), & (\operatorname{B}, \, ^{\backprime\backprime} \operatorname{i} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{B} ^{\prime\prime}), & (\operatorname{B}, \, ^{\backprime\backprime} \operatorname{i} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{i} ^{\prime\prime}) & \}. \end{array}</math> |} The triples in <math>L_\operatorname{B}</math> represent the way that interpreter <math>\operatorname{B}</math> uses signs. For example, the listing of the triple <math>(\operatorname{B}, \, ^{\backprime\backprime} \operatorname{i} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{B} ^{\prime\prime})</math> in <math>L_\operatorname{B}</math> represents the fact that <math>\operatorname{B}</math> uses <math>^{\backprime\backprime} \operatorname{B} ^{\prime\prime}</math> to mean the same thing that <math>\operatorname{B}</math> uses <math>^{\backprime\backprime} \operatorname{i} ^{\prime\prime}</math> to mean, namely, <math>\operatorname{B}.</math> The triples that make up the relations <math>L_\operatorname{A}</math> and <math>L_\operatorname{B}</math> are conveniently arranged in the form of ''[[relational database|relational data tables]]'', as follows: <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:60%" |+ <math>L_\operatorname{A} ~=~ \operatorname{Sign~Relation~of~Interpreter~A}</math> |- style="background:#e6e6ff" ! style="width:33%" | <math>\operatorname{Object}</math> ! style="width:33%" | <math>\operatorname{Sign}</math> ! style="width:33%" | <math>\operatorname{Interpretant}</math> |- | <math>\operatorname{A}</math> | <math>^{\backprime\backprime} \operatorname{A} ^{\prime\prime}</math> | <math>^{\backprime\backprime} \operatorname{A} ^{\prime\prime}</math> |- | <math>\operatorname{A}</math> | <math>^{\backprime\backprime} \operatorname{A} ^{\prime\prime}</math> | <math>^{\backprime\backprime} \operatorname{i} ^{\prime\prime}</math> |- | <math>\operatorname{A}</math> | <math>^{\backprime\backprime} \operatorname{i} ^{\prime\prime}</math> | <math>^{\backprime\backprime} \operatorname{A} ^{\prime\prime}</math> |- | <math>\operatorname{A}</math> | <math>^{\backprime\backprime} \operatorname{i} ^{\prime\prime}</math> | <math>^{\backprime\backprime} \operatorname{i} ^{\prime\prime}</math> |- | <math>\operatorname{B}</math> | <math>^{\backprime\backprime} \operatorname{B} ^{\prime\prime}</math> | <math>^{\backprime\backprime} \operatorname{B} ^{\prime\prime}</math> |- | <math>\operatorname{B}</math> | <math>^{\backprime\backprime} \operatorname{B} ^{\prime\prime}</math> | <math>^{\backprime\backprime} \operatorname{u} ^{\prime\prime}</math> |- | <math>\operatorname{B}</math> | <math>^{\backprime\backprime} \operatorname{u} ^{\prime\prime}</math> | <math>^{\backprime\backprime} \operatorname{B} ^{\prime\prime}</math> |- | <math>\operatorname{B}</math> | <math>^{\backprime\backprime} \operatorname{u} ^{\prime\prime}</math> | <math>^{\backprime\backprime} \operatorname{u} ^{\prime\prime}</math> |} <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:60%" |+ <math>L_\operatorname{B} ~=~ \operatorname{Sign~Relation~of~Interpreter~B}</math> |- style="background:#e6e6ff" ! style="width:33%" | <math>\operatorname{Object}</math> ! style="width:33%" | <math>\operatorname{Sign}</math> ! style="width:33%" | <math>\operatorname{Interpretant}</math> |- | <math>\operatorname{A}</math> | <math>^{\backprime\backprime} \operatorname{A} ^{\prime\prime}</math> | <math>^{\backprime\backprime} \operatorname{A} ^{\prime\prime}</math> |- | <math>\operatorname{A}</math> | <math>^{\backprime\backprime} \operatorname{A} ^{\prime\prime}</math> | <math>^{\backprime\backprime} \operatorname{u} ^{\prime\prime}</math> |- |<math>\operatorname{A}</math> | <math>^{\backprime\backprime} \operatorname{u} ^{\prime\prime}</math> | <math>^{\backprime\backprime} \operatorname{A} ^{\prime\prime}</math> |- | <math>\operatorname{A}</math> | <math>^{\backprime\backprime} \operatorname{u} ^{\prime\prime}</math> | <math>^{\backprime\backprime} \operatorname{u} ^{\prime\prime}</math> |- | <math>\operatorname{B}</math> | <math>^{\backprime\backprime} \operatorname{B} ^{\prime\prime}</math> | <math>^{\backprime\backprime} \operatorname{B} ^{\prime\prime}</math> |- | <math>\operatorname{B}</math> | <math>^{\backprime\backprime} \operatorname{B} ^{\prime\prime}</math> | <math>^{\backprime\backprime} \operatorname{i} ^{\prime\prime}</math> |- | <math>\operatorname{B}</math> | <math>^{\backprime\backprime} \operatorname{i} ^{\prime\prime}</math> | <math>^{\backprime\backprime} \operatorname{B} ^{\prime\prime}</math> |- | <math>\operatorname{B}</math> | <math>^{\backprime\backprime} \operatorname{i} ^{\prime\prime}</math> | <math>^{\backprime\backprime} \operatorname{i} ^{\prime\prime}</math> |} <br> ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://mywikibiz.com/Triadic_relation Triadic Relation @ MyWikiBiz] * [http://mathweb.org/wiki/Triadic_relation Triadic Relation @ MathWeb Wiki] * [http://netknowledge.org/wiki/Triadic_relation Triadic Relation @ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Triadic_relation Triadic Relation @ OER Commons] {{col-break}} * [http://p2pfoundation.net/Triadic_Relation Triadic Relation @ P2P Foundation] * [http://semanticweb.org/wiki/Triadic_relation Triadic Relation @ SemanticWeb] * [http://ref.subwiki.org/wiki/Triadic_relation Triadic Relation @ Subject Wikis] * [http://beta.wikiversity.org/wiki/Triadic_relation Triadic Relation @ Wikiversity Beta] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;] * [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;] ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Triadic_relation Triadic Relation], [http://mywikibiz.com/ MyWikiBiz] * [http://mathweb.org/wiki/Triadic_relation Triadic Relation], [http://mathweb.org/ MathWeb Wiki] * [http://netknowledge.org/wiki/Triadic_relation Triadic Relation], [http://netknowledge.org/ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Triadic_relation Triadic Relation], [http://wiki.oercommons.org/ OER Commons] * [http://p2pfoundation.net/Triadic_Relation Triadic Relation], [http://p2pfoundation.net/ P2P Foundation] * [http://semanticweb.org/wiki/Triadic_relation Triadic Relation], [http://semanticweb.org/ SemanticWeb] {{col-break}} * [http://planetmath.org/encyclopedia/TriadicRelation.html Triadic Relation], [http://planetmath.org/ PlanetMath] * [http://beta.wikiversity.org/wiki/Triadic_relation Triadic Relation], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://getwiki.net/-Triadic_Relation Triadic Relation], [http://getwiki.net/ GetWiki] * [http://wikinfo.org/index.php/Triadic_relation Triadic Relation], [http://wikinfo.org/ Wikinfo] * [http://textop.org/wiki/index.php?title=Triadic_relation Triadic Relation], [http://textop.org/wiki/ Textop Wiki] * [http://en.wikipedia.org/w/index.php?title=Triadic_relation&oldid=108548758 Triadic Relation], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Artificial Intelligence]] [[Category:Cognitive Sciences]] [[Category:Computer Science]] [[Category:Formal Sciences]] [[Category:Hermeneutics]] [[Category:Information Systems]] [[Category:Information Theory]] [[Category:Intelligence Amplification]] [[Category:Knowledge Representation]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Philosophy]] [[Category:Pragmatics]] [[Category:Semantics]] [[Category:Semiotics]] [[Category:Syntax]] 0c0c7e3aac0b698aca2fafde1677e39592e89949 0 314 535 2010-06-18T20:44:20Z Jon Awbrey 3 + article wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''sign relation''' is the basic construct in the theory of signs, also known as [[semeiotic]] or [[semiotics]], as developed by [[Charles Sanders Peirce]]. ==Anthesis== <blockquote> Thus, if a sunflower, in turning towards the sun, becomes by that very act fully capable, without further condition, of reproducing a sunflower which turns in precisely corresponding ways toward the sun, and of doing so with the same reproductive power, the sunflower would become a Representamen of the sun. (C.S. Peirce, "Syllabus" (''c''.&nbsp;1902), ''Collected Papers'', CP 2.274). </blockquote> In his picturesque illustration of a sign relation, along with his tracing of a corresponding sign process, or ''[[semiosis]]'', Peirce uses the technical term ''representamen'' for his concept of a sign, but the shorter word is precise enough, so long as one recognizes that its meaning in a particular theory of signs is given by a specific definition of what it means to be a sign. ==Definition== One of Peirce's clearest and most complete definitions of a sign is one that he gives, not incidentally, in the context of defining "[[logic]]", and so it is informative to view it in that setting. <blockquote> Logic will here be defined as ''formal semiotic''. A definition of a sign will be given which no more refers to human thought than does the definition of a line as the place which a particle occupies, part by part, during a lapse of time. Namely, a sign is something, ''A'', which brings something, ''B'', its ''interpretant'' sign determined or created by it, into the same sort of correspondence with something, ''C'', its ''object'', as that in which itself stands to ''C''. It is from this definition, together with a definition of "formal", that I deduce mathematically the principles of logic. I also make a historical review of all the definitions and conceptions of logic, and show, not merely that my definition is no novelty, but that my non-psychological conception of logic has ''virtually'' been quite generally held, though not generally recognized. (C.S. Peirce, NEM 4, 20–21). </blockquote> In the general discussion of diverse theories of signs, the question frequently arises whether signhood is an absolute, essential, indelible, or ''[[ontology|ontological]]'' property of a thing, or whether it is a relational, interpretive, and mutable role that a thing can be said to have only within a particular context of relationships. Peirce's definition of a ''sign'' defines it in relation to its ''object'' and its ''interpretant sign'', and thus it defines signhood in ''[[logic of relatives|relative terms]]'', by means of a predicate with three places. In this definition, signhood is a role in a [[triadic relation]], a role that a thing bears or plays in a given context of relationships — it is not as an ''absolute'', ''non-relative'' property of a thing-in-itself, one that it possesses independently of all relationships to other things. Some of the terms that Peirce uses in his definition of a sign may need to be elaborated for the contemporary reader. * '''Correspondence'''. From the way that Peirce uses this term throughout his work, it is clear that he means what he elsewhere calls a "triple correspondence", and thus this is just another way of referring to the whole triadic sign relation itself. In particular, his use of this term should not be taken to imply a dyadic corresponence, like the kinds of "mirror image" correspondence between realities and representations that are bandied about in contemporary controversies about "[[correspondence theories of truth]]". * '''Determination'''. Peirce's concept of determination is broader in several directions than the sense of the word that refers to strictly deterministic causal-temporal processes. First, and especially in this context, he is invoking a more general concept of determination, what is called a ''formal'' or ''informational'' determination, as in saying "two points determine a line", rather than the more special cases of causal and temporal determinisms. Second, he characteristically allows for what is called ''determination in measure'', that is, an order of determinism that admits a full spectrum of more and less determined relationships. * '''Non-psychological'''. Peirce's "non-psychological conception of logic" must be distinguished from any variety of ''anti-psychologism''. He was quite interested in matters of psychology and had much of import to say about them. But logic and psychology operate on different planes of study even when they have occasion to view the same data, as logic is a ''[[normative science]]'' where psychology is a ''[[descriptive science]]'', and so they have very different aims, methods, and rationales. ==Signs and inquiry== : ''Main article : [[Inquiry]]'' There is a close relationship between the pragmatic theory of signs and the pragmatic theory of [[inquiry]]. In fact, the correspondence between the two studies exhibits so many congruences and parallels that it is often best to treat them as integral parts of one and the same subject. In a very real sense, inquiry is the process by which sign relations come to be established and continue to evolve. In other words, inquiry, "thinking" in its best sense, "is a term denoting the various ways in which things acquire significance" ([[John Dewey]]). Thus, there is an active and intricate form of cooperation that needs to be appreciated and maintained between these converging modes of investigation. Its proper character is best understood by realizing that the theory of inquiry is adapted to study the developmental aspects of sign relations, a subject which the theory of signs is specialized to treat from structural and comparative points of view. ==Examples of sign relations== Because the examples to follow have been artificially constructed to be as simple as possible, their detailed elaboration can run the risk of trivializing the whole theory of sign relations. Despite their simplicity, however, these examples have subtleties of their own, and their careful treatment will serve to illustrate many important issues in the general theory of signs. Imagine a discussion between two people, Ann and Bob, and attend only to that aspect of their interpretive practice that involves the use of the following nouns and pronouns: "Ann", "Bob", "I", "you". The ''object domain'' of this discussion fragment is the set of two people {Ann, Bob}. The ''syntactic domain'' or the ''sign system'' that is involved in their discussion is limited to the ''[[set]]'' of four signs {"Ann", "Bob", "I", "You"}. In their discussion, Ann and Bob are not only the passive objects of nominative and accusative references but also the active interpreters of the language that they use. The ''system of interpretation'' (SOI) associated with each language user can be represented in the form of an individual ''[[triadic relation|three-place relation]]'' called the ''sign relation'' of that interpreter. Understood in terms of its ''[[set theory|set-theoretic]] [[extension (semantics)|extension]]'', a sign relation '''L''' is a ''[[subset]]'' of a ''[[cartesian product]]'' '''O''' × '''S''' × '''I'''. Here, '''O''', '''S''', '''I''' are three sets that are known as the ''object domain'', the ''sign domain'', and the ''interpretant domain'', respectively, of the sign relation '''L'''&nbsp;⊆&nbsp;'''O'''&nbsp;×&nbsp;'''S'''&nbsp;×&nbsp;'''I'''. Broadly speaking, the three domains of a sign relation can be any sets whatsoever, but the kinds of sign relations that are typically contemplated in a computational setting are usually constrained to having '''I''' ⊆ '''S'''. In this case, interpretants are just a special variety of signs, and this makes it convenient to lump signs and interpretants together into a single class called the ''syntactic domain''. In the forthcoming examples, '''S''' and '''I''' are identical as sets, so the same elements manifest themselves in two different roles of the sign relations in question. When it is necessary to refer to the whole set of objects and signs in the union of the domains '''O''', '''S''', '''I''' for a given sign relation '''L''', one may refer to this set as the ''world of '''L''''' and write '''W''' = '''W'''_'''L''' = '''O''' ∪ '''S''' ∪ '''I'''. To facilitate an interest in the abstract structures of sign relations, and to keep the notations as brief as possible as the examples become more complicated, it serves to introduce the following general notations: :{| cellpadding="4" | align="center" | '''O''' || = || Object Domain |- | align="center" | '''S''' || = || Sign Domain |- | align="center" | '''I''' || = || Interpretant Domain |} Introducing a few abbreviations for use in considering the present Example, we have the following data: :{| cellpadding="4" | align="center" | '''O''' | = | {Ann, Bob} | = | {A, B} |- | align="center" | '''S''' | = | {"Ann", "Bob", "I", "You"} | = | {"A", "B", "i", "u"} |- | align="center" | '''I''' | = | {"Ann", "Bob", "I", "You"} | = | {"A", "B", "i", "u"} |} In the present Example, '''S''' = '''I''' = Syntactic Domain. The next two Tables give the sign relations associated with the interpreters A and B, respectively, putting them in the form of ''[[relational database]]s''. Thus, the rows of each Table list the ordered triples of the form (''o'', ''s'', ''i'') that make up the corresponding sign relations, '''L'''_A and '''L'''_B ⊆ '''O''' × '''S''' × '''I'''. It is often tempting to use the same names for objects and for relations involving these objects, but it is best to avoid this in a first approach, taking up the issues that this practice raises after the less problematic features of these relations have been treated. {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:60%" |+ '''L'''_A = Sign Relation of Interpreter A |- style="background:#e6e6ff" ! style="width:20%" | Object ! style="width:20%" | Sign ! style="width:20%" | Interpretant |- | '''A''' || '''"A"''' || '''"A"''' |- | '''A''' || '''"A"''' || '''"i"''' |- | '''A''' || '''"i"''' || '''"A"''' |- | '''A''' || '''"i"''' || '''"i"''' |- | '''B''' || '''"B"''' || '''"B"''' |- | '''B''' || '''"B"''' || '''"u"''' |- | '''B''' || '''"u"''' || '''"B"''' |- | '''B''' || '''"u"''' || '''"u"''' |} <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:60%" |+ '''L'''_B = Sign Relation of Interpreter B |- style="background:#e6e6ff" ! style="width:20%" | Object ! style="width:20%" | Sign ! style="width:20%" | Interpretant |- | '''A''' || '''"A"''' || '''"A"''' |- | '''A''' || '''"A"''' || '''"u"''' |- | '''A''' || '''"u"''' || '''"A"''' |- | '''A''' || '''"u"''' || '''"u"''' |- | '''B''' || '''"B"''' || '''"B"''' |- | '''B''' || '''"B"''' || '''"i"''' |- | '''B''' || '''"i"''' || '''"B"''' |- | '''B''' || '''"i"''' || '''"i"''' |} <br> These Tables codify a rudimentary level of interpretive practice for the agents A and B, and provide a basis for formalizing the initial semantics that is appropriate to their common syntactic domain. Each row of a Table names an object and two co-referent signs, making up an ordered triple of the form (''o'', ''s'', ''i'') that is called an ''elementary relation'', that is, one element of the relation's set-theoretic extension. Already in this elementary context, there are several different meanings that might attach to the project of a ''formal semiotics'', or a formal theory of meaning for signs. In the process of discussing these alternatives, it is useful to introduce a few terms that are occasionally used in the philosophy of language to point out the needed distinctions. ==Dyadic aspects of sign relations== For an arbitrary triadic relation '''L''' &sube; '''O''' × '''S''' × '''I''', whether it is a sign relation or not, there are six ''[[binary relation|dyadic relation]]s'' that can be obtained by ''[[projection (set theory)|projecting]]'' '''L''' on one of the planes of the '''OSI'''-space '''O'''&nbsp;×&nbsp;'''S'''&nbsp;×&nbsp;'''I'''. The six dyadic projections of a triadic relation '''L''' are defined and notated as follows: :{| cellpadding="4" | '''L'''_OS | = | ''proj''_OS('''L''') | = | { (''o'', ''s'') &isin; '''O''' × '''S''' : (''o'', ''s'', ''i'') &isin; '''L''' for some ''i'' &isin; '''I''' } |- | '''L'''_SO | = | ''proj''_SO('''L''') | = | { (''s'', ''o'') &isin; '''S''' × '''O''' : (''o'', ''s'', ''i'') &isin; '''L''' for some ''i'' &isin; '''I''' } |- | '''L'''_IS | = | ''proj''_IS('''L''') | = | { (''i'', ''s'') &isin; '''I''' × '''S''' : (''o'', ''s'', ''i'') &isin; '''L''' for some ''o'' &isin; '''O''' } |- | '''L'''_SI | = | ''proj''_SI('''L''') | = | { (''s'', ''i'') &isin; '''S''' × '''I''' : (''o'', ''s'', ''i'') &isin; '''L''' for some ''o'' &isin; '''O''' } |- | '''L'''_OI | = | ''proj''_OI('''L''') | = | { (''o'', ''i'') &isin; '''O''' × '''I''' : (''o'', ''s'', ''i'') &isin; '''L''' for some ''s'' &isin; '''S''' } |- | '''L'''_IO | = | ''proj''_IO('''L''') | = | { (''i'', ''o'') &isin; '''I''' × '''O''' : (''o'', ''s'', ''i'') &isin; '''L''' for some ''s'' &isin; '''S''' } |} By way of unpacking the set-theoretic notation, here is what the first definition says in ordinary language. The dyadic relation that results from the projection of '''L''' on the '''OS'''-plane '''O'''&nbsp;×&nbsp;'''S''' is written briefly as '''L'''_OS or written more fully as ''proj''_OS('''L'''), and it is defined as the set of all ordered pairs (''o'',&nbsp;''s'') in the cartesian product '''O'''&nbsp;×&nbsp;'''S''' for which there exists an ordered triple (''o'',&nbsp;''s'',&nbsp;''i'') in '''L''' for some interpretant ''i'' in the interpretant domain '''I'''. In the case where '''L''' is a sign relation, which it becomes by satisfying one of the definitions of a sign relation, some of the dyadic aspects of '''L''' can be recognized as formalizing aspects of sign meaning that have received their share of attention from students of signs over the centuries, and thus they can be associated with traditional concepts and terminology. Of course, traditions may vary as to the precise formation and usage of such concepts and terms. Other aspects of meaning have not received their fair share of attention, and thus remain anonymous on the contemporary scene of sign studies. ===Denotation=== One aspect of a sign's complete meaning is concerned with the reference that a sign has to its objects, which objects are collectively known as the ''denotation'' of the sign. In the pragmatic theory of sign relations, denotative references fall within the projection of the sign relation on the plane that is spanned by its object domain and its sign domain. The dyadic relation that makes up the ''denotative'', ''referential'', or ''semantic'' aspect or component of a sign relation '''L''' is notated as ''Den''('''L'''). Information about the denotative aspect of meaning is obtained from '''L''' by taking its projection on the object-sign plane, in other words, on the 2-dimensional space that is generated by the object domain '''O''' and the sign domain '''S'''. This semantic component of a sign relation '''L''' is written in any one of the forms, '''L'''_OS, ''proj''_OS'''L''', '''L'''₁₂, ''proj''₁₂'''L''', and it is defined as follows: : ''Den''('''L''') = ''proj''_OS'''L''' = { (''o'', ''s'') &isin; '''O''' × '''S''' : (''o'', ''s'', ''i'') &isin; '''L''' for some ''i'' &isin; '''I''' }. Looking to the denotative aspects of '''L'''_A and '''L'''_B, various rows of the Tables specify, for example, that A uses "i" to denote A and "u" to denote B, whereas B uses "i" to denote B and "u" to denote A. All of these denotative references are summed up in the projections on the '''OS'''-plane, as shown in the following Tables: {| align="center" style="width:90%" | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%" |+ ''proj''_OS('''L'''_A) |- style="background:#e6e6ff" ! style="width:50%" | Object ! style="width:50%" | Sign |- | '''A''' || '''"A"''' |- | '''A''' || '''"i"''' |- | '''B''' || '''"B"''' |- | '''B''' || '''"u"''' |} | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%" |+ ''proj''_OS('''L'''_B) |- style="background:#e6e6ff" ! style="width:50%" | Object ! style="width:50%" | Sign |- | '''A''' || '''"A"''' |- | '''A''' || '''"u"''' |- | '''B''' || '''"B"''' |- | '''B''' || '''"i"''' |} |} <br> ===Connotation=== Another aspect of meaning concerns the connection that a sign has to its interpretants within a given sign relation. As before, this type of connection can be vacuous, singular, or plural in its collection of terminal points, and it can be formalized as the dyadic relation that is obtained as a planar projection of the triadic sign relation in question. The connection that a sign makes to an interpretant is here referred to as its ''connotation''. In the full theory of sign relations, this aspect of meaning includes the links that a sign has to affects, concepts, ideas, impressions, intentions, and the whole realm of an agent's mental states and allied activities, broadly encompassing intellectual associations, emotional impressions, motivational impulses, and real conduct. Taken at the full, in the natural setting of semiotic phenomena, this complex system of references is unlikely ever to find itself mapped in much detail, much less completely formalized, but the tangible warp of its accumulated mass is commonly alluded to as the connotative import of language. Formally speaking, however, the connotative aspect of meaning presents no additional difficulty. For a given sign relation '''L''', the dyadic relation that constitutes the ''connotative aspect'' or ''connotative component'' of '''L''' is notated as ''Con''('''L'''). The connotative aspect of a sign relation '''L''' is given by its projection on the plane of signs and interpretants, and is thus defined as follows: : ''Con''('''L''') = ''proj''_SI'''L''' = { (''s'', ''i'') ∈ '''S''' × '''I''' : (''o'', ''s'', ''i'') ∈ '''L''' for some ''o'' ∈ '''O''' }. All of these connotative references are summed up in the projections on the '''SI'''-plane, as shown in the following Tables: {| align="center" style="width:90%" | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%" |+ ''proj''_SI('''L'''_A) |- style="background:#e6e6ff" ! style="width:50%" | Sign ! style="width:50%" | Interpretant |- | '''"A"''' || '''"A"''' |- | '''"A"''' || '''"i"''' |- | '''"i"''' || '''"A"''' |- | '''"i"''' || '''"i"''' |- | '''"B"''' || '''"B"''' |- | '''"B"''' || '''"u"''' |- | '''"u"''' || '''"B"''' |- | '''"u"''' || '''"u"''' |} | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%" |+ ''proj''_SI('''L'''_B) |- style="background:#e6e6ff" ! style="width:50%" | Sign ! style="width:50%" | Interpretant |- | '''"A"''' || '''"A"''' |- | '''"A"''' || '''"u"''' |- | '''"u"''' || '''"A"''' |- | '''"u"''' || '''"u"''' |- | '''"B"''' || '''"B"''' |- | '''"B"''' || '''"i"''' |- | '''"i"''' || '''"B"''' |- | '''"i"''' || '''"i"''' |} |} <br> ===Ennotation=== The aspect of a sign's meaning that arises from the dyadic relation of its objects to its interpretants has no standard name. If an interpretant is considered to be a sign in its own right, then its independent reference to an object can be taken as belonging to another moment of denotation, but this neglects the mediational character of the whole transaction in which this occurs. Denotation and connotation have to do with dyadic relations in which the sign plays an active role, but here we have to consider a dyadic relation between objects and interpretants that is mediated by the sign from an off-stage position, as it were. As a relation between objects and interpretants that is mediated by a sign, this aspect of meaning may be referred to as the ''ennotation'' of a sign, and the dyadic relation that constitutes the ''ennotative aspect'' of a sign relation '''L''' may be notated as ''Enn''('''L'''). The ennotational component of meaning for a sign relation '''L''' is captured by its projection on the plane of the object and interpretant domains, and it is thus defined as follows: : ''Enn''('''L''') = ''proj''_OI'''L''' = { (''o'', ''i'') ∈ '''O''' × '''I''' : (''o'', ''s'', ''i'') ∈ '''L''' for some ''s'' ∈ '''S''' }. As it happens, the sign relations '''L'''_A and '''L'''_B are fully symmetric with respect to exchanging signs and interpretants, so all of the data of ''proj''_OS'''L'''_A is echoed unchanged in ''proj''_OI'''L'''_A and all of the data of ''proj''_OS'''L'''_B is echoed unchanged in ''proj''_OI'''L'''_B. {| align="center" style="width:90%" | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%" |+ ''proj''_OI('''L'''_A) |- style="background:#e6e6ff" ! style="width:50%" | Object ! style="width:50%" | Interpretant |- | '''A''' || '''"A"''' |- | '''A''' || '''"i"''' |- | '''B''' || '''"B"''' |- | '''B''' || '''"u"''' |} | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%" |+ ''proj''_OI('''L'''_B) |- style="background:#e6e6ff" ! style="width:50%" | Object ! style="width:50%" | Interpretant |- | '''A''' || '''"A"''' |- | '''A''' || '''"u"''' |- | '''B''' || '''"B"''' |- | '''B''' || '''"i"''' |} |} <br> ==Semiotic equivalence relations== (Text in Progress) ==Six ways of looking at a sign relation== In the context of 3-adic relations in general, Peirce provides the following illustration of the six ''converses'' of a 3-adic relation, that is, the six differently ordered ways of stating what is logically the same 3-adic relation: : So in a triadic fact, say, the example <br> {| align="center" cellspacing="8" style="width:72%" | align="center" | ''A'' gives ''B'' to ''C'' |} : we make no distinction in the ordinary logic of relations between the ''[[subject (grammar)|subject]] [[nominative]]'', the ''[[direct object]]'', and the ''[[indirect object]]''. We say that the proposition has three ''logical subjects''. We regard it as a mere affair of English grammar that there are six ways of expressing this: <br> {| align="center" cellspacing="8" style="width:72%" | style="width:36%" | ''A'' gives ''B'' to ''C'' | style="width:36%" | ''A'' benefits ''C'' with ''B'' |- | ''B'' enriches ''C'' at expense of ''A'' | ''C'' receives ''B'' from ''A'' |- | ''C'' thanks ''A'' for ''B'' | ''B'' leaves ''A'' for ''C'' |} : These six sentences express one and the same indivisible phenomenon. (C.S. Peirce, "The Categories Defended", MS 308 (1903), EP 2, 170-171). ===IOS=== (Text in Progress) ===ISO=== (Text in Progress) ===OIS=== <blockquote> Words spoken are symbols or signs (σύμβολα) of affections or impressions (παθήματα) of the soul (ψυχή); written words are the signs of words spoken. As writing, so also is speech not the same for all races of men. But the mental affections themselves, of which these words are primarily signs (σημεια), are the same for the whole of mankind, as are also the objects (πράγματα) of which those affections are representations or likenesses, images, copies (ομοιώματα). ([[Aristotle]], ''[[De Interpretatione]]'', 1.16^a4). </blockquote> ===OSI=== (Text in Progress) ===SIO=== <blockquote> Logic will here be defined as ''formal semiotic''. A definition of a sign will be given which no more refers to human thought than does the definition of a line as the place which a particle occupies, part by part, during a lapse of time. Namely, a sign is something, ''A'', which brings something, ''B'', its ''interpretant'' sign determined or created by it, into the same sort of correspondence with something, ''C'', its ''object'', as that in which itself stands to ''C''. It is from this definition, together with a definition of "formal", that I deduce mathematically the principles of logic. I also make a historical review of all the definitions and conceptions of logic, and show, not merely that my definition is no novelty, but that my non-psychological conception of logic has ''virtually'' been quite generally held, though not generally recognized. (C.S. Peirce, "Application to the [[Carnegie Institution]]", L75 (1902), NEM 4, 20-21). </blockquote> ===SOI=== <blockquote> A ''Sign'' is anything which is related to a Second thing, its ''Object'', in respect to a Quality, in such a way as to bring a Third thing, its ''Interpretant'', into relation to the same Object, and that in such a way as to bring a Fourth into relation to that Object in the same form, ''ad infinitum''. (CP 2.92; quoted in Fisch 1986: 274) </blockquote> ==References== ==Bibliography== ===Primary sources=== * [[Charles Sanders Peirce (Bibliography)]] ===Secondary sources=== * Awbrey, Jon, and Awbrey, Susan (1995), "Interpretation as Action : The Risk of Inquiry", ''Inquiry : Critical Thinking Across the Disciplines'' 15, 40–52. [http://www.chss.montclair.edu/inquiry/fall95/awbrey.html Eprint]. * Deledalle, Gérard (2000), ''C.S. Peirce's Philosophy of Signs'', Indiana University Press, Bloomington, IN. * Eisele, Carolyn (1979), in ''Studies in the Scientific and Mathematical Philosophy of C.S. Peirce'', [[Richard Milton Martin]] (ed.), Mouton, The Hague. * Esposito, Joseph (1980), ''Evolutionary Metaphysics: The Development of Peirce's Theory of Categories'', Ohio University Press (?). * Fisch, Max (1986), ''Peirce, Semeiotic, and Pragmatism'', Indiana University Press, Bloomington, IN. * Houser, N., Roberts, D.D., and Van Evra, J. (eds.)(1997), ''Studies in the Logic of C.S. Peirce'', Indiana University Press, Bloomington, IN. * Liszka, J.J. (1996), ''A General Introduction to the Semeiotic of C.S. Peirce'', Indiana University Press, Bloomington, IN. * Misak, C. (ed.)(2004), ''Cambridge Companion to C.S. Peirce'', Cambridge University Press. * Moore, E., and Robin, R. (1964), ''Studies in the Philosophy of C.S. Peirce, Second Series'', University of Massachusetts Press, Amherst, MA. * Murphey, M. (1961), ''The Development of Peirce's Thought''. Reprinted, Hackett, Indianapolis, IN, 1993. * [[Walker Percy]] (2000), pp. 271-291 in ''Signposts in a Strange Land'', P. Samway (ed.), Saint Martin's Press. ==Resources== * [http://www.helsinki.fi/science/commens/dictionary.html The Commens Dictionary of Peirce's Terms] ** [http://www.helsinki.fi/science/commens/terms/semeiotic.html Entry for ''Semeiotic''] ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://mywikibiz.com/Sign_relation Sign Relation @ MyWikiBiz] * [http://mathweb.org/wiki/Sign_relation Sign Relation @ MathWeb Wiki] * [http://netknowledge.org/wiki/Sign_relation Sign Relation @ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Sign_relation Sign Relation @ OER Commons] {{col-break}} * [http://p2pfoundation.net/Sign_Relation Sign Relation @ P2P Foundation] * [http://semanticweb.org/wiki/Sign_relation Sign Relation @ SemanticWeb] * [http://ref.subwiki.org/wiki/Sign_relation Sign Relation @ Subject Wikis] * [http://beta.wikiversity.org/wiki/Sign_relation Sign Relation @ Wikiversity Beta] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;] * [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;] ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Sign_relation Sign Relation], [http://mywikibiz.com/ MyWikiBiz] * [http://mathweb.org/wiki/Sign_relation Sign Relation], [http://mathweb.org/ MathWeb Wiki] * [http://netknowledge.org/wiki/Sign_relation Sign Relation], [http://netknowledge.org/ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Sign_relation Sign Relation], [http://wiki.oercommons.org/ OER Commons] * [http://p2pfoundation.net/Sign_Relation Sign Relation], [http://p2pfoundation.net/ P2P Foundation] * [http://semanticweb.org/wiki/Sign_relation Sign Relation], [http://semanticweb.org/ SemanticWeb] {{col-break}} * [http://planetmath.org/encyclopedia/SignRelation.html Sign Relation], [http://planetmath.org/ PlanetMath] * [http://beta.wikiversity.org/wiki/Sign_relation Sign Relation], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://getwiki.net/-Sign_Relation Sign Relation], [http://getwiki.net/ GetWiki] * [http://wikinfo.org/index.php/Sign_relation Sign Relation], [http://wikinfo.org/ Wikinfo] * [http://textop.org/wiki/index.php?title=Sign_relation Sign Relation], [http://textop.org/wiki/ Textop Wiki] * [http://en.wikipedia.org/w/index.php?title=Sign_relation&oldid=161631069 Sign Relation], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Artificial Intelligence]] [[Category:Cognitive Sciences]] [[Category:Computer Science]] [[Category:Formal Sciences]] [[Category:Hermeneutics]] [[Category:Information Systems]] [[Category:Information Theory]] [[Category:Intelligence Amplification]] [[Category:Knowledge Representation]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Philosophy]] [[Category:Pragmatics]] [[Category:Semantics]] [[Category:Semiotics]] [[Category:Syntax]] c1e2b0cfb170c9112eb133e1009b49897edc9607 0 315 536 2010-06-20T16:57:16Z Jon Awbrey 3 + article wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. The '''minimal negation operator''' <math>\nu\!</math> is a [[multigrade operator]] <math>(\nu_k)_{k \in \mathbb{N}}</math> where each <math>\nu_k\!</math> is a <math>k\!</math>-ary [[boolean function]] defined in such a way that <math>\nu_k (x_1, \ldots , x_k) = 1</math> in just those cases where exactly one of the arguments <math>x_j\!</math> is <math>0.\!</math> In contexts where the initial letter <math>\nu\!</math> is understood, the minimal negation operators can be indicated by argument lists in parentheses. In the following text a distinctive typeface will be used for logical expressions based on minimal negation operators, for example, <math>\texttt{(x, y, z)}</math> = <math>\nu (x, y, z).\!</math> The first four members of this family of operators are shown below, with paraphrases in a couple of other notations, where tildes and primes, respectively, indicate logical negation. {| align="center" cellpadding="8" style="text-align:center" | <math>\begin{matrix} \texttt{()} & = & \nu_0 & = & 0 & = & \operatorname{false} \\[6pt] \texttt{(x)} & = & \nu_1 (x) & = & \tilde{x} & = & x^\prime \\[6pt] \texttt{(x, y)} & = & \nu_2 (x, y) & = & \tilde{x}y \lor x\tilde{y} & = & x^\prime y \lor x y^\prime \\[6pt] \texttt{(x, y, z)} & = & \nu_3 (x, y, z) & = & \tilde{x}yz \lor x\tilde{y}z \lor xy\tilde{z} & = & x^\prime y z \lor x y^\prime z \lor x y z^\prime \end{matrix}</math> |} ==Definition== To express the general case of <math>\nu_k\!</math> in terms of familiar operations, it helps to introduce an intermediary concept: '''Definition.''' Let the function <math>\lnot_j : \mathbb{B}^k \to \mathbb{B}</math> be defined for each integer <math>j\!</math> in the interval <math>[1, k]\!</math> by the following equation: {| align="center" cellpadding="8" | <math>\lnot_j (x_1, \ldots, x_j, \ldots, x_k) ~=~ x_1 \land \ldots \land x_{j-1} \land \lnot x_j \land x_{j+1} \land \ldots \land x_k.</math> |} Then <math>\nu_k : \mathbb{B}^k \to \mathbb{B}</math> is defined by the following equation: {| align="center" cellpadding="8" | <math>\nu_k (x_1, \ldots, x_k) ~=~ \lnot_1 (x_1, \ldots, x_k) \lor \ldots \lor \lnot_j (x_1, \ldots, x_k) \lor \ldots \lor \lnot_k (x_1, \ldots, x_k).</math> |} If we think of the point <math>x = (x_1, \ldots, x_k) \in \mathbb{B}^k</math> as indicated by the boolean product <math>x_1 \cdot \ldots \cdot x_k</math> or the logical conjunction <math>x_1 \land \ldots \land x_k,</math> then the minimal negation <math>\texttt{(} x_1, \ldots, x_k \texttt{)}</math> indicates the set of points in <math>\mathbb{B}^k</math> that differ from <math>x\!</math> in exactly one coordinate. This makes <math>\texttt{(} x_1, \ldots, x_k \texttt{)}</math> a discrete functional analogue of a ''point omitted neighborhood'' in analysis, more exactly, a ''point omitted distance one neighborhood''. In this light, the minimal negation operator can be recognized as a differential construction, an observation that opens a very wide field. It also serves to explain a variety of other names for the same concept, for example, ''logical boundary operator'', ''limen operator'', ''least action operator'', or ''hedge operator'', to name but a few. The rationale for these names is visible in the venn diagrams of the corresponding operations on sets. The remainder of this discussion proceeds on the ''algebraic boolean convention'' that the plus sign <math>(+)\!</math> and the summation symbol <math>(\textstyle\sum)</math> both refer to addition modulo 2. Unless otherwise noted, the boolean domain <math>\mathbb{B} = \{ 0, 1 \}</math> is interpreted so that <math>0 = \operatorname{false}</math> and <math>1 = \operatorname{true}.</math> This has the following consequences: {| align="center" cellpadding="4" width="90%" | valign="top" | <big>&bull;</big> | The operation <math>x + y\!</math> is a function equivalent to the exclusive disjunction of <math>x\!</math> and <math>y,\!</math> while its fiber of 1 is the relation of inequality between <math>x\!</math> and <math>y.\!</math> |- | valign="top" | <big>&bull;</big> | The operation <math>\textstyle\sum_{j=1}^k x_j</math> maps the bit sequence <math>(x_1, \ldots, x_k)\!</math> to its ''parity''. |} The following properties of the minimal negation operators <math>\nu_k : \mathbb{B}^k \to \mathbb{B}</math> may be noted: {| align="center" cellpadding="4" width="90%" | valign="top" | <big>&bull;</big> | The function <math>\texttt{(x, y)}</math> is the same as that associated with the operation <math>x + y\!</math> and the relation <math>x \ne y.</math> |- | valign="top" | <big>&bull;</big> | In contrast, <math>\texttt{(x, y, z)}</math> is not identical to <math>x + y + z.\!</math> |- | valign="top" | <big>&bull;</big> | More generally, the function <math>\nu_k (x_1, \dots, x_k)</math> for <math>k > 2\!</math> is not identical to the boolean sum <math>\textstyle\sum_{j=1}^k x_j.</math> |- | valign="top" | <big>&bull;</big> | The inclusive disjunctions indicated for the <math>\nu_k\!</math> of more than one argument may be replaced with exclusive disjunctions without affecting the meaning, since the terms disjoined are already disjoint. |} ==Truth tables== Table&nbsp;1 is a [[truth table]] for the sixteen boolean functions of type <math>f : \mathbb{B}^3 \to \mathbb{B}</math> whose fibers of 1 are either the boundaries of points in <math>\mathbb{B}^3</math> or the complements of those boundaries. <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ <math>\text{Table 1.}~~\text{Logical Boundaries and Their Complements}</math> |- style="background:#f0f0ff" | <math>\mathcal{L}_1</math> | <math>\mathcal{L}_2</math> | <math>\mathcal{L}_3</math> | <math>\mathcal{L}_4</math> |- style="background:#f0f0ff" | &nbsp; | align="right" | <math>p\colon\!</math> | <math>1~1~1~1~0~0~0~0</math> | &nbsp; |- style="background:#f0f0ff" | &nbsp; | align="right" | <math>q\colon\!</math> | <math>1~1~0~0~1~1~0~0</math> | &nbsp; |- style="background:#f0f0ff" | &nbsp; | align="right" | <math>r\colon\!</math> | <math>1~0~1~0~1~0~1~0</math> | &nbsp; |- | <math>\begin{matrix} f_{104} \\[4pt] f_{148} \\[4pt] f_{146} \\[4pt] f_{97} \\[4pt] f_{134} \\[4pt] f_{73} \\[4pt] f_{41} \\[4pt] f_{22} \end{matrix}</math> | <math>\begin{matrix} f_{01101000} \\[4pt] f_{10010100} \\[4pt] f_{10010010} \\[4pt] f_{01100001} \\[4pt] f_{10000110} \\[4pt] f_{01001001} \\[4pt] f_{00101001} \\[4pt] f_{00010110} \end{matrix}</math> | <math>\begin{matrix} 0~1~1~0~1~0~0~0 \\[4pt] 1~0~0~1~0~1~0~0 \\[4pt] 1~0~0~1~0~0~1~0 \\[4pt] 0~1~1~0~0~0~0~1 \\[4pt] 1~0~0~0~0~1~1~0 \\[4pt] 0~1~0~0~1~0~0~1 \\[4pt] 0~0~1~0~1~0~0~1 \\[4pt] 0~0~0~1~0~1~1~0 \end{matrix}</math> | <math>\begin{matrix} \texttt{(~p~,~q~,~r~)} \\[4pt] \texttt{(~p~,~q~,(r))} \\[4pt] \texttt{(~p~,(q),~r~)} \\[4pt] \texttt{(~p~,(q),(r))} \\[4pt] \texttt{((p),~q~,~r~)} \\[4pt] \texttt{((p),~q~,(r))} \\[4pt] \texttt{((p),(q),~r~)} \\[4pt] \texttt{((p),(q),(r))} \end{matrix}</math> |- | <math>\begin{matrix} f_{233} \\[4pt] f_{214} \\[4pt] f_{182} \\[4pt] f_{121} \\[4pt] f_{158} \\[4pt] f_{109} \\[4pt] f_{107} \\[4pt] f_{151} \end{matrix}</math> | <math>\begin{matrix} f_{11101001} \\[4pt] f_{11010110} \\[4pt] f_{10110110} \\[4pt] f_{01111001} \\[4pt] f_{10011110} \\[4pt] f_{01101101} \\[4pt] f_{01101011} \\[4pt] f_{10010111} \end{matrix}</math> | <math>\begin{matrix} 1~1~1~0~1~0~0~1 \\[4pt] 1~1~0~1~0~1~1~0 \\[4pt] 1~0~1~1~0~1~1~0 \\[4pt] 0~1~1~1~1~0~0~1 \\[4pt] 1~0~0~1~1~1~1~0 \\[4pt] 0~1~1~0~1~1~0~1 \\[4pt] 0~1~1~0~1~0~1~1 \\[4pt] 1~0~0~1~0~1~1~1 \end{matrix}</math> | <math>\begin{matrix} \texttt{(((p),(q),(r)))} \\[4pt] \texttt{(((p),(q),~r~))} \\[4pt] \texttt{(((p),~q~,(r)))} \\[4pt] \texttt{(((p),~q~,~r~))} \\[4pt] \texttt{((~p~,(q),(r)))} \\[4pt] \texttt{((~p~,(q),~r~))} \\[4pt] \texttt{((~p~,~q~,(r)))} \\[4pt] \texttt{((~p~,~q~,~r~))} \end{matrix}</math> |} <br> ==Charts and graphs== This Section focuses on visual representations of minimal negation operators. A few bits of terminology are useful in describing the pictures, but the formal details are tedious reading, and may be familiar to many readers, so the full definitions of the terms marked in ''italics'' are relegated to a Glossary at the end of the article. Two ways of visualizing the space <math>\mathbb{B}^k</math> of <math>2^k\!</math> points are the [[hypercube]] picture and the [[venn diagram]] picture. The hypercube picture associates each point of <math>\mathbb{B}^k</math> with a unique point of the <math>k\!</math>-dimensional hypercube. The venn diagram picture associates each point of <math>\mathbb{B}^k</math> with a unique "cell" of the venn diagram on <math>k\!</math> "circles". In addition, each point of <math>\mathbb{B}^k</math> is the unique point in the ''[[fiber (mathematics)|fiber]] of truth'' <math>[|s|]\!</math> of a ''singular proposition'' <math>s : \mathbb{B}^k \to \mathbb{B},</math> and thus it is the unique point where a ''singular conjunction'' of <math>k\!</math> ''literals'' is <math>1.\!</math> For example, consider two cases at opposite vertices of the cube: {| align="center" cellpadding="4" width="90%" | valign="top" | <big>&bull;</big> | The point <math>(1, 1, \ldots , 1, 1)</math> with all 1's as coordinates is the point where the conjunction of all posited variables evaluates to <math>1,\!</math> namely, the point where: |- | &nbsp; | align="center" | <math>x_1 ~ x_2 ~\ldots~ x_{n-1} ~ x_n ~=~ 1.</math> |- | valign="top" | <big>&bull;</big> | The point <math>(0, 0, \ldots , 0, 0)</math> with all 0's as coordinates is the point where the conjunction of all negated variables evaluates to <math>1,\!</math> namely, the point where: |- | &nbsp; | align="center" | <math>\texttt{(} x_1 \texttt{)(} x_2 \texttt{)} \ldots \texttt{(} x_{n-1} \texttt{)(} x_n \texttt{)} ~=~ 1.</math> |} To pass from these limiting examples to the general case, observe that a singular proposition <math>s : \mathbb{B}^k \to \mathbb{B}</math> can be given canonical expression as a conjunction of literals, <math>s = e_1 e_2 \ldots e_{k-1} e_k</math>. Then the proposition <math>\nu (e_1, e_2, \ldots, e_{k-1}, e_k)</math> is <math>1\!</math> on the points adjacent to the point where <math>s\!</math> is <math>1,\!</math> and 0 everywhere else on the cube. For example, consider the case where <math>k = 3.\!</math> Then the minimal negation operation <math>\nu (p, q, r)\!</math> &mdash; written more simply as <math>\texttt{(p, q, r)}</math> &mdash; has the following venn diagram: {| align="center" cellpadding="8" style="text-align:center" | <p>[[Image:Venn Diagram (P,Q,R).jpg|500px]]</p> <p><math>\text{Figure 2.}~~\texttt{(p, q, r)}</math></p> |} For a contrasting example, the boolean function expressed by the form <math>\texttt{((p),(q),(r))}</math> has the following venn diagram: {| align="center" cellpadding="8" style="text-align:center" | <p>[[Image:Venn Diagram ((P),(Q),(R)).jpg|500px]]</p> <p><math>\text{Figure 3.}~~\texttt{((p),(q),(r))}</math></p> |} ==Glossary of basic terms== ; Boolean domain : A ''[[boolean domain]]'' <math>\mathbb{B}</math> is a generic 2-element set, for example, <math>\mathbb{B} = \{ 0, 1 \},</math> whose elements are interpreted as logical values, usually but not invariably with <math>0 = \operatorname{false}</math> and <math>1 = \operatorname{true}.</math> ; Boolean variable : A ''[[boolean variable]]'' <math>x\!</math> is a variable that takes its value from a boolean domain, as <math>x \in \mathbb{B}.</math> ; Proposition : In situations where boolean values are interpreted as logical values, a [[boolean-valued function]] <math>f : X \to \mathbb{B}</math> or a [[boolean function]] <math>g : \mathbb{B}^k \to \mathbb{B}</math> is frequently called a ''[[proposition]]''. ; Basis element, Coordinate projection : Given a sequence of <math>k\!</math> boolean variables, <math>x_1, \ldots, x_k,</math> each variable <math>x_j\!</math> may be treated either as a ''basis element'' of the space <math>\mathbb{B}^k</math> or as a ''coordinate projection'' <math>x_j : \mathbb{B}^k \to \mathbb{B}.</math> ; Basic proposition : This means that the set of objects <math>\{ x_j : 1 \le j \le k \}</math> is a set of boolean functions <math>\{ x_j : \mathbb{B}^k \to \mathbb{B} \}</math> subject to logical interpretation as a set of ''basic propositions'' that collectively generate the complete set of <math>2^{2^k}</math> propositions over <math>\mathbb{B}^k.</math> ; Literal : A ''literal'' is one of the <math>2k\!</math> propositions <math>x_1, \ldots, x_k, \texttt{(} x_1 \texttt{)}, \ldots, \texttt{(} x_k \texttt{)},</math> in other words, either a ''posited'' basic proposition <math>x_j\!</math> or a ''negated'' basic proposition <math>\texttt{(} x_j \texttt{)},</math> for some <math>j = 1 ~\text{to}~ k.</math> ; Fiber : In mathematics generally, the ''[[fiber (mathematics)|fiber]]'' of a point <math>y \in Y</math> under a function <math>f : X \to Y</math> is defined as the inverse image <math>f^{-1}(y) \subseteq X.</math> : In the case of a boolean function <math>f : \mathbb{B}^k \to \mathbb{B},</math> there are just two fibers: : The fiber of <math>0\!</math> under <math>f,\!</math> defined as <math>f^{-1}(0),\!</math> is the set of points where the value of <math>f\!</math> is <math>0.\!</math> : The fiber of <math>1\!</math> under <math>f,\!</math> defined as <math>f^{-1}(1),\!</math> is the set of points where the value of <math>f\!</math> is <math>1.\!</math> ; Fiber of truth : When <math>1\!</math> is interpreted as the logical value <math>\operatorname{true},</math> then <math>f^{-1}(1)\!</math> is called the ''fiber of truth'' in the proposition <math>f.\!</math> Frequent mention of this fiber makes it useful to have a shorter way of referring to it. This leads to the definition of the notation <math>[|f|] = f^{-1}(1)\!</math> for the fiber of truth in the proposition <math>f.\!</math> ; Singular boolean function : A ''singular boolean function'' <math>s : \mathbb{B}^k \to \mathbb{B}</math> is a boolean function whose fiber of <math>1\!</math> is a single point of <math>\mathbb{B}^k.</math> ; Singular proposition : In the interpretation where <math>1\!</math> equals <math>\operatorname{true},</math> a singular boolean function is called a ''singular proposition''. : Singular boolean functions and singular propositions serve as functional or logical representatives of the points in <math>\mathbb{B}^k.</math> ; Singular conjunction : A ''singular conjunction'' in <math>\mathbb{B}^k \to \mathbb{B}</math> is a conjunction of <math>k\!</math> literals that includes just one conjunct of the pair <math>\{ x_j, ~\nu(x_j) \}</math> for each <math>j = 1 ~\text{to}~ k.</math> : A singular proposition <math>s : \mathbb{B}^k \to \mathbb{B}</math> can be expressed as a singular conjunction: {| align="center" cellspacing"10" width="90%" | height="36" | <math>s ~=~ e_1 e_2 \ldots e_{k-1} e_k</math>, |- | <math>\begin{array}{llll} \text{where} & e_j & = & x_j \\[6pt] \text{or} & e_j & = & \nu (x_j), \\[6pt] \text{for} & j & = & 1 ~\text{to}~ k. \end{array}</math> |} ==Resources== * [http://planetmath.org/encyclopedia/MinimalNegationOperator.html Minimal Negation Operator] @ [http://planetmath.org/ PlanetMath] * [http://planetphysics.org/encyclopedia/MinimalNegationOperator.html Minimal Negation Operator] @ [http://planetphysics.org/ PlanetPhysics] * [http://www.proofwiki.org/wiki/Definition:Minimal_Negation_Operator Minimal Negation Operator] @ [http://www.proofwiki.org/ ProofWiki] * [http://atlas.wolfram.com/ Wolfram Atlas of Simple Programs] ** [http://atlas.wolfram.com/01/01/ Elementary Cellular Automata Rules (ECARs)] ** [http://atlas.wolfram.com/01/01/rulelist.html ECAR Index] ** [http://atlas.wolfram.com/01/01/views/3/TableView.html ECAR Icons] ** [http://atlas.wolfram.com/01/01/views/87/TableView.html ECAR Examples] ** [http://atlas.wolfram.com/01/01/views/172/TableView.html ECAR Formulas] ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://mywikibiz.com/Minimal_negation_operator Minimal Negation Operator @ MyWikiBiz] * [http://mathweb.org/wiki/Minimal_negation_operator Minimal Negation Operator @ MathWeb Wiki] * [http://netknowledge.org/wiki/Minimal_negation_operator Minimal Negation Operator @ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Minimal_negation_operator Minimal Negation Operator @ OER Commons] {{col-break}} * [http://p2pfoundation.net/Minimal_Negation_Operator Minimal Negation Operator @ P2P Foundation] * [http://semanticweb.org/wiki/Minimal_negation_operator Minimal Negation Operator @ SemanticWeb] * [http://ref.subwiki.org/wiki/Minimal_negation_operator Minimal Negation Operator @ Subject Wikis] * [http://beta.wikiversity.org/wiki/Minimal_negation_operator Minimal Negation Operator @ Wikiversity Beta] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;] * [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;] ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Minimal_negation_operator Minimal Negation Operator], [http://mywikibiz.com/ MyWikiBiz] * [http://planetmath.org/encyclopedia/MinimalNegationOperator.html Minimal Negation Operator], [http://planetmath.org/ PlanetMath] * [http://proofwiki.org/wiki/Definition:Minimal_Negation_Operator Minimal Negation Operator], [http://proofwiki.org/ ProofWiki] * [http://beta.wikiversity.org/wiki/Minimal_negation_operator Minimal Negation Operator], [http://beta.wikiversity.org/ Wikiversity Beta] {{col-break}} * [http://getwiki.net/-Minimal_Negation_Operator Minimal Negation Operator], [http://getwiki.net/ GetWiki] * [http://wikinfo.org/index.php/Minimal_negation_operator Minimal Negation Operator], [http://wikinfo.org/ Wikinfo] * [http://textop.org/wiki/index.php?title=Minimal_negation_operator Minimal Negation Operator], [http://textop.org/wiki/ Textop Wiki] * [http://en.wikipedia.org/w/index.php?title=Minimal_negation_operator&oldid=75156728 Minimal Negation Operator], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Automata Theory]] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Differential Logic]] [[Category:Equational Reasoning]] [[Category:Formal Languages]] [[Category:Formal Sciences]] [[Category:Formal Systems]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Neural Networks]] [[Category:Philosophy]] [[Category:Semiotics]] b16e11a398788fb346cc3e73c6b17e96d28aa623 0 316 537 2010-06-20T17:22:18Z Jon Awbrey 3 + article wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. In [[logic]] and [[mathematics]], a '''multigrade operator''' <math>\Omega</math> is a ''[[parametric operator]]'' with ''parameter'' ''k'' in the set '''N''' of non-negative integers. The application of a multigrade operator <math>\Omega</math> to a finite sequence of operands (''x''₁,&nbsp;…,&nbsp;''x''_''k'') is typically denoted with the parameter ''k'' left tacit, as the appropriate application is implicit in the number of operands listed. Thus <math>\Omega</math>(''x''₁,&nbsp;…,&nbsp;''x''_''k'') may be taken for <math>\Omega</math>_''k''(''x''₁,&nbsp;…,&nbsp;''x''_''k''). ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://mywikibiz.com/Multigrade_operator Multigrade Operator @ MyWikiBiz] * [http://mathweb.org/wiki/Multigrade_operator Multigrade Operator @ MathWeb Wiki] * [http://netknowledge.org/wiki/Multigrade_operator Multigrade Operator @ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Multigrade_operator Multigrade Operator @ OER Commons] {{col-break}} * [http://p2pfoundation.net/Multigrade_Operator Multigrade Operator @ P2P Foundation] * [http://semanticweb.org/wiki/Multigrade_operator Multigrade Operator @ SemanticWeb] * [http://ref.subwiki.org/wiki/Multigrade_operator Multigrade Operator @ Subject Wikis] * [http://beta.wikiversity.org/wiki/Multigrade_operator Multigrade Operator @ Wikiversity Beta] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;] * [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;] ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Multigrade_operator Multigrade Operator], [http://mywikibiz.com/ MyWikiBiz] * [http://mathweb.org/wiki/Multigrade_operator Multigrade Operator], [http://mathweb.org/wiki/ MathWeb Wiki] * [http://planetmath.org/encyclopedia/MultigradeOperator.html Multigrade Operator], [http://planetmath.org/ PlanetMath] * [http://planetphysics.org/encyclopedia/MultigradeOperator.html Multigrade Operator], [http://planetphysics.org/ PlanetPhysics] * [http://proofwiki.org/wiki/Definition:Multigrade_Operator Multigrade Operator], [http://proofwiki.org/ ProofWiki] {{col-break}} * [http://beta.wikiversity.org/wiki/Multigrade_operator Multigrade Operator], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://getwiki.net/-Parametric_Operator&r=multigrade_operator Multigrade Operator], [http://getwiki.net/ GetWiki] * [http://wikinfo.org/index.php/Multigrade_operator Multigrade Operator], [http://wikinfo.org/ Wikinfo] * [http://textop.org/wiki/index.php?title=Multigrade_operator Multigrade Operator], [http://textop.org/wiki/ Textop Wiki] * [http://en.wikipedia.org/w/index.php?title=Multigrade_operator&oldid=40451309 Multigrade Operator], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Automata Theory]] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Differential Logic]] [[Category:Equational Reasoning]] [[Category:Formal Languages]] [[Category:Formal Sciences]] [[Category:Formal Systems]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Neural Networks]] [[Category:Philosophy]] [[Category:Semiotics]] a1e19344e9131d61edf5bc32c34c01d4b616f5dd 0 317 538 2010-06-20T17:44:24Z Jon Awbrey 3 + article wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. In [[logic]] and [[mathematics]], a '''parametric operator''' <math>\Omega\!</math> with '''parameter''' <math>\alpha\!</math> in the '''parameter set''' <math>\Alpha\!</math> is an [[indexed family]] of operators <math>(\Omega_{\alpha})_{\Alpha} = \{\Omega_{\alpha} : \alpha \in \Alpha\}</math> with index <math>\alpha\!</math> in the index set <math>\Alpha\!</math>. ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://mywikibiz.com/Parametric_operator Parametric Operator @ MyWikiBiz] * [http://mathweb.org/wiki/Parametric_operator Parametric Operator @ MathWeb Wiki] * [http://netknowledge.org/wiki/Parametric_operator Parametric Operator @ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Parametric_operator Parametric Operator @ OER Commons] {{col-break}} * [http://p2pfoundation.net/Parametric_Operator Parametric Operator @ P2P Foundation] * [http://semanticweb.org/wiki/Parametric_operator Parametric Operator @ SemanticWeb] * [http://ref.subwiki.org/wiki/Parametric_operator Parametric Operator @ Subject Wikis] * [http://beta.wikiversity.org/wiki/Parametric_operator Parametric Operator @ Wikiversity Beta] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;] * [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;] ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Parametric_operator Parametric Operator], [http://mywikibiz.com/ MyWikiBiz] * [http://mathweb.org/wiki/Parametric_operator Parametric Operator], [http://mathweb.org/wiki/ MathWeb Wiki] * [http://planetmath.org/encyclopedia/ParametricOperator.html Parametric Operator], [http://planetmath.org/ PlanetMath] * [http://planetphysics.org/encyclopedia/ParametricOperator.html Parametric Operator], [http://planetphysics.org/ PlanetPhysics] * [http://proofwiki.org/wiki/Definition:Parametric_Operator Parametric Operator], [http://proofwiki.org/ ProofWiki] {{col-break}} * [http://beta.wikiversity.org/wiki/Parametric_operator Parametric Operator], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://getwiki.net/-Parametric_Operator Parametric Operator], [http://getwiki.net/ GetWiki] * [http://wikinfo.org/index.php/Parametric_operator Parametric Operator], [http://wikinfo.org/ Wikinfo] * [http://textop.org/wiki/index.php?title=Parametric_operator Parametric Operator], [http://textop.org/wiki/ Textop Wiki] * [http://en.wikipedia.org/w/index.php?title=Parametric_operator&oldid=40451935 Parametric Operator], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Automata Theory]] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Differential Logic]] [[Category:Equational Reasoning]] [[Category:Formal Languages]] [[Category:Formal Sciences]] [[Category:Formal Systems]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Neural Networks]] [[Category:Philosophy]] [[Category:Semiotics]] 9101d21d24457ef1d8fd1013d318f8c6b4a8e01d 6 318 539 2010-06-20T18:02:57Z Jon Awbrey 3 wikitext text/x-wiki da39a3ee5e6b4b0d3255bfef95601890afd80709 6 319 540 2010-06-20T18:04:13Z Jon Awbrey 3 wikitext text/x-wiki da39a3ee5e6b4b0d3255bfef95601890afd80709 0 320 541 2010-06-20T23:08:15Z Jon Awbrey 3 + article wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. '''Ampheck''', from [[Ancient Greek|Greek]] &#945;&#956;&#966;&#942;&#954;&#951;&#962; double-edged, is a term coined by [[Charles Sanders Peirce]] for either one of the pair of logically dual operators, variously referred to as [[Peirce arrow]]s, [[Sheffer stroke]]s, or [[logical NAND|NAND]] and [[logical NNOR|NNOR]]. Either of these logical operators is a ''[[sole sufficient operator]]'' for deriving or generating all of the other operators in the subject matter variously described as [[boolean function]]s, [[monadic predicate calculus]], [[propositional logic]], sentential calculus, or [[zeroth order logic]]. <blockquote> <p>For example, <math>x \curlywedge y</math> signifies that <math>x\!</math> is <math>\mathbf{f}</math> and <math>y\!</math> is <math>\mathbf{f}</math>. Then <math>(x \curlywedge y) \curlywedge z</math>, or <math>\underline {x \curlywedge y} \curlywedge z</math>, will signify that <math>z\!</math> is <math>\mathbf{f}</math>, but that the statement that <math>x\!</math> and <math>y\!</math> are both <math>\mathbf{f}</math> is itself <math>\mathbf{f}</math>, that is, is ''false''. Hence, the value of <math>x \curlywedge x</math> is the same as that of <math>\overline {x}</math>; and the value of <math>\underline {x \curlywedge x} \curlywedge x</math> is <math>\mathbf{f}</math>, because it is necessarily false; while the value of <math>\underline {x \curlywedge y} \curlywedge \underline {x \curlywedge y}</math> is only <math>\mathbf{f}</math> in case <math>x \curlywedge y</math> is <math>\mathbf{v}</math>; and <math>( \underline {x \curlywedge x} \curlywedge x) \curlywedge (x \curlywedge \underline {x \curlywedge x})</math> is necessarily true, so that its value is <math>\mathbf{v}</math>.</p> <p>With these two signs, the [[vinculum]] (with its equivalents, parentheses, brackets, braces, etc.) and the sign <math>\curlywedge</math>, which I will call the ''ampheck'' (from &#945;&#956;&#966;&#951;&#954;&#942;&#962;&nbsp;, cutting both ways), all assertions as to the values of quantities can be expressed. (C.S. Peirce, CP 4.264).</p> </blockquote> In the above passage, Peirce introduces the term ''ampheck'' for the 2-place logical connective or the binary logical operator that is currently called the ''[[joint denial]]'' in logic, the NNOR operator in computer science, or indicated by means of phrases like "neither-nor" or "both not" in ordinary language. For this operation he employs a symbol that the typographer most likely set by inverting the [[zodiac]] symbol for [[Aries]], but set in the text above by means of the ''curly wedge'' symbol. In the same paper, Peirce introduces a symbol for the logically dual operator. This was rendered by the editors of his ''Collected Papers'' as an inverted Aries symbol with a bar or a serif at the top, in this way denoting the connective or logical operator that is currently called the ''[[alternative denial]]'' in logic, the NAND operator in computer science, or invoked by means of phrases like "not-and" or "not both" in ordinary language. It is not clear whether it was Peirce himself or later writers who initiated the practice, but on account of their dual relationship it became common to refer to these two operators in the plural, as the ''amphecks''. ==References and further reading== * [[Glenn Clark|Clark, Glenn]] (1997), "New Light on Peirce's Iconic Notation for the Sixteen Binary Connectives", pp. 304–333 in Houser, Roberts, Van Evra (eds.), ''Studies in the Logic of Charles Sanders Peirce'', Indiana University Press, Bloomington, IN, 1997. * [[Nathan Houser|Houser, N.]], [[Don D. Roberts|Roberts, Don D.]], and [[James Van Evra|Van Evra, James]] (eds., 1997), ''Studies in the Logic of Charles Sanders Peirce'', Indiana University Press, Bloomington, IN. * [[Warren Sturgis McCulloch|McCulloch, W.S.]] (1961), "What Is a Number, that a Man May Know It, and a Man, that He May Know a Number?" (Ninth [[Alfred Korzybski]] Memorial Lecture), ''General Semantics Bulletin'', Nos. 26 & 27, 7–18, Institute of General Semantics, Lakeville, CT, 1961. Reprinted, pp. 1–18 in ''Embodiments of Mind''. * McCulloch, W.S. (1965), ''Embodiments of Mind'', MIT Press, Cambridge, MA. * [[Charles Sanders Peirce (Bibliography)|Peirce, C.S., Bibliography]]. * Peirce, C.S., ''Collected Papers of Charles Sanders Peirce'', vols. 1–6, [[Charles Hartshorne]] and [[Paul Weiss]] (eds.), vols. 7–8, [[Arthur W. Burks]] (ed.), Harvard University Press, Cambridge, MA, 1931–1935, 1958. * Peirce, C.S. (1902), "The Simplest Mathematics". First published as CP 4.227–323 in ''Collected Papers''. * [[Shea Zellweger|Zellweger, Shea]] (1997), "Untapped Potential in Peirce's Iconic Notation for the Sixteen Binary Connectives", pp. 334–386 in Houser, Roberts, Van Evra (eds.), ''Studies in the Logic of Charles Sanders Peirce'', Indiana University Press, Bloomington, IN, 1997. ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://mywikibiz.com/Ampheck Ampheck @ MyWikiBiz] * [http://mathweb.org/wiki/Ampheck Ampheck @ MathWeb Wiki] * [http://netknowledge.org/wiki/Ampheck Ampheck @ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Ampheck Ampheck @ OER Commons] {{col-break}} * [http://p2pfoundation.net/Ampheck Ampheck @ P2P Foundation] * [http://semanticweb.org/wiki/Ampheck Ampheck @ SemanticWeb] * [http://ref.subwiki.org/wiki/Ampheck Ampheck @ Subject Wikis] * [http://beta.wikiversity.org/wiki/Ampheck Ampheck @ Wikiversity Beta] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;] * [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;] ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Ampheck Ampheck], [http://mywikibiz.com/ MyWikiBiz] * [http://mathweb.org/wiki/Ampheck Ampheck], [http://mathweb.org/wiki/ MathWeb Wiki] * [http://netknowledge.org/wiki/Ampheck Ampheck], [http://netknowledge.org/ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Ampheck Ampheck], [http://wiki.oercommons.org/ OER Commons] {{col-break}} * [http://p2pfoundation.net/Ampheck Ampheck], [http://p2pfoundation.net/ P2P Foundation] * [http://semanticweb.org/wiki/Ampheck Ampheck], [http://semanticweb.org/ SemanticWeb] * [http://planetmath.org/encyclopedia/Ampheck.html Ampheck], [http://planetmath.org/ PlanetMath] * [http://beta.wikiversity.org/wiki/Ampheck Ampheck], [http://beta.wikiversity.org/ Wikiversity Beta] {{col-break}} * [http://getwiki.net/-Ampheck Ampheck], [http://getwiki.net/ GetWiki] * [http://wikinfo.org/index.php/Ampheck Ampheck], [http://wikinfo.org/ Wikinfo] * [http://textop.org/wiki/index.php?title=Ampheck Ampheck], [http://textop.org/wiki/ Textop Wiki] * [http://en.wikipedia.org/w/index.php?title=Ampheck&oldid=62218032 Ampheck], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Automata Theory]] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Discrete Mathematics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Neural Networks]] [[Category:Semiotics]] 4a02ca58dd72d46cffcec784b3c5fc4aa89f891d 0 321 542 2010-06-20T23:23:53Z Jon Awbrey 3 + article wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''boolean domain''' <math>\mathbb{B}</math> is a generic 2-element [[set]], say, <math>\mathbb{B} = \{ 0, 1 \},</math> whose elements are interpreted as [[logical value]]s, typically, <math>0 = \operatorname{false}</math> and <math>1 = \operatorname{true}.</math> A '''boolean variable''' <math>x\!</math> is a [[variable]] that takes its value from a boolean domain, as <math>x \in \mathbb{B}.</math> ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://mywikibiz.com/Boolean_domain Boolean Domain @ MyWikiBiz] * [http://mathweb.org/wiki/Boolean_domain Boolean Domain @ MathWeb Wiki] * [http://netknowledge.org/wiki/Boolean_domain Boolean Domain @ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Boolean_domain Boolean Domain @ OER Commons] {{col-break}} * [http://p2pfoundation.net/Boolean_Domain Boolean Domain @ P2P Foundation] * [http://semanticweb.org/wiki/Boolean_domain Boolean Domain @ SemanticWeb] * [http://ref.subwiki.org/wiki/Boolean_domain Boolean Domain @ Subject Wikis] * [http://beta.wikiversity.org/wiki/Boolean_domain Boolean Domain @ Wikiversity Beta] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;] * [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;] ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Boolean_domain Boolean Domain], [http://mywikibiz.com/ MyWikiBiz] * [http://mathweb.org/wiki/Boolean_domain Boolean Domain], [http://mathweb.org/ MathWeb Wiki] * [http://planetmath.org/encyclopedia/BooleanDomain.html Boolean Domain], [http://planetmath.org/ PlanetMath] * [http://planetphysics.org/encyclopedia/BooleanDomain.html Boolean Domain], [http://planetphysics.org/ PlanetPhysics] {{col-break}} * [http://beta.wikiversity.org/wiki/Boolean_domain Boolean Domain], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://wikinfo.org/index.php/Boolean_domain Boolean Domain], [http://wikinfo.org/ Wikinfo] * [http://textop.org/wiki/index.php?title=Boolean_domain Boolean Domain], [http://textop.org/wiki/ Textop Wiki] * [http://en.wikipedia.org/w/index.php?title=Boolean_domain&oldid=71168300 Boolean Domain], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Discrete Mathematics]] [[Category:Logic]] [[Category:Mathematics]] c498f9b659c9e557f77785442cc9caab73f14346 0 322 543 2010-06-21T00:06:17Z Jon Awbrey 3 + article wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''finitary boolean function''' is a [[function (mathematics)|function]] of the form <math>f : \mathbb{B}^k \to \mathbb{B},</math> where <math>\mathbb{B} = \{ 0, 1 \}</math> is a [[boolean domain]] and where <math>k\!</math> is a nonnegative integer. In the case where <math>k = 0,\!</math> the function is simply a constant element of <math>\mathbb{B}.</math> There are <math>2^{2^k}</math> such functions. These play a basic role in questions of [[complexity theory]] as well as the design of circuits and chips for digital computers. ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://mywikibiz.com/Boolean_function Boolean Function @ MyWikiBiz] * [http://mathweb.org/wiki/Boolean_function Boolean Function @ MathWeb Wiki] * [http://netknowledge.org/wiki/Boolean_function Boolean Function @ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Boolean_function Boolean Function @ OER Commons] {{col-break}} * [http://p2pfoundation.net/Boolean_Function Boolean Function @ P2P Foundation] * [http://semanticweb.org/wiki/Boolean_function Boolean Function @ SemanticWeb] * [http://ref.subwiki.org/wiki/Boolean_function Boolean Function @ Subject Wikis] * [http://beta.wikiversity.org/wiki/Boolean_function Boolean Function @ Wikiversity Beta] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;] * [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;] ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Boolean_function Boolean Function], [http://mywikibiz.com/ MyWikiBiz] * [http://mathweb.org/wiki/Boolean_function Boolean Function], [http://mathweb.org/ MathWeb Wiki] * [http://planetmath.org/encyclopedia/BooleanFunction.html Boolean Function], [http://planetmath.org/ PlanetMath] * [http://planetphysics.org/encyclopedia/BooleanFunction.html Boolean Function], [http://planetphysics.org/ PlanetPhysics] {{col-break}} * [http://beta.wikiversity.org/wiki/Boolean_function Boolean Function], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://wikinfo.org/index.php/Boolean_function Boolean Function], [http://wikinfo.org/ Wikinfo] * [http://textop.org/wiki/index.php?title=Boolean_function Boolean Function], [http://textop.org/wiki/ Textop Wiki] * [http://en.wikipedia.org/w/index.php?title=Boolean_function&oldid=60886833 Boolean Function], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Discrete Mathematics]] [[Category:Logic]] [[Category:Mathematics]] bf873e7643823393f495dee9e0f1020e7e57ea8e 0 323 544 2010-06-21T03:25:37Z Jon Awbrey 3 + article wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''boolean-valued function''' is a [[function (mathematics)|function]] of the type <math>f : X \to \mathbb{B},</math> where <math>X\!</math> is an arbitrary [[set]] and where <math>\mathbb{B}</math> is a [[boolean domain]]. In the [[formal science]]s &mdash; [[mathematics]], [[mathematical logic]], [[statistics]] &mdash; and their applied disciplines, a boolean-valued function may also be referred to as a [[characteristic function]], [[indicator function]], [[predicate]], or [[proposition]]. In all of these uses it is understood that the various terms refer to a mathematical object and not the corresponding [[semiotic]] sign or syntactic expression. In [[semantics|formal semantic]] theories of [[truth]], a '''truth predicate''' is a predicate on the [[sentence]]s of a [[formal language]], interpreted for logic, that formalizes the intuitive concept that is normally expressed by saying that a sentence is true. A truth predicate may have additional domains beyond the formal language domain, if that is what is required to determine a final truth value. ==Examples== A '''binary sequence''' is a boolean-valued function <math>f : \mathbb{N}^+ \to \mathbb{B}</math>, where <math>\mathbb{N}^+ = \{ 1, 2, 3, \ldots \},</math>. In other words, <math>f\!</math> is an infinite [[sequence]] of 0's and 1's. A '''binary sequence''' of '''length''' <math>k\!</math> is a boolean-valued function <math>f : [k] \to \mathbb{B}</math>, where <math>[k] = \{ 1, 2, \ldots k \}.</math> ==References== * [[Frank Markham Brown|Brown, Frank Markham]] (2003), ''Boolean Reasoning: The Logic of Boolean Equations'', 1st edition, Kluwer Academic Publishers, Norwell, MA. 2nd edition, Dover Publications, Mineola, NY, 2003. * [[Zvi Kohavi|Kohavi, Zvi]] (1978), ''Switching and Finite Automata Theory'', 1st edition, McGraw–Hill, 1970. 2nd edition, McGraw–Hill, 1978. * [[Robert R. Korfhage|Korfhage, Robert R.]] (1974), ''Discrete Computational Structures'', Academic Press, New York, NY. * [[Mathematical Society of Japan]], ''Encyclopedic Dictionary of Mathematics'', 2nd edition, 2 vols., Kiyosi Itô (ed.), MIT Press, Cambridge, MA, 1993. Cited as EDM. * [[Marvin L. Minsky|Minsky, Marvin L.]], and [[Seymour A. Papert|Papert, Seymour, A.]] (1988), ''[[Perceptrons]], An Introduction to Computational Geometry'', MIT Press, Cambridge, MA, 1969. Revised, 1972. Expanded edition, 1988. ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://mywikibiz.com/Boolean-valued_function Boolean-Valued Function @ MyWikiBiz] * [http://mathweb.org/wiki/Boolean-valued_function Boolean-Valued Function @ MathWeb Wiki] * [http://netknowledge.org/wiki/Boolean-valued_function Boolean-Valued Function @ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Boolean-valued_function Boolean-Valued Function @ OER Commons] {{col-break}} * [http://p2pfoundation.net/Boolean-Valued_Function Boolean-Valued Function @ P2P Foundation] * [http://semanticweb.org/wiki/Boolean-valued_function Boolean-Valued Function @ SemanticWeb] * [http://ref.subwiki.org/wiki/Boolean-valued_function Boolean-Valued Function @ Subject Wikis] * [http://beta.wikiversity.org/wiki/Boolean-valued_function Boolean-Valued Function @ Wikiversity Beta] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;] * [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;] ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Boolean-valued_function Boolean-Valued Function], [http://mywikibiz.com/ MyWikiBiz] * [http://mathweb.org/wiki/Boolean-valued_function Boolean-Valued Function], [http://mathweb.org/ MathWeb Wiki] * [http://planetmath.org/encyclopedia/BooleanValuedFunction.html Boolean-Valued Function], [http://planetmath.org/ PlanetMath] * [http://planetphysics.org/encyclopedia/BooleanValuedFunction.html Boolean-Valued Function], [http://planetphysics.org/ PlanetPhysics] {{col-break}} * [http://beta.wikiversity.org/wiki/Boolean-valued_function Boolean-Valued Function], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://wikinfo.org/index.php/Boolean-valued_function Boolean-Valued Function], [http://wikinfo.org/ Wikinfo] * [http://textop.org/wiki/index.php?title=Boolean-valued_function Boolean-Valued Function], [http://textop.org/wiki/ Textop Wiki] * [http://en.wikipedia.org/w/index.php?title=Boolean-valued_function&oldid=67166584 Boolean-Valued Function], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Discrete Mathematics]] [[Category:Logic]] [[Category:Mathematics]] 989e172d3fd4ab15e8fadde7410e95d71a312748 0 324 545 2010-06-21T11:28:34Z Jon Awbrey 3 + article wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. '''Differential logic''' is the component of logic whose object is the successful description of variation &mdash; for example, the aspects of change, difference, distribution, and diversity &mdash; in [[universes of discourse]] that are subject to logical description. In formal logic, differential logic treats the principles that govern the use of a ''differential logical calculus'', that is, a formal system with the expressive capacity to describe change and diversity in logical universes of discourse. A simple example of a differential logical calculus is furnished by a [[differential propositional calculus]]. This augments ordinary [[propositional calculus]] in the same way that the [[differential calculus]] of [[Leibniz]] and [[Newton]] augments the [[analytic geometry]] of [[Descartes]]. ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://mywikibiz.com/Differential_logic Differential Logic @ MyWikiBiz] * [http://mathweb.org/wiki/Differential_logic Differential Logic @ MathWeb Wiki] * [http://netknowledge.org/wiki/Differential_logic Differential Logic @ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Differential_logic Differential Logic @ OER Commons] {{col-break}} * [http://p2pfoundation.net/Differential_Logic Differential Logic @ P2P Foundation] * [http://semanticweb.org/wiki/Differential_logic Differential Logic @ SemanticWeb] * [http://ref.subwiki.org/wiki/Differential_logic Differential Logic @ Subject Wikis] * [http://beta.wikiversity.org/wiki/Differential_logic Differential Logic @ Wikiversity Beta] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;] * [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;] ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Differential_logic Differential Logic], [http://mywikibiz.com/ MyWikiBiz] * [http://mathweb.org/wiki/Differential_logic Differential Logic], [http://mathweb.org/wiki/ MathWeb Wiki] * [http://netknowledge.org/wiki/Differential_logic Differential Logic], [http://netknowledge.org/ NetKnowledge] * [http://p2pfoundation.net/Differential_logic Differential Logic], [http://p2pfoundation.net/ P2P Foundation] * [http://semanticweb.org/wiki/Differential_logic Differential Logic], [http://semanticweb.org/ Semantic Web] {{col-break}} * [http://knol.google.com/k/jon-awbrey/differential-logic/3fkwvf69kridz/2 Differential Logic], [http://knol.google.com/ Google Knol] * [http://planetmath.org/encyclopedia/DifferentialLogic.html Differential Logic], [http://planetmath.org/ PlanetMath] * [http://planetphysics.org/encyclopedia/DifferentialLogic.html Differential Logic], [http://planetphysics.org/ PlanetPhysics] * [http://beta.wikiversity.org/wiki/Differential_logic Differential Logic], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://getwiki.net/-Differential_Logic Differential Logic], [http://getwiki.net/ GetWiki] {{col-end}} [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Artificial Intelligence]] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Cybernetics]] [[Category:Differential Logic]] [[Category:Dynamical Systems]] [[Category:Equational Reasoning]] [[Category:Formal Languages]] [[Category:Formal Sciences]] [[Category:Formal Systems]] [[Category:Graph Theory]] [[Category:History of Logic]] [[Category:History of Mathematics]] [[Category:Knowledge Representation]] [[Category:Logic]] [[Category:Logical Graphs]] [[Category:Mathematics]] [[Category:Mathematical Systems Theory]] [[Category:Philosophy]] [[Category:Semiotics]] [[Category:Systems Science]] [[Category:Visualization]] 4ce7b1742f99f98b07582176de4a8cef5be0b1d0 0 325 546 2010-06-21T11:54:59Z Jon Awbrey 3 + article wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''relative term''', also called a '''rhema''' or a '''rheme''', is a logical term that requires reference to any number of other objects, called the ''[[correlate]]s'' of the term, in order to [[denotation|denote]] a definite object, called the ''[[relate]]'' (pronounced with the accent on the first syllable) of the relative term in question. A relative term is typically expressed in ordinary language by means of a phrase with explicit or implicit blanks, for example, ''lover&nbsp;of&nbsp;__'', or ''giver&nbsp;of&nbsp;__&nbsp;to&nbsp;__''. ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://mywikibiz.com/Relative_term Relative Term @ MyWikiBiz] * [http://mathweb.org/wiki/Relative_term Relative Term @ MathWeb Wiki] * [http://netknowledge.org/wiki/Relative_term Relative Term @ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Relative_term Relative Term @ OER Commons] {{col-break}} * [http://p2pfoundation.net/Relative_Term Relative Term @ P2P Foundation] * [http://semanticweb.org/wiki/Relative_term Relative Term @ SemanticWeb] * [http://ref.subwiki.org/wiki/Relative_term Relative Term @ Subject Wikis] * [http://beta.wikiversity.org/wiki/Relative_term Relative Term @ Wikiversity Beta] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;] * [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;] ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Relative_term Relative Term], [http://mywikibiz.com/ MyWikiBiz] * [http://mathweb.org/wiki/Relative_term Relative Term], [http://mathweb.org/wiki/ MathWeb Wiki] * [http://netknowledge.org/wiki/Relative_term Relative Term], [http://netknowledge.org/ NetKnowledge] {{col-break}} * [http://wiki.oercommons.org/mediawiki/index.php/Relative_term Relative Term], [http://wiki.oercommons.org/ OER Commons] * [http://p2pfoundation.net/Relative_Term Relative Term], [http://p2pfoundation.net/ P2P Foundation] * [http://semanticweb.org/wiki/Relative_term Relative Term], [http://semanticweb.org/ Semantic Web] {{col-break}} * [http://beta.wikiversity.org/wiki/Relative_term Relative Term], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://getwiki.net/-Rheme&r=Relative_term Relative Term], [http://getwiki.net/ GetWiki] * [http://en.wikipedia.org/w/index.php?title=Relative_term&oldid=35330741 Relative Term], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Formal Languages]] [[Category:Formal Sciences]] [[Category:Formal Systems]] [[Category:Inquiry]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Science]] [[Category:Semiotics]] [[Category:Philosophy]] 38e727c6b23560d691125df55735672a3d35dc9a 0 326 547 2010-06-21T14:02:30Z Jon Awbrey 3 + article wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''continuous predicate''', as described by [[Charles Sanders Peirce]], is a special type of [[relation (mathematics)|relation]]al [[predicate]] that arises as the [[limit (mathematics)|limit]] of an iterated process of [[hypostatic abstraction]]. Here is one of Peirce's definitive discussions of the concept: <blockquote> <p>When we have analyzed a proposition so as to throw into the subject everything that can be removed from the predicate, all that it remains for the predicate to represent is the form of connection between the different subjects as expressed in the propositional ''form''. What I mean by "everything that can be removed from the predicate" is best explained by giving an example of something not so removable.</p> <p>But first take something removable. "Cain kills Abel." Here the predicate appears as "&mdash; kills &mdash;." But we can remove killing from the predicate and make the latter "&mdash; stands in the relation &mdash; to &mdash;." Suppose we attempt to remove more from the predicate and put the last into the form "&mdash; exercises the function of relate of the relation &mdash; to &mdash;" and then putting "the function of relate to the relation" into another subject leave as predicate "&mdash; exercises &mdash; in respect to &mdash; to &mdash;." But this "exercises" expresses "exercises the function". Nay more, it expresses "exercises the function of relate", so that we find that though we may put this into a separate subject, it continues in the predicate just the same.</p> <p>Stating this in another form, to say that "A is in the relation R to B" is to say that A is in a certain relation to R. Let us separate this out thus: "A is in the relation R¹ (where R¹ is the relation of a relate to the relation of which it is the relate) to R to B". But A is here said to be in a certain relation to the relation R¹. So that we can express the same fact by saying, "A is in the relation R¹ to the relation R¹ to the relation R to B", and so on ''ad infinitum''.</p> <p>A predicate which can thus be analyzed into parts all homogeneous with the whole I call a ''continuous predicate''. It is very important in logical analysis, because a continuous predicate obviously cannot be a ''compound'' except of continuous predicates, and thus when we have carried analysis so far as to leave only a continuous predicate, we have carried it to its ultimate elements. (C.S. Peirce, "Letters to Lady Welby" (14 December 1908), ''Selected Writings'', pp. 396&ndash;397).</p> </blockquote> ==References== * [[Charles Sanders Peirce|Peirce, C.S.]], "Letters to Lady Welby", pp. 380&ndash;432 in ''Charles S. Peirce : Selected Writings (Values in a Universe of Chance)'', [[Philip P. Wiener]] (ed.), Dover, New York, NY, 1966. ==Resources== * [http://vectors.usc.edu/thoughtmesh/publish/147.php Continuous Predicate &rarr; ThoughtMesh] ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://mywikibiz.com/Continuous_predicate Continuous Predicate @ MyWikiBiz] * [http://mathweb.org/wiki/Continuous_predicate Continuous Predicate @ MathWeb Wiki] * [http://netknowledge.org/wiki/Continuous_predicate Continuous Predicate @ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Continuous_predicate Continuous Predicate @ OER Commons] {{col-break}} * [http://p2pfoundation.net/Continuous_Predicate Continuous Predicate @ P2P Foundation] * [http://semanticweb.org/wiki/Continuous_predicate Continuous Predicate @ SemanticWeb] * [http://ref.subwiki.org/wiki/Continuous_predicate Continuous Predicate @ Subject Wikis] * [http://beta.wikiversity.org/wiki/Continuous_predicate Continuous Predicate @ Wikiversity Beta] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;] * [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;] ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Continuous_predicate Continuous Predicate], [http://mywikibiz.com/ MyWikiBiz] * [http://beta.wikiversity.org/wiki/Continuous_predicate Continuous Predicate], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://netknowledge.org/wiki/Continuous_predicate Continuous Predicate], [http://netknowledge.org/ NetKnowledge] * [http://planetmath.org/encyclopedia/ContinuousPredicate.html Continuous Predicate], [http://planetmath.org/ PlanetMath] * [http://semanticweb.org/wiki/Continuous_predicate Continuous Predicate], [http://semanticweb.org/ SemanticWeb] {{col-break}} * [http://vectors.usc.edu/thoughtmesh/publish/147.php Continuous Predicate], [http://vectors.usc.edu/thoughtmesh/ ThoughtMesh] * [http://getwiki.net/-Continuous_Predicate Continuous Predicate], [http://getwiki.net/ GetWiki] * [http://wikinfo.org/index.php?title=Continuous_predicate Continuous Predicate], [http://wikinfo.org/ Wikinfo] * [http://textop.org/wiki/index.php?title=Continuous_predicate Continuous Predicate], [http://textop.org/wiki/ Textop Wiki] * [http://en.wikipedia.org/w/index.php?title=Continuous_predicate&oldid=96870273 Continuous Predicate], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Computer Science]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Philosophy]] [[Category:Semiotics]] 6823a28719a294f12dae7449ea9ffe46470ce304 0 327 548 2010-06-21T15:15:43Z Jon Awbrey 3 + article wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. '''Hypostatic abstraction''' is a formal operation that takes an element of information, as expressed in a proposition <math>X ~\operatorname{is}~ Y,</math> and conceives its information to consist in the relation between that subject and another subject, as expressed in the proposition <math>X ~\operatorname{has}~ Y\!\operatorname{-ness}.</math> The existence of the abstract subject <math>Y\!\operatorname{-ness}</math> consists solely in the truth of those propositions that contain the concrete predicate <math>Y.\!</math> Hypostatic abstraction is known under many names, for example, ''hypostasis'', ''objectification'', ''reification'', and ''subjectal abstraction''. The object of discussion or thought thus introduced is termed a ''[[hypostatic object]]''. The above definition is adapted from the one given by introduced [[Charles Sanders Peirce]] (CP 4.235, "[[The Simplest Mathematics]]" (1902), in ''Collected Papers'', CP 4.227&ndash;323). The way that Peirce describes it, the main thing about the formal operation of hypostatic abstraction, insofar as it can be observed to operate on formal linguistic expressions, is that it converts an adjective or some part of a predicate into an extra [[subject]], upping the ''arity'', also called the ''adicity'', of the main predicate in the process. For example, a typical case of hypostatic abstraction occurs in the transformation from "honey is sweet" to "honey possesses sweetness", which transformation can be viewed in the following variety of ways: <br> <p>[[Image:Hypostatic Abstraction Figure 1.png|center]]</p><br> <p>[[Image:Hypostatic Abstraction Figure 2.png|center]]</p><br> <p>[[Image:Hypostatic Abstraction Figure 3.png|center]]</p><br> <p>[[Image:Hypostatic Abstraction Figure 4.png|center]]</p><br> The grammatical trace of this hypostatic transformation tells of a process that abstracts the adjective "sweet" from the main predicate "is sweet", thus arriving at a new, increased-arity predicate "possesses", and as a by-product of the reaction, as it were, precipitating out the substantive "sweetness" as a new second subject of the new predicate, "possesses". ==References== * [[Charles Sanders Peirce|Peirce, C.S.]], ''Collected Papers of Charles Sanders Peirce'', vols. 1&ndash;6, [[Charles Hartshorne]] and [[Paul Weiss]] (eds.), vols. 7&ndash;8, [[Arthur W. Burks]] (ed.), Harvard University Press, Cambridge, MA, 1931&ndash;1935, 1958. ==Resources== * [http://vectors.usc.edu/thoughtmesh/publish/146.php Hypostatic Abstraction &rarr; ThoughtMesh] * [http://web.clas.ufl.edu/users/jzeman/peirce_on_abstraction.htm J. Jay Zeman, ''Peirce on Abstraction''] ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://mywikibiz.com/Hypostatic_abstraction Hypostatic Abstraction @ MyWikiBiz] * [http://mathweb.org/wiki/Hypostatic_abstraction Hypostatic Abstraction @ MathWeb Wiki] * [http://netknowledge.org/wiki/Hypostatic_abstraction Hypostatic Abstraction @ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Hypostatic_abstraction Hypostatic Abstraction @ OER Commons] {{col-break}} * [http://p2pfoundation.net/Hypostatic_Abstraction Hypostatic Abstraction @ P2P Foundation] * [http://semanticweb.org/wiki/Hypostatic_abstraction Hypostatic Abstraction @ SemanticWeb] * [http://ref.subwiki.org/wiki/Hypostatic_abstraction Hypostatic Abstraction @ Subject Wikis] * [http://beta.wikiversity.org/wiki/Hypostatic_abstraction Hypostatic Abstraction @ Wikiversity Beta] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;] * [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;] ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Hypostatic_abstraction Hypostatic Abstraction], [http://mywikibiz.com/ MyWikiBiz] * [http://netknowledge.org/wiki/Hypostatic_abstraction Hypostatic Abstraction], [http://netknowledge.org/ NetKnowledge] * [http://planetmath.org/encyclopedia/HypostaticAbstraction.html Hypostatic Abstraction], [http://planetmath.org/ PlanetMath] * [http://knol.google.com/k/jon-awbrey/hypostatic-abstraction/3fkwvf69kridz/7 Hypostatic Abstraction], [http://knol.google.com/ Google Knol] * [http://vectors.usc.edu/thoughtmesh/publish/146.php Hypostatic Abstraction], [http://vectors.usc.edu/thoughtmesh/ ThoughtMesh] {{col-break}} * [http://beta.wikiversity.org/wiki/Hypostatic_abstraction Hypostatic Abstraction], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://getwiki.net/-Hypostatic_Abstraction Hypostatic Abstraction], [http://getwiki.net/ GetWiki] * [http://wikinfo.org/index.php?title=Hypostatic_abstraction Hypostatic Abstraction], [http://wikinfo.org/ Wikinfo] * [http://textop.org/wiki/index.php?title=Hypostatic_abstraction Hypostatic Abstraction], [http://textop.org/wiki/ Textop Wiki] * [http://en.wikipedia.org/w/index.php?title=Hypostatic_abstraction&oldid=69736615 Hypostatic Abstraction], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Computer Science]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Philosophy]] [[Category:Semiotics]] d1807b35923706f5cbad61d54ee6f569614d0865 6 328 549 2010-06-21T16:04:17Z Jon Awbrey 3 wikitext text/x-wiki da39a3ee5e6b4b0d3255bfef95601890afd80709 6 329 550 2010-06-21T16:06:33Z Jon Awbrey 3 wikitext text/x-wiki da39a3ee5e6b4b0d3255bfef95601890afd80709 6 330 551 2010-06-21T16:08:24Z Jon Awbrey 3 wikitext text/x-wiki da39a3ee5e6b4b0d3255bfef95601890afd80709 6 331 552 2010-06-21T16:10:50Z Jon Awbrey 3 wikitext text/x-wiki da39a3ee5e6b4b0d3255bfef95601890afd80709 0 332 553 2010-06-21T18:04:03Z Jon Awbrey 3 + article wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. The '''logic of relatives''', more precisely, the '''logic of relative terms''', is the study of [[relation (mathematics)|relation]]s as represented in symbolic forms known as ''rhemes'', ''rhemata'', or ''relative terms''. The treatment of relations by way of their corresponding relative terms affords a distinctive perspective on the subject, even though all angles of approach must ultimately converge on the same formal subject matter. The consideration of ''[[relative term]]s'' has its roots in antiquity, but it entered a radically new phase of development with the work of [[Charles Sanders Peirce]], beginning with his paper [[Logic of Relatives (1870)|"Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic" (1870)]]. ==See also== * [[Logic of Relatives (1870)]] * [[Logic of Relatives (1883)]] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Peirce's_1870_Logic_Of_Relatives Notes on Peirce's 1870 Logic Of Relatives] ==References== * [[Charles Sanders Peirce|Peirce, C.S.]], "Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic", ''Memoirs of the American Academy of Arts and Sciences'' 9, 317&ndash;378, 1870. Reprinted, ''Collected Papers'' CP 3.45&ndash;149, ''Chronological Edition'' CE 2, 359&ndash;429. ==Bibliography== * [[Aristotle]], "[[The Categories]]", [[Harold P. Cooke]] (trans.), pp. 1&ndash;109 in ''Aristotle, Vol. 1'', [[Loeb Classical Library]], [[William Heinemann]], London, UK, 1938. * Aristotle, "[[On Interpretation]]", [[Harold P. Cooke]] (trans.), pp. 111&ndash;179 in ''Aristotle, Vol. 1'', [[Loeb Classical Library]], [[William Heinemann]], London, UK, 1938. * Aristotle, "[[Prior Analytics]]", [[Hugh Tredennick]] (trans.), pp. 181&ndash;531 in ''Aristotle, Vol. 1'', [[Loeb Classical Library]], William Heinemann, London, UK, 1938. * [[George Boole|Boole, George]], ''An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities'', [[Macmillan Publishers|Macmillan]], 1854. Reprinted with corrections, [[Dover Publications]], New York, NY, 1958. * Maddux, Roger D., ''Relation Algebras'', vol. 150 in 'Studies in Logic and the Foundations of Mathematics', Elsevier Science, 2006. * Peirce, C.S., ''Collected Papers of Charles Sanders Peirce'', vols. 1&ndash;6, [[Charles Hartshorne]] and [[Paul Weiss]] (eds.), vols. 7&ndash;8, [[Arthur W. Burks]] (ed.), Harvard University Press, Cambridge, MA, 1931&ndash;1935, 1958. Cited as CP volume.paragraph. * Peirce, C.S., ''Writings of Charles S. Peirce : A Chronological Edition, Volume 2, 1867&ndash;1871'', Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1984. Cited as CE 2. ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://mywikibiz.com/Logic_of_relatives Logic of Relatives @ MyWikiBiz] * [http://mathweb.org/wiki/Logic_of_relatives Logic of Relatives @ MathWeb Wiki] * [http://netknowledge.org/wiki/Logic_of_relatives Logic of Relatives @ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Logic_of_relatives Logic of Relatives @ OER Commons] {{col-break}} * [http://p2pfoundation.net/Logic_of_Relatives Logic of Relatives @ P2P Foundation] * [http://semanticweb.org/wiki/Logic_of_relatives Logic of Relatives @ SemanticWeb] * [http://ref.subwiki.org/wiki/Logic_of_relatives Logic of Relatives @ Subject Wikis] * [http://beta.wikiversity.org/wiki/Logic_of_relatives Logic of Relatives @ Wikiversity Beta] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;] * [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;] ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Logic_of_relatives Logic of Relatives], [http://mywikibiz.com/ MyWikiBiz] * [http://mathweb.org/wiki/Logic_of_relatives Logic of Relatives], [http://mathweb.org/wiki/ MathWeb Wiki] * [http://netknowledge.org/wiki/Logic_of_relatives Logic of Relatives], [http://netknowledge.org/ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Logic_of_relatives Logic of Relatives], [http://wiki.oercommons.org/ OER Commons] * [http://p2pfoundation.net/Logic_of_Relatives Logic of Relatives], [http://p2pfoundation.net/ P2P Foundation] * [http://semanticweb.org/wiki/Logic_of_Relatives Logic of Relatives], [http://semanticweb.org/ Semantic Web] {{col-break}} * [http://beta.wikiversity.org/wiki/Logic_of_relatives Logic of Relatives], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://knol.google.com/k/jon-awbrey/logic-of-relatives/3fkwvf69kridz/5 Logic of Relatives], [http://knol.google.com/ Google Knol] * [http://getwiki.net/-Logic_of_Relatives Logic of Relatives], [http://getwiki.net/ GetWiki] * [http://wikinfo.org/index.php/Logic_of_relatives Logic of Relatives], [http://wikinfo.org/ Wikinfo] * [http://textop.org/wiki/index.php?title=Logic_of_relatives Logic of Relatives], [http://textop.org/wiki/ Textop Wiki] * [http://en.wikipedia.org/w/index.php?title=Logic_of_relatives&oldid=43501411 Logic of Relatives], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:EN]] [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Formal Languages]] [[Category:Formal Sciences]] [[Category:Formal Systems]] [[Category:Inquiry]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Science]] [[Category:Semiotics]] [[Category:Philosophy]] cce14109303bd0a16c93072d8d2fb8984c16f612 554 553 2010-06-21T18:06:05Z Jon Awbrey 3 del extraneous category wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. The '''logic of relatives''', more precisely, the '''logic of relative terms''', is the study of [[relation (mathematics)|relation]]s as represented in symbolic forms known as ''rhemes'', ''rhemata'', or ''relative terms''. The treatment of relations by way of their corresponding relative terms affords a distinctive perspective on the subject, even though all angles of approach must ultimately converge on the same formal subject matter. The consideration of ''[[relative term]]s'' has its roots in antiquity, but it entered a radically new phase of development with the work of [[Charles Sanders Peirce]], beginning with his paper [[Logic of Relatives (1870)|"Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic" (1870)]]. ==See also== * [[Logic of Relatives (1870)]] * [[Logic of Relatives (1883)]] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Peirce's_1870_Logic_Of_Relatives Notes on Peirce's 1870 Logic Of Relatives] ==References== * [[Charles Sanders Peirce|Peirce, C.S.]], "Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic", ''Memoirs of the American Academy of Arts and Sciences'' 9, 317&ndash;378, 1870. Reprinted, ''Collected Papers'' CP 3.45&ndash;149, ''Chronological Edition'' CE 2, 359&ndash;429. ==Bibliography== * [[Aristotle]], "[[The Categories]]", [[Harold P. Cooke]] (trans.), pp. 1&ndash;109 in ''Aristotle, Vol. 1'', [[Loeb Classical Library]], [[William Heinemann]], London, UK, 1938. * Aristotle, "[[On Interpretation]]", [[Harold P. Cooke]] (trans.), pp. 111&ndash;179 in ''Aristotle, Vol. 1'', [[Loeb Classical Library]], [[William Heinemann]], London, UK, 1938. * Aristotle, "[[Prior Analytics]]", [[Hugh Tredennick]] (trans.), pp. 181&ndash;531 in ''Aristotle, Vol. 1'', [[Loeb Classical Library]], William Heinemann, London, UK, 1938. * [[George Boole|Boole, George]], ''An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities'', [[Macmillan Publishers|Macmillan]], 1854. Reprinted with corrections, [[Dover Publications]], New York, NY, 1958. * Maddux, Roger D., ''Relation Algebras'', vol. 150 in 'Studies in Logic and the Foundations of Mathematics', Elsevier Science, 2006. * Peirce, C.S., ''Collected Papers of Charles Sanders Peirce'', vols. 1&ndash;6, [[Charles Hartshorne]] and [[Paul Weiss]] (eds.), vols. 7&ndash;8, [[Arthur W. Burks]] (ed.), Harvard University Press, Cambridge, MA, 1931&ndash;1935, 1958. Cited as CP volume.paragraph. * Peirce, C.S., ''Writings of Charles S. Peirce : A Chronological Edition, Volume 2, 1867&ndash;1871'', Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1984. Cited as CE 2. ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://mywikibiz.com/Logic_of_relatives Logic of Relatives @ MyWikiBiz] * [http://mathweb.org/wiki/Logic_of_relatives Logic of Relatives @ MathWeb Wiki] * [http://netknowledge.org/wiki/Logic_of_relatives Logic of Relatives @ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Logic_of_relatives Logic of Relatives @ OER Commons] {{col-break}} * [http://p2pfoundation.net/Logic_of_Relatives Logic of Relatives @ P2P Foundation] * [http://semanticweb.org/wiki/Logic_of_relatives Logic of Relatives @ SemanticWeb] * [http://ref.subwiki.org/wiki/Logic_of_relatives Logic of Relatives @ Subject Wikis] * [http://beta.wikiversity.org/wiki/Logic_of_relatives Logic of Relatives @ Wikiversity Beta] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;] * [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;] ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Logic_of_relatives Logic of Relatives], [http://mywikibiz.com/ MyWikiBiz] * [http://mathweb.org/wiki/Logic_of_relatives Logic of Relatives], [http://mathweb.org/wiki/ MathWeb Wiki] * [http://netknowledge.org/wiki/Logic_of_relatives Logic of Relatives], [http://netknowledge.org/ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Logic_of_relatives Logic of Relatives], [http://wiki.oercommons.org/ OER Commons] * [http://p2pfoundation.net/Logic_of_Relatives Logic of Relatives], [http://p2pfoundation.net/ P2P Foundation] * [http://semanticweb.org/wiki/Logic_of_Relatives Logic of Relatives], [http://semanticweb.org/ Semantic Web] {{col-break}} * [http://beta.wikiversity.org/wiki/Logic_of_relatives Logic of Relatives], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://knol.google.com/k/jon-awbrey/logic-of-relatives/3fkwvf69kridz/5 Logic of Relatives], [http://knol.google.com/ Google Knol] * [http://getwiki.net/-Logic_of_Relatives Logic of Relatives], [http://getwiki.net/ GetWiki] * [http://wikinfo.org/index.php/Logic_of_relatives Logic of Relatives], [http://wikinfo.org/ Wikinfo] * [http://textop.org/wiki/index.php?title=Logic_of_relatives Logic of Relatives], [http://textop.org/wiki/ Textop Wiki] * [http://en.wikipedia.org/w/index.php?title=Logic_of_relatives&oldid=43501411 Logic of Relatives], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Formal Languages]] [[Category:Formal Sciences]] [[Category:Formal Systems]] [[Category:Inquiry]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Science]] [[Category:Semiotics]] [[Category:Philosophy]] 831c3ed83b56c9fd758be42774bff954eacae5b0 0 333 555 2010-06-21T18:34:25Z Jon Awbrey 3 + article wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''logical matrix''', in the finite dimensional case, is a ''k''-dimensional [[array]] with entries from the [[boolean domain]] '''B'''&nbsp;=&nbsp;{0,&nbsp;1}. Such a [[matrix]] affords a [[matrix representation]] of a ''k''-adic [[relation (mathematics)|relation]]. ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://mywikibiz.com/Logical_matrix Logical Matrix @ MyWikiBiz] * [http://mathweb.org/wiki/Logical_matrix Logical Matrix @ MathWeb Wiki] * [http://netknowledge.org/wiki/Logical_matrix Logical Matrix @ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Logical_matrix Logical Matrix @ OER Commons] {{col-break}} * [http://p2pfoundation.net/Logical_Matrix Logical Matrix @ P2P Foundation] * [http://semanticweb.org/wiki/Logical_matrix Logical Matrix @ SemanticWeb] * [http://ref.subwiki.org/wiki/Logical_matrix Logical Matrix @ Subject Wikis] * [http://beta.wikiversity.org/wiki/Logical_matrix Logical Matrix @ Wikiversity Beta] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;] * [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;] ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Logical_matrix Logical Matrix], [http://mywikibiz.com/ MyWikiBiz] * [http://planetmath.org/encyclopedia/LogicalMatrix.html Logical Matrix], [http://planetmath.org/ PlanetMath] * [http://beta.wikiversity.org/wiki/Logical_matrix Logical Matrix], [http://beta.wikiversity.org/ Wikiversity Beta] {{col-break}} * [http://wikinfo.org/index.php/Logical_matrix Logical Matrix], [http://wikinfo.org/ Wikinfo] * [http://textop.org/wiki/index.php?title=Logical_matrix Logical Matrix], [http://textop.org/wiki/ Textop Wiki] * [http://en.wikipedia.org/w/index.php?title=Logical_matrix&oldid=43606082 Logical Matrix], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Discrete Mathematics]] [[Category:Logic]] [[Category:Mathematics]] bfc73f3cc6851eca65fe3284abc8bfad3aeada1e 0 334 556 2010-06-21T20:10:31Z Jon Awbrey 3 + article wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. In mathematics, a '''finitary relation''' is defined by one of the formal definitions given below. * The basic idea is to generalize the concept of a ''[[binary relation|2-place relation]]'', such as the relation of ''[[equality (mathematics)|equality]]'' denoted by the sign "=" in a statement like "5&nbsp;+&nbsp;7&nbsp;=&nbsp;12" or the relation of ''[[order theory|order]]'' denoted by the sign "<" in a statement like "5&nbsp;<&nbsp;12". Relations that involve two 'places' or 'roles' are called ''[[binary relation]]s'' by some and ''dyadic relations'' by others, the latter being historically prior but also useful when necessary to avoid confusion with [[binary numeral system|binary (base 2) numerals]]. * The next step up is to consider relations that involve increasing but still finite numbers of places or roles. These are called ''finite place'' or ''finitary'' relations. A finitary relation that involves ''k'' places is variously called a ''k-ary'', a ''k-adic'', or a ''k-dimensional'' relation. The number ''k'' is then called the ''[[arity]]'', the ''adicity'', or the ''[[dimension]]'' of the relation, respectively. ==Informal introduction== The definition of ''relation'' given in the next Section formally captures a concept that is actually quite familiar from everyday life. For example, consider the relationship, involving three roles that people might play, expressed in a statement of the form "''X'' suspects that ''Y'' likes ''Z''&nbsp;". The facts of a concrete situation could be organized in a Table like the following: {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:#f8f8ff; text-align:center; width:60%" |+ '''Relation S : X suspects that Y likes Z''' |- style="background:#e6e6ff" ! Person X !! Person Y !! Person Z |- | Alice || Bob || Denise |- | Charles || Alice || Bob |- | Charles || Charles || Alice |- | Denise || Denise || Denise |} <br> Each row of the Table records a fact or makes an assertion of the form "''X'' suspects that ''Y'' likes ''Z''&nbsp;". For instance, the first row says, in effect, "Alice suspects that Bob likes Denise". The Table represents a relation ''S'' over the set ''P'' of people under discussion: : ''P'' = {Alice, Bob, Charles, Denise}. The data of the Table are equivalent to the following set of ordered triples: : ''S'' = {(Alice,&nbsp;Bob,&nbsp;Denise), (Charles,&nbsp;Alice,&nbsp;Bob), (Charles,&nbsp;Charles,&nbsp;Alice), (Denise,&nbsp;Denise,&nbsp;Denise)}. By a slight overuse of notation, it is usual to write ''S''(Alice,&nbsp;Bob,&nbsp;Denise) to say the same thing as the first row of the Table. The relation ''S'' is a ''ternary'' relation, since there are ''three'' items involved in each row. The relation itself is a mathematical object, defined in terms of concepts from [[set theory]], that carries all of the information from the Table in one neat package. The Table for relation ''S'' is an extremely simple example of a [[relational database]]. The theoretical aspects of databases are the specialty of one branch of [[computer science]], while their practical impacts have become all too familiar in our everyday lives. Computer scientists, logicians, and mathematicians, however, tend to see different things when they look at these concrete examples and samples of the more general concept of a relation. For one thing, databases are designed to deal with empirical data, and experience is always finite, whereas mathematics is nothing if not concerned with infinity, at the very least, potential infinity. This difference in perspective brings up a number of ideas that are usefully introduced at this point, if by no means covered in depth. ==Example: divisibility== A more typical example of a 2-place relation in mathematics is the relation of ''[[divisor|divisibility]]'' between two positive integers ''n'' and ''m'' that is expressed in statements like "''n'' divides ''m''" or "''n'' goes into ''m''". This is a relation that comes up so often that a special symbol "|" is reserved to express it, allowing one to write "''n''|''m''" for "''n'' divides ''m''". To express the binary relation of divisibility in terms of sets, we have the set ''P'' of positive integers, ''P'' = {1, 2, 3, …}, and we have the binary relation ''D'' on ''P'' such that the ordered pair (''n'', ''m'') is in the relation ''D'' just in case ''n''|''m''. In other turns of phrase that are frequently used, one says that the number ''n'' is related by ''D'' to the number ''m'' just in case ''n'' is a factor of ''m'', that is, just in case ''n'' divides ''m'' with no remainder. The relation ''D'', regarded as a set of ordered pairs, consists of all pairs of numbers (''n'', ''m'') such that ''n'' divides ''m''. For example, 2 is a factor of 4, and 6 is a factor of 72, which two facts can be written either as 2|4 and 6|72 or as ''D''(2, 4) and ''D''(6, 72). ==Formal definitions== There are two definitions of k-place relations that are commonly encountered in mathematics. In order of simplicity, the first of these definitions is as follows: '''Definition 1.''' A '''relation''' ''L'' over the [[set]]s ''X''₁, …, ''X''_''k'' is a [[subset]] of their [[cartesian product]], written ''L'' &sube; ''X''₁ &times; … &times; ''X''_''k''. Under this definition, then, a ''k''-ary relation is simply a set of ''k''-[[tuple]]s. The second definition makes use of an idiom that is common in mathematics, stipulating that "such and such is an ''n''-tuple" in order to ensure that such and such a mathematical object is determined by the specification of ''n'' component mathematical objects. In the case of a relation ''L'' over ''k'' sets, there are ''k'' + 1 things to specify, namely, the ''k'' sets plus a subset of their cartesian product. In the idiom, this is expressed by saying that ''L'' is a (''k''+1)-tuple. '''Definition 2.''' A '''relation''' ''L'' over the sets ''X''₁, …, ''X''_''k'' is a (''k''+1)-tuple ''L'' = (''X''₁, …, ''X''_''k'', ''G''(''L'')), where ''G''(''L'') is a subset of the cartesian product ''X''₁ &times; … &times; ''X''_''k''. ''G''(''L'') is called the ''[[graph]]'' of ''L''. Elements of a relation are more briefly denoted by using boldface characters, for example, the constant element <math>\mathbf{a}</math> = (a₁,&nbsp;…,&nbsp;a_''k'') or the variable element <math>\mathbf{x}</math> = (''x''₁,&nbsp;…,&nbsp;''x''_''k''). A statement of the form "<math>\mathbf{a}</math> is in the relation ''L''&nbsp;" is taken to mean that <math>\mathbf{a}</math> is in ''L'' under the first definition and that <math>\mathbf{a}</math> is in ''G''(''L'') under the second definition. The following considerations apply under either definition: :* The sets ''X''_''j'' for ''j'' = 1 to ''k'' are called the ''[[domain]]s'' of the relation. In the case of the first definition, the relation itself does not uniquely determine a given sequence of domains. :* If all of the domains ''X''_''j'' are the same set ''X'', then ''L'' is more simply referred to as a ''k''-ary relation over ''X''. :* If any of the domains ''X''_''j'' is empty, then the cartesian product is empty, and the only relation over such a sequence of domains is the empty relation ''L'' = <math>\varnothing</math>. As a result, naturally occurring applications of the relation concept typically involve a stipulation that all of the domains be nonempty. As a rule, whatever definition best fits the application at hand will be chosen for that purpose, and anything that falls under it will be called a 'relation' for the duration of that discussion. If it becomes necessary to distinguish the two alternatives, the latter type of object can be referred to as an ''embedded'' or ''included'' relation. If ''L'' is a relation over the domains ''X''₁, …, ''X''_''k'', it is conventional to consider a sequence of terms called ''variables'', ''x''₁, …, ''x''_''k'', that are said to ''range over'' the respective domains. A ''[[boolean domain]]'' '''B''' is a generic 2-element set, say, '''B''' = {0, 1}, whose elements are interpreted as logical values, typically 0 = false and 1 = true. The ''[[characteristic function]]'' of the relation ''L'', written ''f''_''L'' or &chi;(''L''), is the [[boolean-valued function]] ''f''_''L''&nbsp;:&nbsp;''X''₁&nbsp;&times;&nbsp;…&nbsp;&times;&nbsp;''X''_''k''&nbsp;→&nbsp;'''B''', defined in such a way that ''f''_''L''(<math>\mathbf{x}</math>) = 1 just in case the ''k''-tuple <math>\mathbf{x}</math> is in the relation ''L''. The characteristic function of a relation may also be called its ''[[indicator function]]'', especially in probability and statistics. It is conventional in applied mathematics, computer science, and statistics to refer to a boolean-valued function like ''f''_''L'' as a ''k''-place ''[[predicate]]''. From the more abstract viewpoints of [[formal logic]] and [[model theory]], the relation ''L'' is seen as constituting a ''logical model'' or a ''relational structure'' that serves as one of many possible [[interpretation]]s of a corresponding ''k''-place ''predicate symbol'', as that term is used in ''[[first-order logic|predicate calculus]]''. Due to the convergence of many different styles of study on the same areas of interest, the reader will find much variation in usage here. The variation presented in this article treats a relation as the [[set theory|set-theoretic]] ''[[extension (semantics)|extension]]'' of a relational concept or term. Another variation reserves the term 'relation' to the corresponding logical entity, either the ''[[comprehension (logic)|logical comprehension]]'', which is the totality of ''[[intension]]s'' or abstract ''[[property (philosophy)|properties]]'' that all of the elements of the relation in extension have in common, or else the symbols that are taken to denote these elements and intensions. Further, but hardly finally, some writers of the latter persuasion introduce terms with more concrete connotations, like 'relational structure', for the set-theoretic extension of a given relational concept. ==Example: coplanarity== For lines ''L'' in three-dimensional space, there is a ternary relation picking out the triples of lines that are [[coplanar]]. This ''does not'' reduce to the binary [[symmetric relation]] of coplanarity of pairs of lines. In other words, writing ''P''(''L'', ''M'', ''N'') when the lines ''L'', ''M'', and ''N'' lie in a plane, and ''Q''(''L'', ''M'') for the binary relation, it is not true that ''Q''(''L'', ''M''), ''Q''(''M'', ''N'') and ''Q''(''N'', ''L'') together imply ''P''(''L'', ''M'', ''N''); although the converse is certainly true (any pair out of three coplanar lines is coplanar, ''a fortiori''). There are two geometrical reasons for this. In one case, for example taking the ''x''-axis, ''y''-axis and ''z''-axis, the three lines are concurrent, i.e. intersect at a single point. In another case, ''L'', ''M'', and ''N'' can be three edges of an infinite [[triangular prism]]. What is true is that if each pair of lines intersects, and the points of intersection are distinct, then pairwise coplanarity implies coplanarity of the triple. ==Remarks== Relations are classified according to the number of sets in the cartesian product, in other words the number of terms in the expression: :* Unary relation or [[property (philosophy)|property]]: ''L''(''u'') :* Binary relation: ''L''(''u'', ''v'') or ''u'' ''L'' ''v'' :* Ternary relation: ''L''(''u'', ''v'', ''w'') :* Quaternary relation: ''L''(''u'', ''v'', ''w'', ''x'') Relations with more than four terms are usually referred to as ''k''-ary, for example, "a 5-ary relation". ==References== * [[Charles Sanders Peirce|Peirce, C.S.]] (1870), "Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic", ''Memoirs of the American Academy of Arts and Sciences'' 9, 317–378, 1870. Reprinted, ''Collected Papers'' CP 3.45–149, ''Chronological Edition'' CE 2, 359–429. * [[Stanisław Marcin Ulam|Ulam, S.M.]] and [[Al Bednarek|Bednarek, A.R.]] (1990), "On the Theory of Relational Structures and Schemata for Parallel Computation", pp. 477–508 in A.R. Bednarek and Françoise Ulam (eds.), ''Analogies Between Analogies: The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators'', University of California Press, Berkeley, CA. ==Bibliography== * [[Nicolas Bourbaki|Bourbaki, N.]] (1994), ''Elements of the History of Mathematics'', John Meldrum (trans.), Springer-Verlag, Berlin, Germany. * [[Paul Richard Halmos|Halmos, P.R.]] (1960), ''Naive Set Theory'', D. Van Nostrand Company, Princeton, NJ. * [[Francis William Lawvere|Lawvere, F.W.]], and [[Robert Rosebrugh|Rosebrugh, R.]] (2003), ''Sets for Mathematics'', Cambridge University Press, Cambridge, UK. * Maddux, R.D. (2006), ''Relation Algebras'', vol.&nbsp;150 in 'Studies in Logic and the Foundations of Mathematics', Elsevier Science. * Mili, A., Desharnais, J., Mili, F., with Frappier, M. (1994), ''Computer Program Construction'', Oxford University Press, New York, NY. * [[Marvin L. Minsky|Minsky, M.L.]], and [[Seymour A. Papert|Papert, S.A.]] (1969/1988), ''[[Perceptron]]s, An Introduction to Computational Geometry'', MIT Press, Cambridge, MA, 1969. Expanded edition, 1988. * [[Charles Sanders Peirce|Peirce, C.S.]] (1984), ''Writings of Charles S. Peirce : A Chronological Edition, Volume 2, 1867-1871'', Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN. * [[Josiah Royce|Royce, J.]] (1961), ''The Principles of Logic'', Philosophical Library, New York, NY. * [[Alfred Tarski|Tarski, A.]] (1956/1983), ''Logic, Semantics, Metamathematics, Papers from 1923 to 1938'', J.H. Woodger (trans.), 1st edition, Oxford University Press, 1956. 2nd edition, J. Corcoran (ed.), Hackett Publishing, Indianapolis, IN, 1983. * [[Stanisław Marcin Ulam|Ulam, S.M.]] (1990), ''Analogies Between Analogies : The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators'', A.R. Bednarek and Françoise Ulam (eds.), University of California Press, Berkeley, CA. * [[Paulus Venetus|Venetus, P.]] (1984), ''Logica Parva, Translation of the 1472 Edition with Introduction and Notes'', Alan R. Perreiah (trans.), Philosophia Verlag, Munich, Germany. ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://mywikibiz.com/Relation_(mathematics) Relation @ MyWikiBiz] * [http://mathweb.org/wiki/Relation_(mathematics) Relation @ MathWeb Wiki] * [http://netknowledge.org/wiki/Relation_(mathematics) Relation @ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Relation_(mathematics) Relation @ OER Commons] {{col-break}} * [http://p2pfoundation.net/Relation_(mathematics) Relation @ P2P Foundation] * [http://semanticweb.org/wiki/Relation_(mathematics) Relation @ SemanticWeb] * [http://ref.subwiki.org/wiki/Relation_(mathematics) Relation @ Subject Wikis] * [http://beta.wikiversity.org/wiki/Relation_(mathematics) Relation @ Wikiversity Beta] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;] * [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;] ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Relation_(mathematics) Relation], [http://mywikibiz.com/ MyWikiBiz] * [http://beta.wikiversity.org/wiki/Relation_(mathematics) Relation], [http://beta.wikiversity.org/ Wikiversity Beta] {{col-break}} * [http://www.getwiki.net/-Relation_(mathematics) Relation], [http://www.getwiki.net/ GetWiki] * [http://wikinfo.org/index.php/Relation_(mathematics) Relation], [http://wikinfo.org/index.php/Main_Page Wikinfo] {{col-break}} * [http://textop.org/wiki/index.php?title=Relation_(mathematics) Relation], [http://textop.org/wiki/ Textop Wiki] * [http://en.wikipedia.org/w/index.php?title=Relation_(mathematics)&oldid=73324659 Relation], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Relation Theory]] [[Category:Set Theory]] 0326f0814616c91287c1f02a42eaafef806ac6ad 0 335 557 2010-06-22T15:24:20Z Jon Awbrey 3 + article wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. In [[logic]] and [[mathematics]], '''relation composition''', or the composition of [[relation (mathematics)|relations]], is the generalization of [[function composition]], or the composition of [[function (mathematics)|functions]]. ==Preliminaries== The first order of business is to define the operation on [[relation (mathematics)|relations]] that is variously known as the ''composition of relations'', ''relational composition'', or ''relative multiplication''. In approaching the more general constructions, it pays to begin with the composition of 2-adic and 3-adic relations. As an incidental observation on usage, there are many different conventions of [[syntax]] for denoting the application and composition of relations, with perhaps even more options in general use than are common for the application and composition of [[function (mathematics)|functions]]. In this case there is little chance of standardization, since the convenience of conventions is relative to the context of use, and the same writers use different styles of syntax in different settings, depending on the ease of analysis and computation. {| align="center" cellpadding="4" width="90%" | <p>The first dimension of variation in syntax has to do with the correspondence between the order of operation and the linear order of terms on the page.</p> |- | <p>The second dimension of variation in syntax has to do with the automatic assumptions in place about the associations of terms in the absence of associations marked by parentheses. This becomes a significant factor with relations in general because the usual property of [[associativity]] is lost as both the complexities of compositions and the dimensions of relations increase.</p> |} These two factors together generate the following four styles of syntax: {| align="center" cellpadding="4" width="90%" | LALA = left application, left association. |- | LARA = left application, right association. |- | RALA = right application, left association. |- | RARA = right application, right association. |} ==Definition== A notion of relational composition is to be defined that generalizes the usual notion of functional composition: {| align="center" cellpadding="4" width="90%" | <p>Composing ''on the right'', <math>f : X \to Y</math> followed by <math>g : Y \to Z</math></p> <p>results in a ''composite function'' formulated as <math>fg : X \to Z.</math></p> |- | <p>Composing ''on the left'', <math>f : X \to Y</math> followed by <math>g : Y \to Z</math></p> <p>results in a ''composite function'' formulated as <math>gf : X \to Z.</math></p> |} Note on notation. The ordinary symbol for functional composition is the ''[[composition sign]]'', a small circle "<math>\circ</math>" written between the names of the functions being composed, as <math>f \circ g,</math> but the sign is often omitted if there is no risk of confusing the composition of functions with their algebraic product. In contexts where both compositions and products occur, either the composition is marked on each occasion or else the product is marked by means of a ''center dot'' "<math>\cdot</math>", as <math>f \cdot g.</math> Generalizing the paradigm along parallel lines, the ''composition'' of a pair of 2-adic relations is formulated in the following two ways: {| align="center" cellpadding="4" width="90%" | <p>Composing ''on the right'', <math>P \subseteq X \times Y</math> followed by <math>Q \subseteq Y \times Z</math></p> <p>results in a ''composite relation'' formulated as <math>PQ \subseteq X \times Z.</math></p> |- | <p>Composing ''on the left'', <math>P \subseteq X \times Y</math> followed by <math>Q \subseteq Y \times Z</math></p> <p>results in a ''composite relation'' formulated as <math>QP \subseteq X \times Z.</math></p> |} ==Geometric construction== There is a neat way of defining relational compositions in geometric terms, not only showing their relationship to the [[projection (set theory)|projection]] operations that come with any [[cartesian product]], but also suggesting natural directions for generalizing relational compositions beyond the 2-adic case, and even beyond relations that have any fixed [[arity]], in effect, to the general case of [[formal language]]s as generalized relations. This way of looking at relational compositions is sometimes referred to as [[Alfred Tarski|Tarski]]'s Trick (T²), on account of his having put it to especially good use in his work (Ulam and Bednarek, 1977). It supplies the imagination with a geometric way of visualizing the relational composition of a pair of 2-adic relations, doing this by attaching concrete imagery to the basic [[set theory|set-theoretic]] operations, namely, [[intersection (set theory)|intersection]]s, [[projection (set theory)|projection]]s, and a certain class of operations [[inverse relation|inverse]] to projections, here called ''[[tacit extension]]s''. The stage is set for Tarski's trick by highlighting the links between two topics that are likely to appear wholly unrelated at first, namely: :* The use of logical [[conjunction]], as denoted by the symbol "&and;" in expressions of the form ''F''(''x'',&nbsp;''y'',&nbsp;''z'') = ''G''(''x'',&nbsp;''y'')&nbsp;&and;&nbsp;''H''(''y'',&nbsp;''z''), to define a 3-adic relation ''F'' in terms of a pair of 2-adic relations ''G'' and ''H''. :* The concepts of 2-adic ''[[projection (set theory)|projection]]'' and ''projective determination'', that are invoked in the 'weak' notion of ''projective reducibility''. The relational composition ''G''&nbsp;&omicron;&nbsp;''H'' of a pair of 2-adic relations ''G'' and ''H'' will be constructed in three stages, first, by taking the tacit extensions of ''G'' and ''H'' to 3-adic relations that reside in the same space, next, by taking the intersection of these extensions, tantamount to the maximal 3-adic relation that is consistent with the ''prima facie'' 2-adic relation data, finally, by projecting this intersection on a suitable plane to form a third 2-adic relation, constituting in fact the relational composition ''G''&nbsp;&omicron;&nbsp;''H'' of the relations ''G'' and ''H''. The construction of a relational composition in a specifically mathematical setting normally begins with [[relation (mathematics)|mathematical relations]] at a higher level of abstraction than the corresponding objects in linguistic or logical settings. This is due to the fact that mathematical objects are typically specified only ''up to [[isomorphism]]'' as the conventional saying goes, that is, any objects that have the 'same form' are generally regarded as the being the same thing, for most all intents and mathematical purposes. Thus, the mathematical construction of a relational composition begins by default with a pair of 2-adic relations that reside, without loss of generality, in the same plane, say, ''G'', ''H''&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''Y'', as depicted in Figure 1. o-------------------------------------------------o | | | o o | | |\ |\ | | | \ | \ | | | \ | \ | | | \ | \ | | | \ | \ | | | \ | \ | | | * \ | * \ | | X * Y X * Y | | \ * | \ * | | | \ G | \ H | | | \ | \ | | | \ | \ | | | \ | \ | | | \ | \ | | | \| \| | | o o | | | o-------------------------------------------------o Figure 1. Dyadic Relations G, H c X x Y The 2-adic relations ''G'' and ''H'' cannot be composed at all at this point, not without additional information or further stipulation. In order for their relational composition to be possible, one of two types of cases has to happen: :* The first type of case occurs when ''X''&nbsp;=&nbsp;''Y''. In this case, both of the compositions ''G''&nbsp;&omicron;&nbsp;''H'' and ''H''&nbsp;&omicron;&nbsp;''G'' are defined. :* The second type of case occurs when ''X'' and ''Y'' are distinct, but when it nevertheless makes sense to speak of a 2-adic relation ''&#292;'' that is isomorphic to ''H'', but living in the plane ''YZ'', that is, in the space of the cartesian product ''Y''&nbsp;&times;&nbsp;''Z'', for some set ''Z''. Whether you view isomorphic things to be the same things or not, you still have to specify the exact isomorphisms that are needed to transform any given representation of a thing into a required representation of the same thing. Let us imagine that we have done this, and say how later: o-------------------------------------------------o | | | o o | | |\ /| | | | \ / | | | | \ / | | | | \ / | | | | \ / | | | | \ / | | | | * \ / * | | | X * Y Y * Z | | \ * | | * / | | \ G | | &#292; / | | \ | | / | | \ | | / | | \ | | / | | \ | | / | | \| |/ | | o o | | | o-------------------------------------------------o Figure 2. Dyadic Relations G c X x Y and &#292; c Y x Z With the required spaces carefully swept out, the stage is set for the presentation of Tarski's trick, and the invocation of the following symbolic formula, claimed to be a definition of the relational composition ''P''&nbsp;&omicron;&nbsp;''Q'' of a pair of 2-adic relations ''P'',&nbsp;''Q''&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''X''. : '''Definition.''' ''P''&nbsp;&omicron;&nbsp;''Q'' = ''proj''₁₃ (''P''&nbsp;&times;&nbsp;''X''&nbsp;&cap;&nbsp;''X''&nbsp;&times;&nbsp;''Q''). To get this drift of this definition, one needs to understand that it's written within a school of thought that holds that all 2-adic relations are, 'without loss of generality', covered well enough, 'for all practical purposes', under the aegis of subsets of a suitable cartesian square, and thus of the form ''L''&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''X''. So, if one has started out with a 2-adic relation of the shape ''L''&nbsp;&sube;&nbsp;''U''&nbsp;&times;&nbsp;''V'', one merely lets ''X''&nbsp;=&nbsp;''U''&nbsp;&cup;&nbsp;''V'', trading in the initial ''L'' for a new ''L''&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''X'' as need be. The projection ''proj''₁₃ is just the projection of the cartesian cube ''X''&nbsp;&times;&nbsp;''X''&nbsp;&times;&nbsp;''X'' on the space of shape ''X''&nbsp;&times;&nbsp;''X'' that is spanned by the first and the third domains, but since they now have the same names and the same contents it is necessary to distinguish them by numbering their relational places. Finally, the notation of the cartesian product sign "&times;" is extended to signify two other products with respect to a 2-adic relation ''L''&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''X'' and a subset ''W''&nbsp;&sube;&nbsp;''X'', as follows: : '''Definition.''' ''L'' &times; ''W'' = {(''x'', ''y'', ''z'') &isin; ''X''³ : (''x'', ''y'') &isin; ''L'' &and; ''z'' &isin; ''W''}. : '''Definition.''' ''W'' &times; ''L'' = {(''x'', ''y'', ''z'') &isin; ''X''³ : ''x'' &isin; ''W'' &and; (''y'', ''z'') &isin; ''L''}. Applying these definitions to the case ''P'',&nbsp;''Q''&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''X'', the two 2-adic relations whose relational composition ''P''&nbsp;&omicron;&nbsp;''Q''&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''X'' is about to be defined, one finds: : ''P'' &times; ''X'' = {(''x'', ''y'', ''z'') &isin; ''X''³ : (''x'', ''y'') &isin; ''P'' &and; ''z'' &isin; ''X''}, : ''X'' &times; ''Q'' = {(''x'', ''y'', ''z'') &isin; ''X''³ : ''x'' &isin; ''X'' &and; (''y'', ''z'') &isin; ''Q''}. These are just the appropriate special cases of the tacit extensions already defined. : ''P'' &times; ''X'' = ''te''₁₂³(''P''), : ''X'' &times; ''Q'' = ''te''₂₃¹(''Q''). In summary, then, the expression: : ''proj''₁₃(''P'' &times; ''X'' &cap; ''X'' &times; ''Q'') is equivalent to the expression: : ''proj''₁₃(''te''₁₂³(''P'') &cap; ''te''₂₃¹(''Q'')) and this form is generalized — although, relative to one's school of thought, perhaps inessentially so — by the form that was given above as follows: : '''Definition.''' ''P''&nbsp;&omicron;&nbsp;''Q'' = ''proj''_''XZ''(''te''_''XY''^''Z''(''P'')&nbsp;&cap;&nbsp;''te''_''YZ''^''X''(''Q'')). Figure 3 presents a geometric picture of what is involved in formulating a definition of the 3-adic relation ''F''&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''Y''&nbsp;&times;&nbsp;''Z'' by way of a conjunction of the 2-adic relation ''G''&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''Y'' and the 2-adic relation ''H''&nbsp;&sube;&nbsp;''Y''&nbsp;&times;&nbsp;''Z'', as done for example by means of an expression of the following form: :* ''F''(''x'', ''y'', ''z'') = ''G''(''x'', ''y'') &and; ''H''(''y'', ''z''). o-------------------------------------------------o | | | o | | /|\ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | o o o | | |\ / \ /| | | | \ / F \ / | | | | \ / * \ / | | | | \ /*\ / | | | | / \//*\\/ \ | | | | / /\/ \/\ \ | | | |/ ///\ /\\\ \| | | o X /// Y \\\ Z o | | |\ \/// | \\\/ /| | | | \ /// | \\\ / | | | | \ ///\ | /\\\ / | | | | \ /// \ | / \\\ / | | | | \/// \ | / \\\/ | | | | /\/ \ | / \/\ | | | | *//\ \|/ /\\* | | | X */ Y o Y \* Z | | \ * | | * / | | \ G | | H / | | \ | | / | | \ | | / | | \ | | / | | \ | | / | | \| |/ | | o o | | | o-------------------------------------------------o Figure 3. Projections of F onto G and H To interpret the Figure, visualize the 3-adic relation ''F''&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''Y''&nbsp;&times;&nbsp;''Z'' as a body in ''XYZ''-space, while ''G'' is a figure in ''XY''-space and ''H'' is a figure in ''YZ''-space. The 2-adic '''projections''' that accompany a 3-adic relation over ''X'', ''Y'', ''Z'' are defined as follows: :* ''proj''_''XY''(''L'') = {(''x'', ''y'') &isin; ''X'' &times; ''Y'' : (&exist; ''z'' &isin; ''Z'') (''x'', ''y'', ''z'') &isin; ''L''}, :* ''proj''_''XZ''(''L'') = {(''x'', ''z'') &isin; ''X'' &times; ''Z'' : (&exist; ''y'' &isin; ''Y'') (''x'', ''y'', ''z'') &isin; ''L''}, :* ''proj''_''YZ''(''L'') = {(''y'', ''z'') &isin; ''Y'' &times; ''Z'' : (&exist; ''x'' &isin; ''X'') (''x'', ''y'', ''z'') &isin; ''L''}. For many purposes it suffices to indicate the 2-adic projections of a 3-adic relation ''L'' by means of the briefer equivalents listed here: :* ''L''_''XY'' = ''proj''_''XY''(''L''), :* ''L''_''XZ'' = ''proj''_''XZ''(''L''), :* ''L''_''YZ'' = ''proj''_''YZ''(''L''). In light of these definitions, ''proj''_''XY'' is a mapping from the set <font face=signature>L</font>_''XYZ'' of 3-adic relations over ''X'', ''Y'', ''Z'' to the set <font face=signature>L</font>_''XY'' of 2-adic relations over ''X'' and ''Y'', with similar relationships holding for the other projections. To formalize these relationships in a concise but explicit manner, it serves to add a few more definitions. The set <font face=signature>L</font>_''XYZ'', whose membership is just the 3-adic relations over ''X'', ''Y'', ''Z'', can be recognized as the set of all subsets of the cartesian product ''X''&nbsp;&times;&nbsp;''Y''&nbsp;&times;&nbsp;''Z'', also known as the '''power set''' of ''X''&nbsp;&times;&nbsp;''Y''&nbsp;&times;&nbsp;''Z'', and notated here as ''Pow''(''X''&nbsp;&times;&nbsp;''Y''&nbsp;&times;&nbsp;''Z''). :* <font face=signature>L</font>_''XYZ'' = {''L''&nbsp;:&nbsp;''L''&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''Y''&nbsp;&times;&nbsp;''Z''} = ''Pow''(''X''&nbsp;&times;&nbsp;''Y''&nbsp;&times;&nbsp;''Z''). Likewise, the power sets of the pairwise cartesian products encompass all of the 2-adic relations on pairs of distinct domains that can be chosen from {''X'', ''Y'', ''Z''}. :* <font face=signature>L</font>_''XY'' = {''L'' : ''L'' &sube; ''X'' &times; ''Y''} = ''Pow''(''X'' &times; ''Y''), :* <font face=signature>L</font>_''XZ'' = {''L'' : ''L'' &sube; ''X'' &times; ''Z''} = ''Pow''(''X'' &times; ''Z''), :* <font face=signature>L</font>_''YZ'' = {''L'' : ''L'' &sube; ''Y'' &times; ''Z''} = ''Pow''(''Y'' &times; ''Z''). In mathematics, the inverse relation corresponding to a projection map is usually called an '''extension'''. To avoid confusion with other senses of the word, however, it is probably best for the sake of this discussion to stick with the more specific term '''tacit extension'''. The ''tacit extensions'' ''te''_''XY''^''Z'', ''te''_''XZ''^''Y'', ''te''_''YZ''^''X'', of the 2-adic relations ''U''&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''Y'', ''V''&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''Z'', ''W''&nbsp;&sube;&nbsp;''Y''&nbsp;&times; ''Z'', respectively, are defined in the following way: :* ''te''_''XY''^''Z''(''U'') = {(''x'', ''y'', ''z'') : (''x'', ''y'') &isin; ''U''} :* ''te''_''XZ''^''Y''(''V'') = {(''x'', ''y'', ''z'') : (''x'', ''z'') &isin; ''V''} :* ''te''_''YZ''^''X''(''W'') = {(''x'', ''y'', ''z'') : (''y'', ''z'') &isin; ''W''} So long as the intended indices attaching to the tacit extensions can be gathered from context, it is usually clear enough to use the abbreviated forms, ''te''(''U''), ''te''(''V''), ''te''(''W''). The definition and illustration of relational composition presently under way makes use of the tacit extension of ''G''&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''Y'' to ''te''(''G'')&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''Y''&nbsp;&times;&nbsp;''Z'' and the tacit extension of ''H''&nbsp;&sube;&nbsp;''Y''&nbsp;&times;&nbsp;''Z'' to ''te''(''H'')&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''Y''&nbsp;&times; ''Z'', only. Geometric illustrations of ''te''(''G'') and ''te''(''H'') are afforded by Figures 4 and 5, respectively. o-------------------------------------------------o | | | o | | /|\ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | / | * \ | | o o ** o | | |\ / \*** /| | | | \ / *** / | | | | \ / ***\ / | | | | \ *** / | | | | / \*** / \ | | | | / *** / \ | | | |/ ***\ / \| | | o X /** Y Z o | | |\ \//* | / /| | | | \ /// | / / | | | | \ ///\ | / / | | | | \ /// \ | / / | | | | \/// \ | / / | | | | /\/ \ | / / | | | | *//\ \|/ / * | | | X */ Y o Y * Z | | \ * | | * / | | \ G | | H / | | \ | | / | | \ | | / | | \ | | / | | \ | | / | | \| |/ | | o o | | | o-------------------------------------------------o Figure 4. Tacit Extension of G to X x Y x Z <br> o-------------------------------------------------o | | | o | | /|\ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | / * | \ | | o ** o o | | |\ ***/ \ /| | | | \ *** \ / | | | | \ /*** \ / | | | | \ *** / | | | | / \ ***/ \ | | | | / \ *** \ | | | |/ \ /*** \| | | o X Y **\ Z o | | |\ \ | *\\/ /| | | | \ \ | \\\ / | | | | \ \ | /\\\ / | | | | \ \ | / \\\ / | | | | \ \ | / \\\/ | | | | \ \ | / \/\ | | | | * \ \|/ /\\* | | | X * Y o Y \* Z | | \ * | | * / | | \ G | | H / | | \ | | / | | \ | | / | | \ | | / | | \ | | / | | \| |/ | | o o | | | o-------------------------------------------------o Figure 5. Tacit Extension of H to X x Y x Z A geometric interpretation can now be given that fleshes out in graphic form the meaning of a formula like the following: :* ''F''(''x'', ''y'', ''z'') = ''G''(''x'', ''y'') &and; ''H''(''y'', ''z''). The conjunction that is indicated by "&and;" corresponds as usual to an intersection of two sets, however, in this case it is the intersection of the tacit extensions ''te''(''G'') and ''te''(''H''). o-------------------------------------------------o | | | o | | /|\ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | o o o | | |\ / \ /| | | | \ / F \ / | | | | \ / * \ / | | | | \ /*\ / | | | | / \//*\\/ \ | | | | / /\/ \/\ \ | | | |/ ///\ /\\\ \| | | o X /// Y \\\ Z o | | |\ \/// | \\\/ /| | | | \ /// | \\\ / | | | | \ ///\ | /\\\ / | | | | \ /// \ | / \\\ / | | | | \/// \ | / \\\/ | | | | /\/ \ | / \/\ | | | | *//\ \|/ /\\* | | | X */ Y o Y \* Z | | \ * | | * / | | \ G | | H / | | \ | | / | | \ | | / | | \ | | / | | \ | | / | | \| |/ | | o o | | | o-------------------------------------------------o Figure 6. F as the Intersection of te(G) and te(H) ==Algebraic construction== The transition from a geometric picture of relation composition to an algebraic formulation is accomplished through the introduction of coordinates, in other words, identifiable names for the objects that are related through the various forms of relations, 2-adic and 3-adic in the present case. Adding coordinates to the running Example produces the following Figure: o-------------------------------------------------o | | | o | | /|\ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | o o o | | |\ / \ /| | | | \ / F \ / | | | | \ / * \ / | | | | \ /*\ / | | | | / \//*\\/ \ | | | | / /\/ \/\ \ | | | |/ ///\ /\\\ \| | | o X /// Y \\\ Z o | | |\ 7\/// | \\\/7 /| | | | \ 6// | \\6 / | | | | \ //5\ | /5\\ / | | | | \ /// 4\ | /4 \\\ / | | | | \/// 3\ | /3 \\\/ | | | | G/\/ 2\ | /2 \/\H | | | | *//\ 1\|/1 /\\* | | | X *\ Y o Y /* Z | | 7\ *\\ |7 7| //* /7 | | 6\ |\\\|6 6|///| /6 | | 5\| \\@5 5@// |/5 | | 4@ \@4 4@/ @4 | | 3\ @3 3@ /3 | | 2\ |2 2| /2 | | 1\|1 1|/1 | | o o | | | o-------------------------------------------------o Figure 7. F as the Intersection of te(G) and te(H) Thinking of relations in operational terms is facilitated by using a variant notation for tuples and sets of tuples, namely, the ordered pair (''x'',&nbsp;''y'') is written ''x'':''y'', the ordered triple (''x'',&nbsp;''y'',&nbsp;''z'') is written ''x'':''y'':''z'', and so on, and a set of tuples is conceived as a logical-algebraic sum, which can be written out in the smaller finite cases in forms like ''a'':''b''&nbsp;+&nbsp;''b'':''c''&nbsp;+&nbsp;''c'':''d'' and so on. For example, translating the relations ''F''&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''Y''&nbsp;&times;&nbsp;''Z'', ''G''&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''Y'', ''H''&nbsp;&sube;&nbsp;''Y''&nbsp;&times;&nbsp;''Z'' into this notation produces the following summary of the data: {| cellpadding=8 style="text-align:center" | &nbsp; || ''F'' || = || 4:3:4 || + || 4:4:4 || + || 4:5:4 |- | &nbsp; || ''G'' || = || 4:3 || + || 4:4 || + || 4:5 |- | &nbsp; || ''H'' || = || 3:4 || + || 4:4 || + || 5:4 |} As often happens with abstract notations for functions and relations, the ''type information'', in this case, the fact that ''G'' and ''H'' live in different spaces, is left implicit in the context of use. Let us now verify that all of the proposed definitions, formulas, and other relationships check out against the concrete data of the current composition example. The ultimate goal is to develop a clearer picture of what is going on in the formula that expresses the relational composition of a couple of 2-adic relations in terms of the medial projection of the intersection of their tacit extensions: : ''G'' &omicron; ''H'' = ''proj''_''XZ''(''te''_''XY''^''Z''(''G'') &cap; ''te''_''YZ''^''X''(''H'')). Here is the big picture, with all of the pieces in place: o-------------------------------------------------o | | | o | | / \ | | / \ | | / \ | | / \ | | / \ | | / \ | | / G o H \ | | X * Z | | 7\ /|\ /7 | | 6\ / | \ /6 | | 5\ / | \ /5 | | 4@ | @4 | | 3\ | /3 | | 2\ | /2 | | 1\|/1 | | | | | | | | | | | /|\ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | o | o | | |\ /|\ /| | | | \ / F \ / | | | | \ / * \ / | | | | \ /*\ / | | | | / \//*\\/ \ | | | | / /\/ \/\ \ | | | |/ ///\ /\\\ \| | | o X /// Y \\\ Z o | | |\ \/// | \\\/ /| | | | \ /// | \\\ / | | | | \ ///\ | /\\\ / | | | | \ /// \ | / \\\ / | | | | \/// \ | / \\\/ | | | | G/\/ \ | / \/\H | | | | *//\ \|/ /\\* | | | X *\ Y o Y /* Z | | 7\ *\\ |7 7| //* /7 | | 6\ |\\\|6 6|///| /6 | | 5\| \\@5 5@// |/5 | | 4@ \@4 4@/ @4 | | 3\ @3 3@ /3 | | 2\ |2 2| /2 | | 1\|1 1|/1 | | o o | | | o-------------------------------------------------o Figure 8. G o H = proj_XZ (te(G) |^| te(H)) All that remains is to check the following collection of data and derivations against the situation represented in Figure 8. {| cellpadding=8 style="text-align:center" | &nbsp; || ''F'' || = || 4:3:4 || + || 4:4:4 || + || 4:5:4 |- | &nbsp; || ''G'' || = || 4:3 || + || 4:4 || + || 4:5 |- | &nbsp; || ''H'' || = || 3:4 || + || 4:4 || + || 5:4 |} {| cellpadding=8 style="text-align:center" | &nbsp; || ''G'' &omicron; ''H'' || = || (4:3 + 4:4 + 4:5)(3:4 + 4:4 + 5:4) |- | &nbsp; || &nbsp; || = || 4:4 |} {| cellpadding=8 style="text-align:center" | &nbsp; || ''te''(''G'') || = || ''te''_''XY''^''Z''(''G'') |- | &nbsp; || &nbsp; || = || &sum;_''z''=1..7 (4:3:''z'' + 4:4:''z'' + 4:5:''z'') |} {| cellpadding=8 style="text-align:center" | &nbsp; || ''te''(''G'') || = || 4:3:1 || + || 4:4:1 || + || 4:5:1 || + |- | &nbsp; || &nbsp; || &nbsp; || 4:3:2 || + || 4:4:2 || + || 4:5:2 || + |- | &nbsp; || &nbsp; || &nbsp; || 4:3:3 || + || 4:4:3 || + || 4:5:3 || + |- | &nbsp; || &nbsp; || &nbsp; || 4:3:4 || + || 4:4:4 || + || 4:5:4 || + |- | &nbsp; || &nbsp; || &nbsp; || 4:3:5 || + || 4:4:5 || + || 4:5:5 || + |- | &nbsp; || &nbsp; || &nbsp; || 4:3:6 || + || 4:4:6 || + || 4:5:6 || + |- | &nbsp; || &nbsp; || &nbsp; || 4:3:7 || + || 4:4:7 || + || 4:5:7 |} {| cellpadding=8 style="text-align:center" | &nbsp; || ''te''(''H'') || = || ''te''_''YZ''^''X''(''H'') |- | &nbsp; || &nbsp; || = || &sum;_''x''=1..7 (''x'':3:4 + ''x'':4:4 + ''x'':5:4) |} {| cellpadding=8 style="text-align:center" | &nbsp; || ''te''(''H'') || = || 1:3:4 || + || 1:4:4 || + || 1:5:4 || + |- | &nbsp; || &nbsp; || &nbsp; || 2:3:4 || + || 2:4:4 || + || 2:5:4 || + |- | &nbsp; || &nbsp; || &nbsp; || 3:3:4 || + || 3:4:4 || + || 3:5:4 || + |- | &nbsp; || &nbsp; || &nbsp; || 4:3:4 || + || 4:4:4 || + || 4:5:4 || + |- | &nbsp; || &nbsp; || &nbsp; || 5:3:4 || + || 5:4:4 || + || 5:5:4 || + |- | &nbsp; || &nbsp; || &nbsp; || 6:3:4 || + || 6:4:4 || + || 6:5:4 || + |- | &nbsp; || &nbsp; || &nbsp; || 7:3:4 || + || 7:4:4 || + || 7:5:4 |} {| |- | align="center" | ''te''(''G'') &cap; ''te''(''H'') | = | 4:3:4 + 4:4:4 + 4:5:4 |- | align="center" | ''G'' &omicron; ''H'' | = | ''proj''_''XZ''(''te''(''G'') &cap; ''te''(''H'')) |- | &nbsp; | = | ''proj''_''XZ''(4:3:4 + 4:4:4 + 4:5:4) |- | &nbsp; | = | 4:4 |} ==Matrix representation== We have it within our reach to pick up another way of representing 2-adic relations, namely, the representation as [[logical matrix|logical matrices]], and also to grasp the analogy between relational composition and ordinary [[matrix multiplication]] as it appears in [[linear algebra]]. First of all, while we still have the data of a very simple concrete case in mind, let us reflect on what we did in our last Example in order to find the composition ''G''&nbsp;&omicron;&nbsp;''H'' of the 2-adic relations ''G''&nbsp;and&nbsp;''H''. Here is the setup that we had before: {| cellpadding=8 style="text-align:center" | &nbsp; || ''X'' || = || {1, 2, 3, 4, 5, 6, 7} |} {| cellpadding=8 style="text-align:center" | &nbsp; || ''G'' || = || 4:3 || + || 4:4 || + || 4:5 || &sube; || ''X'' &times; ''X'' |- | &nbsp; || ''H'' || = || 3:4 || + || 4:4 || + || 5:4 || &sube; || ''X'' &times; ''X '' |} Let us recall the rule for finding the relational composition of a pair of 2-adic relations. Given the 2-adic relations ''P''&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''Y'', ''Q''&nbsp;&sube;&nbsp;''Y''&nbsp;&times;&nbsp;''Z'', the relational composition of ''P'' and ''Q'', in that order, is written as ''P''&nbsp;&omicron;&nbsp;''Q'' or more simply as ''PQ'' and obtained as follows: To compute ''PQ'', in general, where ''P'' and ''Q'' are 2-adic relations, simply multiply out the two sums in the ordinary distributive algebraic way, but subject to the following rule for finding the product of two elementary relations of shapes ''a'':''b'' and ''c'':''d''. {| cellpadding=8 style="text-align:center" | &nbsp; || (a:b)(c:d) || = || (a:d) || if b = c |- | &nbsp; || (a:b)(c:d) || = || 0 || otherwise |} To find the relational composition ''G'' &omicron; ''H'', one may begin by writing it as a quasi-algebraic product: {| cellpadding=8 style="text-align:center" | &nbsp; || ''G'' &omicron; ''H'' || = || (4:3 + 4:4 + 4:5)(3:4 + 4:4 + 5:4) |} Multiplying this out in accord with the applicable form of distributive law one obtains the following expansion: {| cellpadding=8 style="text-align:center" | &nbsp; || ''G'' &omicron; ''H'' || = || (4:3)(3:4) || + || (4:3)(4:4) || + || (4:3)(5:4) || + |- | &nbsp; || &nbsp; || &nbsp; || (4:4)(3:4) || + || (4:4)(4:4) || + || (4:4)(5:4) || + |- | &nbsp; || &nbsp; || &nbsp; ||(4:5)(3:4) || + || (4:5)(4:4) || + || (4:5)(5:4) |} Applying the rule that determines the product of elementary relations produces the following array: {| cellpadding=8 style="text-align:center" | &nbsp; || ''G'' &omicron; ''H'' || = || (4:4) || + || 0 || + || 0 || + |- | &nbsp; || &nbsp; || &nbsp; || 0 || + || (4:4) || + || 0 || + |- | &nbsp; || &nbsp; || &nbsp; || 0 || + || 0 || + || (4:4) |} Since the plus sign in this context represents an operation of logical disjunction or set-theoretic aggregation, all of the positive multiplicites count as one, and this gives the ultimate result: {| cellpadding=8 style="text-align:center" | &nbsp; || ''G'' &omicron; ''H'' || = || (4:4) |} With an eye toward extracting a general formula for relation composition, viewed here on analogy to algebraic multiplication, let us examine what we did in multiplying the 2-adic relations ''G'' and ''H'' together to obtain their relational composite ''G'' o ''H''. Given the space ''X'' = {1, 2, 3, 4, 5, 6, 7}, whose cardinality |''X''| is 7, there are |''X'' &times; ''X''| = |''X''| &middot; |''X''| = 7 &middot; 7 = 49 elementary relations of the form ''i'':''j'', where ''i'' and ''j'' range over the space ''X''. Although they might be organized in many different ways, it is convenient to regard the collection of elementary relations as being arranged in a lexicographic block of the following form: {| cellpadding=8 style="text-align:center" | &nbsp; || 1:1 || 1:2 || 1:3 || 1:4 || 1:5 || 1:6 || 1:7 |- | &nbsp; || 2:1 || 2:2 || 2:3 || 2:4 || 2:5 || 2:6 || 2:7 |- | &nbsp; || 3:1 || 3:2 || 3:3 || 3:4 || 3:5 || 3:6 || 3:7 |- | &nbsp; || 4:1 || 4:2 || 4:3 || 4:4 || 4:5 || 4:6 || 4:7 |- | &nbsp; || 5:1 || 5:2 || 5:3 || 5:4 || 5:5 || 5:6 || 5:7 |- | &nbsp; || 6:1 || 6:2 || 6:3 || 6:4 || 6:5 || 6:6 || 6:7 |- | &nbsp; || 7:1 || 7:2 || 7:3 || 7:4 || 7:5 || 7:6 || 7:7 |} The relations ''G'' and ''H'' may then be regarded as logical sums of the following forms: {| cellpadding=8 style="text-align:center" | &nbsp; || ''G'' || = || &sum;_''ij'' ''G''_''ij''(''i'':''j'') |- | &nbsp; || ''H'' || = || &sum;_''ij'' ''H''_''ij''(''i'':''j'') |} The notation &sum;_''ij'' indicates a logical sum over the collection of elementary relations ''i'':''j'', while the factors ''G''_''ij'' and ''H''_''ij'' are values in the boolean domain '''B''' = {0,&nbsp;1} that are known as the ''coefficients'' of the relations ''G'' and ''H'', respectively, with regard to the corresponding elementary relations ''i'':''j''. In general, for a 2-adic relation ''L'', the coefficient ''L''_''ij'' of the elementary relation ''i'':''j'' in the relation ''L'' will be 0 or 1, respectively, as ''i'':''j'' is excluded from or included in ''L''. With these conventions in place, the expansions of ''G'' and ''H'' may be written out as follows: {| cellpadding=6 style="text-align:center" | &nbsp; || ''G'' || = || 4:3 || + || 4:4 || + || 4:5 || = |} {| | style="width:20px" | &nbsp; | 0(1:1) + | 0(1:2) + | 0(1:3) + | 0(1:4) + | 0(1:5) + | 0(1:6) + | 0(1:7) + |- | &nbsp; | 0(2:1) + | 0(2:2) + | 0(2:3) + | 0(2:4) + | 0(2:5) + | 0(2:6) + | 0(2:7) + |- | &nbsp; | 0(3:1) + | 0(3:2) + | 0(3:3) + | 0(3:4) + | 0(3:5) + | 0(3:6) + | 0(3:7) + |- | &nbsp; | 0(4:1) + | 0(4:2) + | '''1'''(4:3) + | '''1'''(4:4) + | '''1'''(4:5) + | 0(4:6) + | 0(4:7) + |- | &nbsp; | 0(5:1) + | 0(5:2) + | 0(5:3) + | 0(5:4) + | 0(5:5) + | 0(5:6) + | 0(5:7) + |- | &nbsp; | 0(6:1) + | 0(6:2) + | 0(6:3) + | 0(6:4) + | 0(6:5) + | 0(6:6) + | 0(6:7) + |- | &nbsp; | 0(7:1) + | 0(7:2) + | 0(7:3) + | 0(7:4) + | 0(7:5) + | 0(7:6) + | 0(7:7) |} <br> {| cellpadding=6 style="text-align:center" | &nbsp; || ''H'' || = || 3:4 || + || 4:4 || + || 5:4 || = |} {| | style="width:20px" | &nbsp; | 0(1:1) + | 0(1:2) + | 0(1:3) + | 0(1:4) + | 0(1:5) + | 0(1:6) + | 0(1:7) + |- | &nbsp; | 0(2:1) + | 0(2:2) + | 0(2:3) + | 0(2:4) + | 0(2:5) + | 0(2:6) + | 0(2:7) + |- | &nbsp; | 0(3:1) + | 0(3:2) + | 0(3:3) + | '''1'''(3:4) + | 0(3:5) + | 0(3:6) + | 0(3:7) + |- | &nbsp; | 0(4:1) + | 0(4:2) + | 0(4:3) + | '''1'''(4:4) + | 0(4:5) + | 0(4:6) + | 0(4:7) + |- | &nbsp; | 0(5:1) + | 0(5:2) + | 0(5:3) + | '''1'''(5:4) + | 0(5:5) + | 0(5:6) + | 0(5:7) + |- | &nbsp; | 0(6:1) + | 0(6:2) + | 0(6:3) + | 0(6:4) + | 0(6:5) + | 0(6:6) + | 0(6:7) + |- | &nbsp; | 0(7:1) + | 0(7:2) + | 0(7:3) + | 0(7:4) + | 0(7:5) + | 0(7:6) + | 0(7:7) |} Stripping down to the bare essentials, one obtains the following matrices of coefficients for the relations ''G''&nbsp;and&nbsp;''H''. {| style="text-align:center; width=30%" | style="width:20px" | &nbsp; || ''G'' || = |- | &nbsp; || 0 || 0 || 0 || 0 || 0 || 0 || 0 |- | &nbsp; || 0 || 0 || 0 || 0 || 0 || 0 || 0 |- | &nbsp; || 0 || 0 || 0 || 0 || 0 || 0 || 0 |- | &nbsp; || 0 || 0 || 1 || 1 || 1 || 0 || 0 |- | &nbsp; || 0 || 0 || 0 || 0 || 0 || 0 || 0 |- | &nbsp; || 0 || 0 || 0 || 0 || 0 || 0 || 0 |- | &nbsp; || 0 || 0 || 0 || 0 || 0 || 0 || 0 |} <br> {| style="text-align:center; width=30%" | style="width:20px" | &nbsp; || ''H'' || = |- | &nbsp; || 0 || 0 || 0 || 0 || 0 || 0 || 0 |- | &nbsp; || 0 || 0 || 0 || 0 || 0 || 0 || 0 |- | &nbsp; || 0 || 0 || 0 || 1 || 0 || 0 || 0 |- | &nbsp; || 0 || 0 || 0 || 1 || 0 || 0 || 0 |- | &nbsp; || 0 || 0 || 0 || 1 || 0 || 0 || 0 |- | &nbsp; || 0 || 0 || 0 || 0 || 0 || 0 || 0 |- | &nbsp; || 0 || 0 || 0 || 0 || 0 || 0 || 0 |} These are the logical matrix representations of the 2-adic relations ''G''&nbsp;and&nbsp;''H''. If the 2-adic relations ''G'' and ''H'' are viewed as logical sums, then their relational composition ''G''&nbsp;&omicron;&nbsp;''H'' can be regarded as a product of sums, a fact that can be indicated as follows: : ''G''&nbsp;&omicron;&nbsp;''H'' = (&sum;_''ij'' ''G''_''ij''(''i'':''j''))(&sum;_''ij'' ''H''_''ij''(''i'':''j'')). The composite relation ''G''&nbsp;&omicron;&nbsp;''H'' is itself a 2-adic relation over the same space ''X'', in other words, ''G''&nbsp;&omicron;&nbsp;''H''&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''X'', and this means that ''G''&nbsp;&omicron;&nbsp;''H'' must be amenable to being written as a logical sum of the following form: : ''G''&nbsp;&omicron;&nbsp;''H'' = &sum;_''ij'' (''G''&nbsp;&omicron;&nbsp;''H'')_''ij''(''i'':''j''). In this formula, (''G''&nbsp;&omicron;&nbsp;''H'')_''ij'' is the coefficient of ''G''&nbsp;&omicron;&nbsp;''H'' with respect to the elementary relation ''i'':''j''. One of the best ways to reason out what ''G''&nbsp;&omicron;&nbsp;''H'' should be is to ask oneself what its coefficient (''G''&nbsp;&omicron;&nbsp;''H'')_''ij'' should be for each of the elementary relations ''i'':''j'' in turn. So let us pose the question: : (''G''&nbsp;&omicron;&nbsp;''H'')_''ij'' = ? In order to answer this question, it helps to realize that the indicated product given above can be written in the following equivalent form: : ''G''&nbsp;&omicron;&nbsp;''H'' = (&sum;_''ik'' ''G''_''ik''(''i'':''k''))(&sum;_''kj'' ''H''_''kj''(''k'':''j'')). A moment's thought will tell us that (''G''&nbsp;&omicron;&nbsp;''H'')_''ij''&nbsp;=&nbsp;1 if and only if there is an element ''k'' in ''X'' such that ''G''_''ik''&nbsp;=&nbsp;1 and ''H''_''kj''&nbsp;=&nbsp;1. Consequently, we have the result: : (''G''&nbsp;&omicron;&nbsp;''H'')_''ij'' = &sum;_k ''G''_''ik''''H''_''kj''. This follows from the properties of boolean arithmetic, specifically, from the fact that the product ''G''_''ik''''H''_''kj'' is 1 if and only if both ''G''_''ik'' and ''H''_''kj'' are 1, and from the fact that &sum;_''k'' ''F''_''k'' is equal to 1 just in case some ''F''_''k'' is 1. All that remains in order to obtain a computational formula for the relational composite ''G''&nbsp;&omicron;&nbsp;''H'' of the 2-adic relations ''G'' and ''H'' is to collect the coefficients (''G''&nbsp;&omicron;&nbsp;''H'')_''ij'' over the appropriate basis of elementary relations ''i'':''j'', as ''i'' and ''j'' range through ''X''. : ''G''&nbsp;&omicron;&nbsp;''H'' = &sum;_''ij'' (''G''&nbsp;&omicron;&nbsp;''H'')_''ij''(''i'':''j'') = &sum;_''ij''(&sum;_''k'' ''G''_''ik''''H''_''kj'')(''i'':''j''). This is the logical analogue of matrix multiplication in linear algebra, the difference in the logical setting being that all of the operations performed on coefficients take place in a system of boolean arithmetic where summation corresponds to logical disjunction and multiplication corresponds to logical conjunction. By way of disentangling this formula, one may notice that the form &sum;_''k'' (''G''_''ik''''H''_''kj'') is what is usually called a "scalar product". In this case it is the scalar product of the ''i''^th row of ''G'' with the ''j''^th column of ''H''. To make this statement more concrete, let us go back to the particular examples of ''G'' and ''H'' that we came in with: {| style="text-align:center; width=30%" | style="width:20px" | &nbsp; || ''G'' || = |- | &nbsp; || 0 || 0 || 0 || 0 || 0 || 0 || 0 |- | &nbsp; || 0 || 0 || 0 || 0 || 0 || 0 || 0 |- | &nbsp; || 0 || 0 || 0 || 0 || 0 || 0 || 0 |- | &nbsp; || 0 || 0 || 1 || 1 || 1 || 0 || 0 |- | &nbsp; || 0 || 0 || 0 || 0 || 0 || 0 || 0 |- | &nbsp; || 0 || 0 || 0 || 0 || 0 || 0 || 0 |- | &nbsp; || 0 || 0 || 0 || 0 || 0 || 0 || 0 |} <br> {| style="text-align:center; width=30%" | style="width:20px" | &nbsp; || ''H'' || = |- | &nbsp; || 0 || 0 || 0 || 0 || 0 || 0 || 0 |- | &nbsp; || 0 || 0 || 0 || 0 || 0 || 0 || 0 |- | &nbsp; || 0 || 0 || 0 || 1 || 0 || 0 || 0 |- | &nbsp; || 0 || 0 || 0 || 1 || 0 || 0 || 0 |- | &nbsp; || 0 || 0 || 0 || 1 || 0 || 0 || 0 |- | &nbsp; || 0 || 0 || 0 || 0 || 0 || 0 || 0 |- | &nbsp; || 0 || 0 || 0 || 0 || 0 || 0 || 0 |} The formula for computing ''G''&nbsp;&omicron;&nbsp;''H'' says the following: {| cellpadding="2" |- | style="width:20px" | &nbsp; | align="center" | (''G''&nbsp;&omicron;&nbsp;''H'')_''ij'' | &nbsp; |- | &nbsp; | align="center" | = | the ''ij''^th entry in the matrix representation for ''G''&nbsp;&omicron;&nbsp;''H'' |- | &nbsp; | align="center" | = | the entry in the ''i''^th row and the ''j''^th column of ''G''&nbsp;&omicron;&nbsp;''H'' |- | &nbsp; | align="center" | = | the scalar product of the ''i''^th row of ''G'' with the ''j''^th column of ''H'' |- | &nbsp; | align="center" | = | &sum;_''k'' ''G''_''ik''''H''_''kj'' |} As it happens, it is possible to make exceedingly light work of this example, since there is only one row of ''G'' and one column of ''H'' that are not all zeroes. Taking the scalar product, in a logical way, of the fourth row of ''G'' with the fourth column of ''H'' produces the sole non-zero entry for the matrix of ''G''&nbsp;&omicron;&nbsp;''H''. {| cellpadding="2px" style="text-align:center" | style="width:20px" | &nbsp; | ''G''&nbsp;&omicron;&nbsp;''H'' | = |} {| style="text-align:center; width=30%" | style="width:20px" | &nbsp; | 0 || 0 || 0 || 0 || 0 || 0 || 0 |- | &nbsp; | 0 || 0 || 0 || 0 || 0 || 0 || 0 |- | &nbsp; | 0 || 0 || 0 || 0 || 0 || 0 || 0 |- | &nbsp; | 0 || 0 || 0 || 1 || 0 || 0 || 0 |- | &nbsp; | 0 || 0 || 0 || 0 || 0 || 0 || 0 |- | &nbsp; | 0 || 0 || 0 || 0 || 0 || 0 || 0 |- | &nbsp; | 0 || 0 || 0 || 0 || 0 || 0 || 0 |} ==Graph-theoretic picture== There is another form of representation for 2-adic relations that is useful to keep in mind, especially for its ability to render the logic of many complex formulas almost instantly understandable to the mind's eye. This is the representation in terms of ''[[bipartite graph]]s'', or ''bigraphs'' for short. Here is what ''G'' and ''H'' look like in the bigraph picture: o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | /|\ | | / | \ G | | / | \ | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 9. G = 4:3 + 4:4 + 4:5 o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | \ | / | | \ | / H | | \|/ | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 10. H = 3:4 + 4:4 + 5:4 These graphs may be read to say: :* ''G'' puts 4 in relation to 3, 4, 5. :* ''H'' puts 3, 4, 5 in relation to 4. To form the composite relation ''G''&nbsp;&omicron;&nbsp;''H'', one simply follows the bigraph for ''G'' by the bigraph for ''H'', here arranging the bigraphs in order down the page, and then treats any non-empty set of paths of length two between two nodes as being equivalent to a single directed edge between those nodes in the composite bigraph for ''G''&nbsp;&omicron;&nbsp;''H''. Here's how it looks in pictures: o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | /|\ | | / | \ G | | / | \ | | o o o o o o o X | | \ | / | | \ | / H | | \|/ | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 11. G Followed By H o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | | | | | G o H | | | | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 12. G Composed With H Once again we find that ''G''&nbsp;&omicron;&nbsp;''H'' = 4:4. We have now seen three different representations of 2-adic relations. If one has a strong preference for letters, or numbers, or pictures, then one may be tempted to take one or another of these as being canonical, but each of them will be found to have its peculiar advantages and disadvantages in any given application, and the maximum advantage is therefore approached by keeping all three of them in mind. To see the promised utility of the bigraph picture of 2-adic relations, let us devise a slightly more complex example of a composition problem, and use it to illustrate the logic of the matrix multiplication formula. Keeping to the same space ''X'' = {1, 2, 3, 4, 5, 6, 7}, define the 2-adic relations ''M'',&nbsp;''N''&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''X'' as follows: {| cellpadding="2px" style="text-align:center" | style="width:20px" | &nbsp; | ''M'' | &nbsp; | = | &nbsp; | 2:1 | + | 2:2 | + | 2:3 | + | 4:3 | + | 4:4 | + | 4:5 | + | 6:5 | + | 6:6 | + | 6:7 |- | &nbsp; | ''N'' | &nbsp; | = | &nbsp; | 1:1 | + | 2:1 | + | 3:3 | + | 4:3 | &nbsp; | + | &nbsp; | 4:5 | + | 5:5 | + | 6:7 | + | 7:7 |} Here are the bigraph pictures: o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | /|\ /|\ /|\ | | / | \ / | \ / | \ M | | / | \ / | \ / | \ | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 13. Dyadic Relation M o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | | / | / \ | \ | | | | / | / \ | \ | N | | |/ |/ \| \| | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 14. Dyadic Relation N To form the composite relation ''M''&nbsp;&omicron;&nbsp;''N'', one simply follows the bigraph for ''M'' by the bigraph for ''N'', here arranging the bigraphs in order down the page, and then counts any non-empty set of paths of length two between two nodes as being equivalent to a single directed edge between those two nodes in the composite bigraph for ''M''&nbsp;&omicron;&nbsp;''N''. Here's how it looks in pictures: o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | /|\ /|\ /|\ | | / | \ / | \ / | \ M | | / | \ / | \ / | \ | | o o o o o o o X | | | / | / \ | \ | | | | / | / \ | \ | N | | |/ |/ \| \| | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 15. M Followed By N o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | / \ / \ / \ | | / \ / \ / \ M o N | | / \ / \ / \ | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 16. M Composed With N Let us hark back to that mysterious matrix multiplication formula, and see how it appears in the light of the bigraph representation. The coefficient of the composition ''M''&nbsp;&omicron;&nbsp;''N'' between ''i'' and ''j'' in ''X'' is given as follows: : (''M''&nbsp;&omicron;&nbsp;''N'')_''ij'' = &sum;_''k''(''M''_''ik''''N''_''kj'') Graphically interpreted, this is a ''sum over paths''. Starting at the node ''i'', ''M''_''ik'' being 1 indicates that there is an edge in the bigraph of ''M'' from node ''i'' to node ''k'', and ''N''_''kj'' being 1 indicates that there is an edge in the bigraph of ''N'' from node ''k'' to node ''j''. So the &sum;_''k'' ranges over all possible intermediaries ''k'', ascending from 0 to 1 just as soon as there happens to be some path of length two between nodes ''i'' and ''j''. It is instructive at this point to compute the other possible composition that can be formed from ''M'' and ''N'', namely, the composition ''N''&nbsp;&omicron;&nbsp;''M'', that takes ''M'' and ''N'' in the opposite order. Here is the graphic computation: o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | | / | / \ | \ | | | | / | / \ | \ | N | | |/ |/ \| \| | | o o o o o o o X | | /|\ /|\ /|\ | | / | \ / | \ / | \ M | | / | \ / | \ / | \ | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 17. N Followed By M o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | | | N o M | | | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 18. N Composed With M In sum, ''N''&nbsp;&omicron;&nbsp;''M'' = 0. This example affords sufficient evidence that relational composition, just like its kindred, matrix multiplication, is a ''[[non-commutative]]'' algebraic operation. ==References== * [[Stanislaw Marcin Ulam|Ulam, S.M.]] and [[Al Bednarek|Bednarek, A.R.]], "On the Theory of Relational Structures and Schemata for Parallel Computation" (1977), pp. 477-508 in A.R. Bednarek and Françoise Ulam (eds.), ''Analogies Between Analogies: The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators'', University of California Press, Berkeley, CA, 1990. ==Bibliography== * [[Mathematical Society of Japan]], ''Encyclopedic Dictionary of Mathematics'', 2nd edition, 2 vols., Kiyosi Itô (ed.), MIT Press, Cambridge, MA, 1993. * Mili, A., Desharnais, J., Mili, F., with Frappier, M., ''Computer Program Construction'', Oxford University Press, New York, NY, 1994. — Introduction to Tarskian relation theory and its applications within the relational programming paradigm. * [[Stanislaw Marcin Ulam|Ulam, S.M.]], ''Analogies Between Analogies: The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators'', A.R. Bednarek and Françoise Ulam (eds.), University of California Press, Berkeley, CA, 1990. ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://mywikibiz.com/Relation_composition Relation Composition @ MyWikiBiz] * [http://mathweb.org/wiki/Relation_composition Relation Composition @ MathWeb Wiki] * [http://netknowledge.org/wiki/Relation_composition Relation Composition @ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Relation_composition Relation Composition @ OER Commons] {{col-break}} * [http://p2pfoundation.net/Relation_Composition Relation Composition @ P2P Foundation] * [http://semanticweb.org/wiki/Relation_composition Relation Composition @ SemanticWeb] * [http://ref.subwiki.org/wiki/Relation_composition Relation Composition @ Subject Wikis] * [http://beta.wikiversity.org/wiki/Relation_composition Relation Composition @ Wikiversity Beta] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;] * [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;] ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Relation_composition Relation Composition], [http://mywikibiz.com/ MyWikiBiz] * [http://mathweb.org/wiki/Relation_composition Relation Composition], [http://mathweb.org/wiki/ MathWeb Wiki] * [http://netknowledge.org/wiki/Relation_composition Relation Composition], [http://netknowledge.org/ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Relation_composition Relation Composition], [http://wiki.oercommons.org/ OER Commons] * [http://p2pfoundation.net/Relation_Composition Relation Composition], [http://p2pfoundation.net/ P2P Foundation] {{col-break}} * [http://semanticweb.org/wiki/Relation_composition Relation Composition], [http://semanticweb.org/ Semantic Web] * [http://planetmath.org/encyclopedia/RelationComposition2.html Relation Composition], [http://planetmath.org/ PlanetMath] * [http://getwiki.net/-Relational_Composition Relation Composition], [http://getwiki.net/ GetWiki] * [http://wikinfo.org/index.php/Relation_composition Relation Composition], [http://wikinfo.org/ Wikinfo] * [http://en.wikipedia.org/w/index.php?title=Relation_composition&oldid=43467878 Relation Composition], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Database Theory]] [[Category:Discrete Mathematics]] [[Category:Formal Sciences]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Relation Theory]] [[Category:Set Theory]] 0857d02757978612e74d236d78faab9fd897740e 0 336 558 2010-06-22T16:54:10Z Jon Awbrey 3 + article wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. In [[logic]] and [[mathematics]], '''relation construction''' and '''relational constructibility''' have to do with the ways that one [[relation (mathematics)|relation]] is determined by an [[indexed family]] or a [[sequence]] of other relations, called the ''relation dataset''. The relation in the focus of consideration is called the ''faciendum''. The relation dataset typically consists of a specified relation over sets of relations, called the ''constructor'', the ''factor'', or the ''method of construction'', plus a specified set of other relations, called the ''faciens'', the ''ingredients'', or the ''makings''. [[Relation composition]] and [[relation reduction]] are special cases of relation constructions. ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://mywikibiz.com/Relation_construction Relation Construction @ MyWikiBiz] * [http://mathweb.org/wiki/Relation_construction Relation Construction @ MathWeb Wiki] * [http://netknowledge.org/wiki/Relation_construction Relation Construction @ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Relation_construction Relation Construction @ OER Commons] {{col-break}} * [http://p2pfoundation.net/Relation_Construction Relation Construction @ P2P Foundation] * [http://semanticweb.org/wiki/Relation_construction Relation Construction @ SemanticWeb] * [http://ref.subwiki.org/wiki/Relation_construction Relation Construction @ Subject Wikis] * [http://beta.wikiversity.org/wiki/Relation_construction Relation Construction @ Wikiversity Beta] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;] * [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;] ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Relation_construction Relation Construction], [http://mywikibiz.com/ MyWikiBiz] * [http://mathweb.org/wiki/Relation_construction Relation Construction], [http://mathweb.org/wiki/ MathWeb Wiki] * [http://netknowledge.org/wiki/Relation_construction Relation Construction], [http://netknowledge.org/ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Relation_construction Relation Construction], [http://wiki.oercommons.org/ OER Commons] * [http://p2pfoundation.net/Relation_Construction Relation Construction], [http://p2pfoundation.net/ P2P Foundation] {{col-break}} * [http://semanticweb.org/wiki/Relation_construction Relation Construction], [http://semanticweb.org/ Semantic Web] * [http://planetmath.org/encyclopedia/RelationConstruction.html Relation Construction], [http://planetmath.org/ PlanetMath] * [http://getwiki.net/-Relational_Construction Relation Construction], [http://getwiki.net/ GetWiki] * [http://wikinfo.org/index.php/Relation_construction Relation Construction], [http://wikinfo.org/ Wikinfo] * [http://en.wikipedia.org/w/index.php?title=Relation_construction&oldid=39070184 Relation Construction], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Database Theory]] [[Category:Discrete Mathematics]] [[Category:Formal Sciences]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Relation Theory]] [[Category:Set Theory]] 6fffd4aa59c74676455605a95f00435915894db1 0 337 559 2010-06-22T17:38:25Z Jon Awbrey 3 + article wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. In [[logic]] and [[mathematics]], '''relation reduction''' and '''relational reducibility''' have to do with the extent to which a given [[relation (mathematics)|relation]] is determined by a set of other relations, called the ''relation dataset''. The relation under examination is called the ''reductandum''. The relation dataset typically consists of a specified relation over sets of relations, called the ''reducer'', the ''method of reduction'', or the ''relational step'', plus a set of other relations, called the ''reduciens'' or the ''relational base'', each of which is properly simpler in a specified way than relation under examination. A question of relation reduction or relational reducibility is sometimes posed as a question of '''relation reconstruction''' or '''relational reconstructibility''', since a useful way of stating the question is to ask whether the reductandum can be reconstructed from the reduciens. See [[Humpty Dumpty]]. A relation that is not uniquely determined by a particular relation dataset is said to be ''irreducible'' in just that respect. A relation that is not uniquely determined by any relation dataset in a particular class of relation datasets is said to be ''irreducible'' in respect of that class. ==Discussion== The main thing that keeps the general problem of relational reducibility from being fully well-defined is that one would have to survey all of the conceivable ways of "getting new relations from old" in order to say precisely what is meant by the claim that the relation <math>L\!</math> is reducible to the set of relations <math>\{ L_j : j \in J \}.</math> This amounts to claiming one can be given a set of ''properly simpler'' relations <math>L_j\!</math> for values <math>j\!</math> in a given index set <math>J\!</math> and that this collection of data would suffice to fix the original relation <math>L\!</math> that one is seeking to analyze, determine, specify, or synthesize. In practice, however, apposite discussion of a particular application typically settles on either one of two different notions of reducibility as capturing the pertinent issues, namely: # Reduction under composition. # Reduction under projections. As it happens, there is an interesting relationship between these two notions of reducibility, the implications of which may be taken up partly in parallel with the discussion of the basic concepts. ==Projective reducibility of relations== It is convenient to begin with the ''projective reduction'' of relations, partly because this type of reduction is simpler and more intuitive (in the visual sense), but also because a number of conceptual tools that are needed in any case arise quite naturally in the projective setting. The work of intuiting how projections operate on multidimensional relations is often facilitated by keeping in mind the following sort of geometric image: * Picture a ''k''-adic relation ''L'' as a body that resides in a ''k''-dimensional space ''X''. If the domains of the relation ''L'' are ''X''₁, …, ''X''_''k'' , then the ''extension'' of the relation ''L'' is a subset of the cartesian product ''X'' = ''X''₁ × … × ''X''_''k'' . In this setting, the interval ''K'' = [1, ''k''] = {1, …, ''k''} is called the ''[[index set]]'' of the ''[[indexed family]]'' of sets ''X''₁, …, ''X''_''k'' . For any subset ''F'' of the index set ''K'', there is the corresponding subfamily of sets, {''X''_''j''&nbsp;:&nbsp;''j''&nbsp;&isin;&nbsp;''F''&nbsp;}, and there is the corresponding cartesian product over this subfamily, notated and defined as ''X''_''F'' = <font size="+2">&Pi;</font>_{''j''&nbsp;&isin;&nbsp;''F''}&nbsp;''X''_''j''. For any point ''x'' in ''X'', the ''projection'' of ''x'' on the subspace ''X''_''F'' is notated as proj_''F''(''x''). More generally, for any relation ''L'' &#8838; ''X'', the projection of ''L'' on the subspace ''X''_''F'' is written as proj_''F''(''L''), or still more simply, as proj_''F'' ''L''. The question of ''projective reduction'' for k-adic relations can be stated with moderate generality in the following way: * Given a set of k-place relations in the same space ''X'' and a set of projections from X to the associated subspaces, do the projections afford sufficient data to tell the different relations apart? ==Projective reducibility of triadic relations== : ''Main article : [[Triadic relation]]'' By way of illustrating the different sorts of things that can occur in considering the projective reducibility of relations, it is convenient to reuse the four examples of 3-adic relations that are discussed in the main article on that subject. ===Examples of projectively irreducible relations=== The 3-adic relations '''L'''₀ and '''L'''₁ are shown in the next two Tables: {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:60%" |+ '''L'''₀ = {(''x'', ''y'', ''z'') &#8712; '''B'''³ : ''x'' + ''y'' + ''z'' = 0} |- style="background:#e6e6ff" ! X !! Y !! Z |- | '''0''' || '''0''' || '''0''' |- | '''0''' || '''1''' || '''1''' |- | '''1''' || '''0''' || '''1''' |- | '''1''' || '''1''' || '''0''' |} <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:60%" |+ '''L'''₁ = {(''x'', ''y'', ''z'') &#8712; '''B'''³ : ''x'' + ''y'' + ''z'' = 1} |- style="background:#e6e6ff" ! X !! Y !! Z |- | '''0''' || '''0''' || '''1''' |- | '''0''' || '''1''' || '''0''' |- | '''1''' || '''0''' || '''0''' |- | '''1''' || '''1''' || '''1''' |} <br> A ''2-adic projection'' of a 3-adic relation ''L'' is the 2-adic relation that results from deleting one column of the table for ''L'' and then deleting all but one row of any resulting rows that happen to be identical in content. In other words, the multiplicity of any repeated row is ignored. In the case of the above two relations, '''L'''₀, '''L'''₁ &#8838; ''X'' × ''Y'' × ''Z'' <u>&#8776;</u> '''B'''³, the 2-adic projections are indexed by the columns or domains that remain, as shown in the following Tables. {| align="center" style="width:90%" | {| border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%" |+ proj_''XY''('''L'''₀) |- style="background:#e6e6ff" ! X !! Y |- | '''0''' || '''0''' |- | '''0''' || '''1''' |- | '''1''' || '''0''' |- | '''1''' || '''1''' |} | {| border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%" |+ proj_''XZ''('''L'''₀) |- style="background:#e6e6ff" ! X !! Z |- | '''0''' || '''0''' |- | '''0''' || '''1''' |- | '''1''' || '''1''' |- | '''1''' || '''0''' |} | {| border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%" |+ proj_''YZ''('''L'''₀) |- style="background:#e6e6ff" ! Y !! Z |- | '''0''' || '''0''' |- | '''1''' || '''1''' |- | '''0''' || '''1''' |- | '''1''' || '''0''' |} |} <br> {| align="center" style="width:90%" | {| border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%" |+ proj_''XY''('''L'''₁) |- style="background:#e6e6ff" ! X !! Y |- | '''0''' || '''0''' |- | '''0''' || '''1''' |- | '''1''' || '''0''' |- | '''1''' || '''1''' |} | {| border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%" |+ proj_''XZ''('''L'''₁) |- style="background:#e6e6ff" ! X !! Z |- | '''0''' || '''1''' |- | '''0''' || '''0''' |- | '''1''' || '''0''' |- | '''1''' || '''1''' |} | {| border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%" |+ proj_''YZ''('''L'''₁) |- style="background:#e6e6ff" ! Y !! Z |- | '''0''' || '''1''' |- | '''1''' || '''0''' |- | '''0''' || '''0''' |- | '''1''' || '''1''' |} |} <br> It is clear on inspection that the following three equations hold: {| align="center" cellpadding="4" style="text-align:center; width:90%" | proj_''XY''('''L'''₀) = proj_''XY''('''L'''₁) | proj_''XZ''('''L'''₀) = proj_''XZ''('''L'''₁) | proj_''YZ''('''L'''₀) = proj_''YZ''('''L'''₁) |} These equations say that '''L'''₀ and '''L'''₁ cannot be distinguished from each other solely on the basis of their 2-adic projection data. In such a case, either relation is said to be ''irreducible with respect to 2-adic projections''. Since reducibility with respect to 2-adic projections is the only interesting case where it concerns the reduction of 3-adic relations, it is customary to say more simply of such a relation that it is ''projectively irreducible'', the 2-adic basis being understood. It is immediate from the definition that projectively irreducible relations always arise in non-trivial multiplets of mutually indiscernible relations. ===Examples of projectively reducible relations=== The 3-adic relations '''L'''_A and '''L'''_B are shown in the next two Tables: {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:60%" |+ '''L'''_A = Sign Relation of Interpreter A |- style="background:#e6e6ff" ! style="width:20%" | Object ! style="width:20%" | Sign ! style="width:20%" | Interpretant |- | '''A''' || '''"A"''' || '''"A"''' |- | '''A''' || '''"A"''' || '''"i"''' |- | '''A''' || '''"i"''' || '''"A"''' |- | '''A''' || '''"i"''' || '''"i"''' |- | '''B''' || '''"B"''' || '''"B"''' |- | '''B''' || '''"B"''' || '''"u"''' |- | '''B''' || '''"u"''' || '''"B"''' |- | '''B''' || '''"u"''' || '''"u"''' |} <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:60%" |+ '''L'''_B = Sign Relation of Interpreter B |- style="background:#e6e6ff" ! style="width:20%" | Object ! style="width:20%" | Sign ! style="width:20%" | Interpretant |- | '''A''' || '''"A"''' || '''"A"''' |- | '''A''' || '''"A"''' || '''"u"''' |- | '''A''' || '''"u"''' || '''"A"''' |- | '''A''' || '''"u"''' || '''"u"''' |- | '''B''' || '''"B"''' || '''"B"''' |- | '''B''' || '''"B"''' || '''"i"''' |- | '''B''' || '''"i"''' || '''"B"''' |- | '''B''' || '''"i"''' || '''"i"''' |} <br> In the case of the two sign relations, '''L'''_A, '''L'''_B &#8838; ''X'' × ''Y'' × ''Z'' <u>&#8776;</u> '''O''' × '''S''' × '''I''', the 2-adic projections are indexed by the columns or domains that remain, as shown in the following Tables. {| align="center" style="width:90%" | {| border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%" |+ proj_''XY''('''L'''_A) |- style="background:#e6e6ff" ! style="width:50%" | Object ! style="width:50%" | Sign |- | '''A''' || '''"A"''' |- | '''A''' || '''"i"''' |- | '''B''' || '''"B"''' |- | '''B''' || '''"u"''' |} | {| border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%" |+ proj_''XZ''('''L'''_A) |- style="background:#e6e6ff" ! style="width:50%" | Object ! style="width:50%" | Interpretant |- | '''A''' || '''"A"''' |- | '''A''' || '''"i"''' |- | '''B''' || '''"B"''' |- | '''B''' || '''"u"''' |} | {| border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%" |+ proj_''YZ''('''L'''_A) |- style="background:#e6e6ff" ! style="width:50%" | Sign ! style="width:50%" | Interpretant |- | '''"A"''' || '''"A"''' |- | '''"A"''' || '''"i"''' |- | '''"i"''' || '''"A"''' |- | '''"i"''' || '''"i"''' |- | '''"B"''' || '''"B"''' |- | '''"B"''' || '''"u"''' |- | '''"u"''' || '''"B"''' |- | '''"u"''' || '''"u"''' |} |} <br> {| align="center" style="width:90%" | {| border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%" |+ proj_''XY''('''L'''_B) |- style="background:#e6e6ff" ! style="width:50%" | Object ! style="width:50%" | Sign |- | '''A''' || '''"A"''' |- | '''A''' || '''"u"''' |- | '''B''' || '''"B"''' |- | '''B''' || '''"i"''' |} | {| border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%" |+ proj_''XZ''('''L'''_B) |- style="background:#e6e6ff" ! style="width:50%" | Object ! style="width:50%" | Interpretant |- | '''A''' || '''"A"''' |- | '''A''' || '''"u"''' |- | '''B''' || '''"B"''' |- | '''B''' || '''"i"''' |} | {| border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%" |+ proj_''YZ''('''L'''_B) |- style="background:#e6e6ff" ! style="width:50%" | Sign ! style="width:50%" | Interpretant |- | '''"A"''' || '''"A"''' |- | '''"A"''' || '''"u"''' |- | '''"u"''' || '''"A"''' |- | '''"u"''' || '''"u"''' |- | '''"B"''' || '''"B"''' |- | '''"B"''' || '''"i"''' |- | '''"i"''' || '''"B"''' |- | '''"i"''' || '''"i"''' |} |} <br> It is clear on inspection that the following three inequations hold: {| align="center" cellpadding="4" style="text-align:center; width:90%" | proj_''XY''('''L'''_A) &#8800; proj_''XY''('''L'''_B) | proj_''XZ''('''L'''_A) &#8800; proj_''XZ''('''L'''_B) | proj_''YZ''('''L'''_A) &#8800; proj_''YZ''('''L'''_B) |} These inequations say that '''L'''_A and '''L'''_B can be distinguished from each other solely on the basis of their 2-adic projection data. But this is not enough to say that either one of them is projectively reducible to their 2-adic projection data. To say that a 3-adic relation is projectively reducible in that respect, one has to show that it can be distinguished from ''every'' other 3-adic relation on the basis of the 2-adic projection data alone. In other words, to show that a 3-adic relation ''L'' on '''O''' × '''S''' × '''I''' is ''reducible'' or ''reconstructible'' in the 2-adic projective sense, it is necessary to show that no distinct ''L&#8242;'' on '''O''' × '''S''' × '''I''' exists such that ''L'' and ''L&#8242;'' have the same set of projections. Proving this takes a much more comprehensive or exhaustive investigation of the space of possible relations on '''O''' × '''S''' × '''I''' than looking merely at one or two relations at a time. '''Fact.''' As it happens, each of the relations '''L'''_A and '''L'''_B is uniquely determined by its 2-adic projections. This can be seen by following the proof that is given below. Before tackling the proof, however, it will speed things along to recall a few ideas and notations from other articles. * If ''L'' is a relation over a set of domains that includes the domains ''U'' and ''V'', then the abbreviated notation ''L''_''UV'' can be used for the projection proj_''UV''(''L''). * The operation of reversing a projection asks what elements of a bigger space project onto given elements of a smaller space. The set of elements that project onto ''x'' under a given projection ''f'' is called the ''[[image (mathematics)|fiber]]'' of ''x'' under ''f'' and is written ''f''^–1(''x'') or ''f''^–1''x''. * If ''X'' is a finite set, the ''cardinality'' of ''X'', written card(''X'') or |''X''|, means the number of elements in ''X''. '''Proof.''' Let ''L'' be either one of the relations '''L'''_A or '''L'''_B. Consider any coordinate position (''s'', ''i'') in the '''SI'''-plane '''S''' × '''I'''. If (''s'', ''i'') is not in ''L''_'''SI''' then there can be no element (''o'', ''s'', ''i'') in ''L'', therefore we may restrict our attention to positions (''s'', ''i'') in ''L''_'''SI''', knowing that there exist at least |''L''_'''SI'''| = 8 elements in ''L'', and seeking only to determine what objects ''o'' exist such that (''o'', ''s'', ''i'') is an element in the ''fiber'' of (''s'', ''i''). In other words, for what ''o'' in '''O''' is (''o'', ''s'', ''i'') in the fiber proj_'''SI'''^–1(''s'', ''i'') ? Now, the circumstance that ''L''_'''OS''' has exactly one element (''o'', ''s'') for each coordinate ''s'' in '''S''' and that ''L''_'''OI''' has exactly one element (''o'', ''i'') for each coordinate ''i'' in '''I''', plus the "coincidence" of it being the same ''o'' at any one choice for (''s'', ''i''), tells us that ''L'' has just the one element (''o'', ''s'', ''i'') over each point of '''S''' × '''I'''. All together, this proves that both '''L'''_A and '''L'''_B are reducible in an informative sense to 3-tuples of 2-adic relations, that is, they are ''projectively 2-adically reducible''. ===Summary=== The ''projective analysis'' of 3-adic relations, illustrated by means of concrete examples, has been pursued just far enough at this point to state this clearly demonstrated result: * Some 3-adic relations are, and other 3-adic relations are not, reducible to, or reconstructible from, their 2-adic projection data. In short, some 3-adic relations are projectively reducible and some 3-adic relations are projectively irreducible. ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://mywikibiz.com/Relation_reduction Relation Reduction @ MyWikiBiz] * [http://mathweb.org/wiki/Relation_reduction Relation Reduction @ MathWeb Wiki] * [http://netknowledge.org/wiki/Relation_reduction Relation Reduction @ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Relation_reduction Relation Reduction @ OER Commons] {{col-break}} * [http://p2pfoundation.net/Relation_Reduction Relation Reduction @ P2P Foundation] * [http://semanticweb.org/wiki/Relation_reduction Relation Reduction @ SemanticWeb] * [http://ref.subwiki.org/wiki/Relation_reduction Relation Reduction @ Subject Wikis] * [http://beta.wikiversity.org/wiki/Relation_reduction Relation Reduction @ Wikiversity Beta] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;] * [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;] ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Relation_reduction Relation Reduction], [http://mywikibiz.com/ MyWikiBiz] * [http://mathweb.org/wiki/Relation_reduction Relation Reduction], [http://mathweb.org/wiki/ MathWeb Wiki] * [http://netknowledge.org/wiki/Relation_reduction Relation Reduction], [http://netknowledge.org/ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Relation_reduction Relation Reduction], [http://wiki.oercommons.org/ OER Commons] * [http://p2pfoundation.net/Relation_Reduction Relation Reduction], [http://p2pfoundation.net/ P2P Foundation] {{col-break}} * [http://semanticweb.org/wiki/Relation_reduction Relation Reduction], [http://semanticweb.org/ Semantic Web] * [http://planetmath.org/encyclopedia/RelationReduction.html Relation Reduction], [http://planetmath.org/ PlanetMath] * [http://getwiki.net/-Relational_Reduction Relation Reduction], [http://getwiki.net/ GetWiki] * [http://wikinfo.org/index.php/Relation_reduction Relation Reduction], [http://wikinfo.org/ Wikinfo] * [http://en.wikipedia.org/w/index.php?title=Relation_reduction&oldid=39828834 Relation Reduction], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Database Theory]] [[Category:Discrete Mathematics]] [[Category:Formal Sciences]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Relation Theory]] [[Category:Set Theory]] 992f932432e25e81afe243ad5886c619afa88c0c 0 338 560 2010-06-22T19:14:38Z Jon Awbrey 3 + article wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''truth table''' is a tabular array that illustrates the computation of a [[boolean function]], that is, a function of the form <math>f : \mathbb{B}^k \to \mathbb{B},</math> where <math>k\!</math> is a non-negative integer and <math>\mathbb{B}</math> is the [[boolean domain]] <math>\{ 0, 1 \}.\!</math> ==Logical negation== ''[[Logical negation]]'' is an [[logical operation|operation]] on one [[logical value]], typically the value of a [[proposition]], that produces a value of ''true'' when its operand is false and a value of ''false'' when its operand is true. The truth table of '''NOT p''' (also written as '''~p''' or '''&not;p''') is as follows: <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:40%" |+ '''Logical Negation''' |- style="background:#e6e6ff" ! style="width:20%" | p ! style="width:20%" | &not;p |- | F || T |- | T || F |} <br> The logical negation of a proposition '''p''' is notated in different ways in various contexts of discussion and fields of application. Among these variants are the following: <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; width:40%" |+ '''Variant Notations''' |- style="background:#e6e6ff" ! style="text-align:center" | Notation ! Vocalization |- | style="text-align:center" | <math>\bar{p}</math> | bar ''p'' |- | style="text-align:center" | <math>p'\!</math> | ''p'' prime,<p> ''p'' complement |- | style="text-align:center" | <math>!p\!</math> | bang ''p'' |} <br> ==Logical conjunction== ''[[Logical conjunction]]'' is an [[logical operation|operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''true'' if and only if both of its operands are true. The truth table of '''p AND q''' (also written as '''p &and; q''', '''p & q''', or '''p<math>\cdot</math>q''') is as follows: <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:45%" |+ '''Logical Conjunction''' |- style="background:#e6e6ff" ! style="width:15%" | p ! style="width:15%" | q ! style="width:15%" | p &and; q |- | F || F || F |- | F || T || F |- | T || F || F |- | T || T || T |} <br> ==Logical disjunction== ''[[Logical disjunction]]'', also called ''logical alternation'', is an [[logical operation|operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''false'' if and only if both of its operands are false. The truth table of '''p OR q''' (also written as '''p &or; q''') is as follows: <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:45%" |+ '''Logical Disjunction''' |- style="background:#e6e6ff" ! style="width:15%" | p ! style="width:15%" | q ! style="width:15%" | p &or; q |- | F || F || F |- | F || T || T |- | T || F || T |- | T || T || T |} <br> ==Logical equality== ''[[Logical equality]]'' is an [[logical operation|operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''true'' if and only if both operands are false or both operands are true. The truth table of '''p EQ q''' (also written as '''p = q''', '''p &harr; q''', or '''p &equiv; q''') is as follows: <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:45%" |+ '''Logical Equality''' |- style="background:#e6e6ff" ! style="width:15%" | p ! style="width:15%" | q ! style="width:15%" | p = q |- | F || F || T |- | F || T || F |- | T || F || F |- | T || T || T |} <br> ==Exclusive disjunction== ''[[Exclusive disjunction]]'', also known as ''logical inequality'' or ''symmetric difference'', is an [[logical operation|operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''true'' just in case exactly one of its operands is true. The truth table of '''p XOR q''' (also written as '''p + q''', '''p &oplus; q''', or '''p &ne; q''') is as follows: <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:45%" |+ '''Exclusive Disjunction''' |- style="background:#e6e6ff" ! style="width:15%" | p ! style="width:15%" | q ! style="width:15%" | p XOR q |- | F || F || F |- | F || T || T |- | T || F || T |- | T || T || F |} <br> The following equivalents can then be deduced: : <math>\begin{matrix} p + q & = & (p \land \lnot q) & \lor & (\lnot p \land q) \\ \\ & = & (p \lor q) & \land & (\lnot p \lor \lnot q) \\ \\ & = & (p \lor q) & \land & \lnot (p \land q) \end{matrix}</math> ==Logical implication== The ''[[logical implication]]'' and the ''[[material conditional]]'' are both associated with an [[logical operation|operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''false'' if and only if the first operand is true and the second operand is false. The truth table associated with the material conditional '''if p then q''' (symbolized as '''p&nbsp;&rarr;&nbsp;q''') and the logical implication '''p implies q''' (symbolized as '''p&nbsp;&rArr;&nbsp;q''') is as follows: <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:45%" |+ '''Logical Implication''' |- style="background:#e6e6ff" ! style="width:15%" | p ! style="width:15%" | q ! style="width:15%" | p &rArr; q |- | F || F || T |- | F || T || T |- | T || F || F |- | T || T || T |} <br> ==Logical NAND== The ''[[logical NAND]]'' is a [[logical operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''false'' if and only if both of its operands are true. In other words, it produces a value of ''true'' if and only if at least one of its operands is false. The truth table of '''p NAND q''' (also written as '''p&nbsp;|&nbsp;q''' or '''p&nbsp;&uarr;&nbsp;q''') is as follows: <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:45%" |+ '''Logical NAND''' |- style="background:#e6e6ff" ! style="width:15%" | p ! style="width:15%" | q ! style="width:15%" | p &uarr; q |- | F || F || T |- | F || T || T |- | T || F || T |- | T || T || F |} <br> ==Logical NNOR== The ''[[logical NNOR]]'' is a [[logical operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''true'' if and only if both of its operands are false. In other words, it produces a value of ''false'' if and only if at least one of its operands is true. The truth table of '''p NNOR q''' (also written as '''p&nbsp;&perp;&nbsp;q''' or '''p&nbsp;&darr;&nbsp;q''') is as follows: <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:45%" |+ '''Logical NNOR''' |- style="background:#e6e6ff" ! style="width:15%" | p ! style="width:15%" | q ! style="width:15%" | p &darr; q |- | F || F || T |- | F || T || F |- | T || F || F |- | T || T || F |} <br> ==Translations== * [http://zh.wikipedia.org/wiki/%E7%9C%9F%E5%80%BC%E8%A1%A8 &#20013;&#25991; : &#30495;&#20540;&#34920;] ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://mywikibiz.com/Truth_table Truth Table @ MyWikiBiz] * [http://mathweb.org/wiki/Truth_table Truth Table @ MathWeb Wiki] * [http://netknowledge.org/wiki/Truth_table Truth Table @ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Truth_table Truth Table @ OER Commons] {{col-break}} * [http://p2pfoundation.net/Truth_Table Truth Table @ P2P Foundation] * [http://semanticweb.org/wiki/Truth_table Truth Table @ SemanticWeb] * [http://ref.subwiki.org/wiki/Truth_table Truth Table @ Subject Wikis] * [http://beta.wikiversity.org/wiki/Truth_table Truth Table @ Wikiversity Beta] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;] * [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;] ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Truth_table Truth Table], [http://mywikibiz.com/ MyWikiBiz] * [http://mathweb.org/wiki/Truth_table Truth Table], [http://mathweb.org/ MathWeb Wiki] * [http://netknowledge.org/wiki/Truth_table Truth Table], [http://netknowledge.org/ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Truth_table Truth Table], [http://wiki.oercommons.org/ OER Commons] {{col-break}} * [http://p2pfoundation.net/Truth_Table Truth Table], [http://p2pfoundation.net/ P2P Foundation] * [http://semanticweb.org/wiki/Truth_table Truth Table], [http://semanticweb.org/ SemanticWeb] * [http://beta.wikiversity.org/wiki/Truth_table Truth Table], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://getwiki.net/-Truth_Table Truth Table], [http://getwiki.net/ GetWiki] {{col-break}} * [http://wikinfo.org/index.php/Truth_table Truth Table], [http://wikinfo.org/ Wikinfo] * [http://textop.org/wiki/index.php?title=Truth_table Truth Table], [http://textop.org/wiki/ Textop Wiki] * [http://en.wikipedia.org/w/index.php?title=Truth_table&oldid=77110085 Truth Table], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Discrete Mathematics]] [[Category:Formal Languages]] [[Category:Formal Sciences]] [[Category:Formal Systems]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Philosophy]] [[Category:Semiotics]] d5a56be8b9cae3bdd291656308d56b1717000ab6 0 339 561 2010-06-22T20:42:34Z Jon Awbrey 3 + article wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. The term '''''universe of discourse''''' is generally attributed to Augustus De&nbsp;Morgan (1846). George Boole (1854) defines it in the following manner: {| align="center" cellpadding="4" width="90%" | <p>In every discourse, whether of the mind conversing with its own thoughts, or of the individual in his intercourse with others, there is an assumed or expressed limit within which the subjects of its operation are confined. &hellip; Now, whatever may be the extent of the field within which all the objects of our discourse are found, that field may properly be termed the universe of discourse. (Boole 1854/1958, p. 42).</p> |} ==References== * Boole, George (1854/1958), ''An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities'', Macmillan Publishers, 1854. Reprinted with corrections, Dover Publications, New York, NY, 1958. * De Morgan, Augustus (1846), ''Cambridge Philosophical Transactions'', ''viii'', p. 380. ==Resources== * [http://www.helsinki.fi/science/commens/dictionary.html Commens Dictionary of Peirce's Terms] ** [http://www.helsinki.fi/science/commens/terms/logicaluniv.html Logical Universe] ** [http://www.helsinki.fi/science/commens/terms/universedisc.html Universe of Discourse] ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://mywikibiz.com/Universe_of_discourse Universe of Discourse @ MyWikiBiz] * [http://mathweb.org/wiki/Universe_of_discourse Universe of Discourse @ MathWeb Wiki] * [http://netknowledge.org/wiki/Universe_of_discourse Universe of Discourse @ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Universe_of_discourse Universe of Discourse @ OER Commons] {{col-break}} * [http://p2pfoundation.net/Universe_of_Discourse Universe of Discourse @ P2P Foundation] * [http://semanticweb.org/wiki/Universe_of_discourse Universe of Discourse @ SemanticWeb] * [http://ref.subwiki.org/wiki/Universe_of_discourse Universe of Discourse @ Subject Wikis] * [http://beta.wikiversity.org/wiki/Universe_of_discourse Universe of Discourse @ Wikiversity Beta] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;] * [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;] ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Universe_of_discourse Universe of Discourse], [http://mywikibiz.com/ MyWikiBiz] * [http://mathweb.org/wiki/Universe_of_discourse Universe of Discourse], [http://mathweb.org/wiki/ MathWeb Wiki] * [http://netknowledge.org/wiki/Universe_of_discourse Universe of Discourse], [http://netknowledge.org/ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Universe_of_discourse Universe of Discourse], [http://wiki.oercommons.org/ OER Commons] {{col-break}} * [http://p2pfoundation.net/Universe_of_Discourse Universe of Discourse], [http://p2pfoundation.net/ P2P Foundation] * [http://semanticweb.org/wiki/Universe_of_discourse Universe of Discourse], [http://semanticweb.org/ SemanticWeb] * [http://planetmath.org/encyclopedia/UniverseOfDiscourse.html Universe of Discourse], [http://planetmath.org/ PlanetMath] * [http://beta.wikiversity.org/wiki/Universe_of_discourse Universe of Discourse], [http://beta.wikiversity.org/ Wikiversity Beta] {{col-end}} [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Computer Science]] [[Category:Formal Grammars]] [[Category:Formal Languages]] [[Category:Formal Sciences]] [[Category:Formal Systems]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Model Theory]] [[Category:Normative Sciences]] [[Category:Pragmatics]] [[Category:Proof Theory]] [[Category:Semantics]] [[Category:Semiotics]] [[Category:Syntax]] 42e69e73c1094c9177d65f3b2d4fee71e25b791d 0 340 562 2010-06-23T02:08:59Z Jon Awbrey 3 + article wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''sole sufficient operator''' or a '''sole sufficient connective''' is an operator that is sufficient by itself to generate all of the operators in a specified class of operators. In [[logic]], it is a logical operator that suffices to generate all of the [[boolean-valued function]]s, <math>f : X \to \mathbb{B} </math>, where <math>X\!</math> is an arbitrary set and where <math>\mathbb{B}</math> is a generic 2-element set, typically <math>\mathbb{B} = \{ 0, 1 \} = \{ false, true \}</math>, in particular, to generate all of the [[finitary boolean function]]s, <math> f : \mathbb{B}^k \to \mathbb{B} </math>. ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://mywikibiz.com/Sole_sufficient_operator Sole Sufficient Operator @ MyWikiBiz] * [http://mathweb.org/wiki/Sole_sufficient_operator Sole Sufficient Operator @ MathWeb Wiki] * [http://netknowledge.org/wiki/Sole_sufficient_operator Sole Sufficient Operator @ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Sole_sufficient_operator Sole Sufficient Operator @ OER Commons] {{col-break}} * [http://p2pfoundation.net/Sole_Sufficient_Operator Sole Sufficient Operator @ P2P Foundation] * [http://semanticweb.org/wiki/Sole_sufficient_operator Sole Sufficient Operator @ SemanticWeb] * [http://ref.subwiki.org/wiki/Sole_sufficient_operator Sole Sufficient Operator @ Subject Wikis] * [http://beta.wikiversity.org/wiki/Sole_sufficient_operator Sole Sufficient Operator @ Wikiversity Beta] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;] * [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;] ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Sole_sufficient_operator Sole Sufficient Operator], [http://mywikibiz.com/ MyWikiBiz] * [http://mathweb.org/wiki/Sole_sufficient_operator Sole Sufficient Operator], [http://mathweb.org/wiki/ MathWeb Wiki] * [http://netknowledge.org/wiki/Sole_sufficient_operator Sole Sufficient Operator], [http://netknowledge.org/ NetKnowledge] * [http://p2pfoundation.net/Sole_sufficient_operator Sole Sufficient Operator], [http://p2pfoundation.net/ P2P Foundation] * [http://http://planetmath.org/encyclopedia/SoleSufficientOperator.html Sole Sufficient Operator], [http://planetmath.org/ PlanetMath] * [http://planetphysics.org/encyclopedia/SoleSufficientOperator.html Sole Sufficient Operator], [http://planetphysics.org/ PlanetPhysics] {{col-break}} * [http://semanticweb.org/wiki/Sole_sufficient_operator Sole Sufficient Operator], [http://semanticweb.org/ SemanticWeb] * [http://beta.wikiversity.org/wiki/Sole_sufficient_operator Sole Sufficient Operator], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://getwiki.net/-Sole_Sufficient_Operator Sole Sufficient Operator], [http://getwiki.net/ GetWiki] * [http://wikinfo.org/index.php/Sole_sufficient_operator Sole Sufficient Operator], [http://wikinfo.org/ Wikinfo] * [http://textop.org/wiki/index.php?title=Sole_sufficient_operator Sole Sufficient Operator], [http://textop.org/wiki/ Textop Wiki] * [http://en.wikipedia.org/w/index.php?title=Sole_sufficient_operator&oldid=156136346 Sole Sufficient Operator], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Semiotics]] c031e70adc4baafb4aaa6122d767b600c2442688 0 341 563 2010-06-23T14:04:42Z Jon Awbrey 3 + article wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. '''''Zeroth order logic''''' is an informal term that is sometimes used to indicate the common principles underlying the [[algebra of sets]], [[boolean algebra]], [[boolean function]]s, [[logical connective]]s, [[monadic predicate calculus]], [[propositional calculus]], and sentential logic. The term serves to mark a level of abstraction in which the more inessential differences among these subjects can be subsumed under the appropriate [[isomorphism]]s. ==Propositional forms on two variables== By way of initial orientation, Table 1 lists equivalent expressions for the sixteen functions of concrete type <math>X \times Y \to \mathbb{B}</math> and abstract type <math>\mathbb{B} \times \mathbb{B} \to \mathbb{B}</math> in a number of different languages for zeroth order logic. <br> {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%" |+ '''Table 1. Propositional Forms on Two Variables''' |- style="background:#e6e6ff" ! style="width:15%" | L₁ ! style="width:15%" | L₂ ! style="width:15%" | L₃ ! style="width:15%" | L₄ ! style="width:15%" | L₅ ! style="width:15%" | L₆ |- style="background:#e6e6ff" | &nbsp; | align="right" | x : | 1 1 0 0 | &nbsp; | &nbsp; | &nbsp; |- style="background:#e6e6ff" | &nbsp; | align="right" | y : | 1 0 1 0 | &nbsp; | &nbsp; | &nbsp; |- | f₀ || f₀₀₀₀ || 0 0 0 0 || ( ) || false || 0 |- | f₁ || f₀₀₀₁ || 0 0 0 1 || (x)(y) || neither x nor y || ¬x &and; ¬y |- | f₂ || f₀₀₁₀ || 0 0 1 0 || (x) y || y and not x || ¬x &and; y |- | f₃ || f₀₀₁₁ || 0 0 1 1 || (x) || not x || ¬x |- | f₄ || f₀₁₀₀ || 0 1 0 0 || x (y) || x and not y || x &and; ¬y |- | f₅ || f₀₁₀₁ || 0 1 0 1 || (y) || not y || ¬y |- | f₆ || f₀₁₁₀ || 0 1 1 0 || (x, y) || x not equal to y || x &ne; y |- | f₇ || f₀₁₁₁ || 0 1 1 1 || (x y) || not both x and y || ¬x &or; ¬y |- | f₈ || f₁₀₀₀ || 1 0 0 0 || x y || x and y || x &and; y |- | f₉ || f₁₀₀₁ || 1 0 0 1 || ((x, y)) || x equal to y || x = y |- | f₁₀ || f₁₀₁₀ || 1 0 1 0 || y || y || y |- | f₁₁ || f₁₀₁₁ || 1 0 1 1 || (x (y)) || not x without y || x &rarr; y |- | f₁₂ || f₁₁₀₀ || 1 1 0 0 || x || x || x |- | f₁₃ || f₁₁₀₁ || 1 1 0 1 || ((x) y) || not y without x || x &larr; y |- | f₁₄ || f₁₁₁₀ || 1 1 1 0 || ((x)(y)) || x or y || x &or; y |- | f₁₅ || f₁₁₁₁ || 1 1 1 1 || (( )) || true || 1 |} <br> These six languages for the sixteen boolean functions are conveniently described in the following order: * Language '''L₃''' describes each boolean function ''f'' : '''B'''² &#8594; '''B''' by means of the sequence of four boolean values (''f''(1,1), ''f''(1,0), ''f''(0,1), ''f''(0,0)). Such a sequence, perhaps in another order, and perhaps with the logical values ''F'' and ''T'' instead of the boolean values 0 and 1, respectively, would normally be displayed as a column in a [[truth table]]. * Language '''L₂''' lists the sixteen functions in the form '''f_i''', where the index '''i''' is a [[bit string]] formed from the sequence of boolean values in '''L₃'''. * Language '''L₁''' notates the boolean functions '''f_i''' with an index '''i''' that is the decimal equivalent of the binary numeral index in '''L₂'''. * Language '''L₄''' expresses the sixteen functions in terms of logical [[conjunction]], indicated by concatenating function names or proposition expressions in the manner of products, plus the family of ''[[minimal negation operator]]s'', the first few of which are given in the following variant notations: : <math>\begin{matrix} (\ ) & = & 0 & = & \mbox{false} \\ (x) & = & \tilde{x} & = & x' \\ (x, y) & = & \tilde{x}y \lor x\tilde{y} & = & x'y \lor xy' \\ (x, y, z) & = & \tilde{x}yz \lor x\tilde{y}z \lor xy\tilde{z} & = & x'yz \lor xy'z \lor xyz' \end{matrix}</math> It may also be noted that <math>(x, y)\!</math> is the same function as <math>x + y\!</math> and <math>x \ne y</math>, and that the inclusive disjunctions indicated for <math>(x, y)\!</math> and for <math>(x, y, z)\!</math> may be replaced with exclusive disjunctions without affecting the meaning, because the terms disjoined are already disjoint. However, the function <math>(x, y, z)\!</math> is not the same thing as the function <math>x + y + z\!</math>. * Language '''L₅''' lists ordinary language expressions for the sixteen functions. Many other paraphrases are possible, but these afford a sample of the simplest equivalents. * Language '''L₆''' expresses the sixteen functions in one of several notations that are commonly used in formal logic. ==Translations== * [http://zh.wikipedia.org/wiki/%E9%9B%B6%E9%98%B6%E9%80%BB%E8%BE%91 &#20013;&#25991; : &#38646;&#38454;&#36923;&#36753;] ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://mywikibiz.com/Zeroth_order_logic Zeroth Order Logic @ MyWikiBiz] * [http://mathweb.org/wiki/Zeroth_order_logic Zeroth Order Logic @ MathWeb Wiki] * [http://netknowledge.org/wiki/Zeroth_order_logic Zeroth Order Logic @ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Zeroth_order_logic Zeroth Order Logic @ OER Commons] {{col-break}} * [http://p2pfoundation.net/Zeroth_Order_Logic Zeroth Order Logic @ P2P Foundation] * [http://semanticweb.org/wiki/Zeroth_order_logic Zeroth Order Logic @ SemanticWeb] * [http://ref.subwiki.org/wiki/Zeroth_order_logic Zeroth Order Logic @ Subject Wikis] * [http://beta.wikiversity.org/wiki/Zeroth_order_logic Zeroth Order Logic @ Wikiversity Beta] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;] * [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;] ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Zeroth_order_logic Zeroth Order Logic], [http://mywikibiz.com/ MyWikiBiz] * [http://planetmath.org/encyclopedia/ZerothOrderLogic.html Zeroth Order Logic], [http://planetmath.org/ PlanetMath] * [http://beta.wikiversity.org/wiki/Zeroth_order_logic Zeroth Order Logic], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://getwiki.net/-Zeroth-Order_Logic Zeroth Order Logic], [http://getwiki.net/ GetWiki] {{col-break}} * [http://wikinfo.org/index.php/Zeroth_order_logic Zeroth Order Logic], [http://wikinfo.org/ Wikinfo] * [http://textop.org/wiki/index.php?title=Zeroth_order_logic Zeroth Order Logic], [http://textop.org/wiki/ Textop Wiki] * [http://en.wikipedia.org/w/index.php?title=Zeroth-order_logic&oldid=77109225 Zeroth Order Logic], [http://en.wikipedia.org/ Wikipedia] * [http://altheim.com/cs/zol.html Zeroth Order Logic], [http://altheim.com/cs Altheim.com] {{col-end}} [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Computer Science]] [[Category:Formal Languages]] [[Category:Formal Sciences]] [[Category:Formal Systems]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Normative Sciences]] 93628b5b0b9e15e57df46a25cca26b88e03ead47 0 342 564 2010-06-23T15:00:08Z Jon Awbrey 3 + article wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. '''Inquiry''' is any proceeding or process that has the aim of augmenting knowledge, resolving doubt, or solving a problem. A theory of inquiry is an account of the various types of inquiry and a treatment of the ways that each type of inquiry achieves its aim. ==Classical sources== ===Deduction=== {| align="center" cellpadding="4" width="90%" <!--QUOTE--> | <p>When three terms are so related to one another that the last is wholly contained in the middle and the middle is wholly contained in or excluded from the first, the extremes must admit of perfect syllogism. By 'middle term' I mean that which both is contained in another and contains another in itself, and which is the middle by its position also; and by 'extremes' (a) that which is contained in another, and (b) that in which another is contained. For if ''A'' is predicated of all ''B'', and ''B'' of all ''C'', ''A'' must necessarily be predicated of all ''C''. &hellip; I call this kind of figure the First. (Aristotle, ''Prior Analytics'', 1.4).</p> |} ===Induction=== {| align="center" cellpadding="4" width="90%" <!--QUOTE--> | <p>Induction, or inductive reasoning, consists in establishing a relation between one extreme term and the middle term by means of the other extreme; for example, if ''B'' is the middle term of ''A'' and ''C'', in proving by means of ''C'' that ''A'' applies to ''B''; for this is how we effect inductions. (Aristotle, ''Prior Analytics'', 2.23).</p> |} ===Abduction=== The ''locus classicus'' for the study of [[abductive reasoning]] is found in [[Aristotle]]'s ''[[Prior Analytics]]'', Book 2, Chapt. 25. It begins this way: {| align="center" cellpadding="4" width="90%" <!--QUOTE--> | <p>We have Reduction (&#945;&#960;&#945;&#947;&#969;&#947;&#951;, [[abductive reasoning|abduction]]):</p> <ol> <li><p>When it is obvious that the first term applies to the middle, but that the middle applies to the last term is not obvious, yet is nevertheless more probable or not less probable than the conclusion;</p></li> <li><p>Or if there are not many intermediate terms between the last and the middle;</p></li> </ol> <p>For in all such cases the effect is to bring us nearer to knowledge.</p> |} By way of explanation, [[Aristotle]] supplies two very instructive examples, one for each of the two varieties of abductive inference steps that he has just described in the abstract: {| align="center" cellpadding="4" width="90%" <!--QUOTE--> | <ol> <li><p>For example, let ''A'' stand for "that which can be taught", ''B'' for "knowledge", and ''C'' for "morality". Then that knowledge can be taught is evident; but whether virtue is knowledge is not clear. Then if ''BC'' is not less probable or is more probable than ''AC'', we have reduction; for we are nearer to knowledge for having introduced an additional term, whereas before we had no knowledge that ''AC'' is true.<p></li> <li><p>Or again we have reduction if there are not many intermediate terms between ''B'' and ''C''; for in this case too we are brought nearer to knowledge. For example, suppose that ''D'' is "to square", ''E'' "rectilinear figure", and ''F'' "circle". Assuming that between ''E'' and ''F'' there is only one intermediate term &mdash; that the circle becomes equal to a rectilinear figure by means of [[lunule]]s &mdash; we should approximate to knowledge.</p></li> </ol> <p>([[Aristotle]], "[[Prior Analytics]]", 2.25, with minor alterations)</p> |} Aristotle's latter variety of abductive reasoning, though it will take some explaining in the sequel, is well worth our contemplation, since it hints already at streams of inquiry that course well beyond the syllogistic source from which they spring, and into regions that Peirce will explore more broadly and deeply. ==Inquiry in the pragmatic paradigm== In the pragmatic philosophies of [[Charles Sanders Peirce]], [[William James]], [[John Dewey]], and others, inquiry is closely associated with the [[normative science]] of [[logic]]. In its inception, the pragmatic model or theory of inquiry was extracted by Peirce from its raw materials in classical logic, with a little bit of help from [[Kant]], and refined in parallel with the early development of symbolic logic by [[Boole]], [[De Morgan]], and Peirce himself to address problems about the nature and conduct of scientific reasoning. Borrowing a brace of concepts from [[Aristotle]], Peirce examined three fundamental modes of reasoning that play a role in inquiry, commonly known as [[abductive reasoning|abductive]], [[deductive reasoning|deductive]], and [[inductive reasoning|inductive]] [[inference]]. In rough terms, ''[[abductive reasoning|abduction]]'' is what we use to generate a likely [[hypothesis]] or an initial [[diagnosis]] in response to a phenomenon of interest or a problem of concern, while ''[[deductive reasoning|deduction]]'' is used to clarify, to derive, and to explicate the relevant consequences of the selected hypothesis, and ''[[inductive reasoning|induction]]'' is used to test the sum of the predictions against the sum of the data. It needs to be observed that the classical and pragmatic treatments of the types of reasoning, dividing the generic territory of inference as they do into three special parts, arrive at a different characterization of the environs of reason than do those accounts that count only two. These three processes typically operate in a cyclic fashion, systematically operating to reduce the uncertainties and the difficulties that initiated the inquiry in question, and in this way, to the extent that inquiry is successful, leading to an increase in knowledge or in skills. In the pragmatic way of thinking everything has a purpose, and the purpose of each thing is the first thing we should try to note about it. The purpose of inquiry is to reduce doubt and lead to a state of belief, which a person in that state will usually call ''[[knowledge]]'' or ''[[certainty]]''. As they contribute to the end of inquiry, we should appreciate that the three kinds of inference describe a cycle that can be understood only as a whole, and none of the three makes complete sense in isolation from the others. For instance, the purpose of abduction is to generate guesses of a kind that deduction can explicate and that induction can evaluate. This places a mild but meaningful constraint on the production of hypotheses, since it is not just any wild guess at explanation that submits itself to reason and bows out when defeated in a match with reality. In a similar fashion, each of the other types of inference realizes its purpose only in accord with its proper role in the whole cycle of inquiry. No matter how much it may be necessary to study these processes in abstraction from each other, the integrity of inquiry places strong limitations on the effective [[modularity]] of its principal components. ===Art and science of inquiry=== For our present purposes, the first feature to note in distinguishing the three principal modes of reasoning from each other is whether each of them is exact or approximate in character. In this light, deduction is the only one of the three types of reasoning that can be made exact, in essence, always deriving true conclusions from true premisses, while abduction and induction are unavoidably approximate in their modes of operation, involving elements of fallible judgment in practice and inescapable error in their application. The reason for this is that deduction, in the ideal limit, can be rendered a purely internal process of the reasoning agent, while the other two modes of reasoning essentially demand a constant interaction with the outside world, a source of phenomena and problems that will no doubt continue to exceed the capacities of any finite resource, human or machine, to master. Situated in this larger reality, approximations can be judged appropriate only in relation to their context of use and can be judged fitting only with regard to a purpose in view. A parallel distinction that is often made in this connection is to call deduction a ''[[demonstrative]]'' form of inference, while abduction and induction are classed as ''[[non-demonstrative]]'' forms of reasoning. Strictly speaking, the latter two modes of reasoning are not properly called inferences at all. They are more like controlled associations of words or ideas that just happen to be successful often enough to be preserved as useful heuristic strategies in the repertoire of the agent. But [[non-demonstrative]] ways of thinking are inherently subject to error, and must be constantly checked out and corrected as needed in practice. In classical terminology, forms of judgment that require attention to the context and the purpose of the judgment are said to involve an element of 'art', in a sense that is judged to distinguish them from 'science', and in their renderings as expressive judgments to implicate arbiters in styles of [[rhetoric]], as contrasted with [[logic]]. In a figurative sense, this means that only deductive logic can be reduced to an exact theoretical science, while the practice of any empirical science will always remain to some degree an art. ===Zeroth order inquiry=== Many aspects of inquiry can be recognized and usefully studied in very basic logical settings, even simpler than the level of [[syllogism]], for example, in the realm of reasoning that is variously known as ''[[boolean algebra]]'', ''[[propositional logic|propositional calculus]]'', ''[[sentential calculus]]'', or ''[[zeroth-order logic]]''. By way of approaching the learning curve on the gentlest availing slope, we may well begin at the level of ''[[zeroth-order inquiry]]'', in effect, taking the syllogistic approach to inquiry only so far as the propositional or sentential aspects of the associated reasoning processes are concerned. One of the bonuses of doing this in the context of Peirce's logical work is that it provides us with doubly instructive exercises in the use of his [[logical graph]]s, taken at the level of his so-called '[[alpha graph]]s'. In the case of propositional calculus or sentential logic, deduction comes down to applications of the [[transitive law]] for conditional implications and the approximate forms of inference hang on the properties that derive from these. In describing the various types of inference I will employ a few old terms of art from classical logic that are still of use in treating these kinds of simple problems in reasoning. : '''Deduction''' takes a Case, the [[minor premiss]] <math>X \Rightarrow Y</math> : and combines it with a Rule,the [[major premiss]] <math>Y \Rightarrow Z</math> : to arrive at a Fact, the demonstrative [[conclusion]] <math>X \Rightarrow Z.</math> : '''Induction''' takes a Case of the form <math>X \Rightarrow Y</math> : and matches it with a Fact of the form <math>X \Rightarrow Z</math> : to infer a Rule of the form <math>Y \Rightarrow Z.</math> : '''Abduction''' takes a Fact of the form <math>X \Rightarrow Z</math> : and matches it with a Rule of the form <math>Y \Rightarrow Z</math> : to infer a Case of the form <math>X \Rightarrow Y.</math> For ease of reference, Figure 1 and the Legend beneath it summarize the classical terminology for the three types of inference and the relationships among them. {| align="center" cellpadding="8" style="text-align:center" | <pre> o-------------------------------------------------o | | | Z | | o | | |\ | | | \ | | | \ | | | \ | | | \ | | | \ R U L E | | | \ | | | \ | | F | \ | | | \ | | A | \ | | | o Y | | C | / | | | / | | T | / | | | / | | | / | | | / C A S E | | | / | | | / | | | / | | | / | | |/ | | o | | X | | | | Deduction takes a Case of the form X => Y, | | matches it with a Rule of the form Y => Z, | | then adverts to a Fact of the form X => Z. | | | | Induction takes a Case of the form X => Y, | | matches it with a Fact of the form X => Z, | | then adverts to a Rule of the form Y => Z. | | | | Abduction takes a Fact of the form X => Z, | | matches it with a Rule of the form Y => Z, | | then adverts to a Case of the form X => Y. | | | | Even more succinctly: | | | | Abduction Deduction Induction | | | | Premiss: Fact Rule Case | | Premiss: Rule Case Fact | | Outcome: Case Fact Rule | | | o-------------------------------------------------o Figure 1. Elementary Structure and Terminology </pre> |} In its original usage a statement of Fact has to do with a deed done or a record made, that is, a type of event that is openly observable and not riddled with speculation as to its very occurrence. In contrast, a statement of Case may refer to a hidden or a hypothetical cause, that is, a type of event that is not immediately observable to all concerned. Obviously, the distinction is a rough one and the question of which mode applies can depend on the points of view that different observers adopt over time. Finally, a statement of a Rule is called that because it states a regularity or a regulation that governs a whole class of situations, and not because of its syntactic form. So far in this discussion, all three types of constraint are expressed in the form of conditional propositions, but this is not a fixed requirement. In practice, these modes of statement are distinguished by the roles that they play within an argument, not by their style of expression. When the time comes to branch out from the syllogistic framework, we will find that propositional constraints can be discovered and represented in arbitrary syntactic forms. ===Kinds of inference=== The three kinds of inference that Peirce would come to refer to as ''abductive'', ''deductive'', and ''inductive'' inference he gives his earliest systematic treatment in two series of lectures on the logic of science: the [[Harvard University]] Lectures of 1865 and the [[Lowell Institute]] Lectures of 1866. There he sums up the characters of the three kinds of reasoning in the following terms: {| align="center" cellpadding="4" width="90%" <!--QUOTE--> | <p>We have then three different kinds of inference:</p> <p>&nbsp;&nbsp;&nbsp;Deduction or inference ''[[a priori and a posteriori (philosophy)|à priori]]'',</p> <p>&nbsp;&nbsp;&nbsp;Induction or inference ''[[à particularis]]'', and</p> <p>&nbsp;&nbsp;&nbsp;Hypothesis or inference ''[[à posteriori]]''.</p> <p>(Peirce, "On the Logic of Science" (1865), CE 1, 267).</p> |} Early in the first series of lectures Peirce gives a very revealing illustration of how he then thinks of the natures, operations, and relationships of this trio of inference types: {| align="center" cellpadding="4" width="90%" <!--QUOTE--> | <p> If I reason that certain conduct is wise <br>because it has a character which belongs <br>''only'' to wise things, I reason ''à priori''.</p> <p> If I think it is wise because it once turned out <br>to be wise, that is, if I infer that it is wise on <br>this occasion because it was wise on that occasion, <br>I reason inductively [''à particularis''].</p> <p> But if I think it is wise because a wise man does it, <br>I then make the pure hypothesis that he does it <br>because he is wise, and I reason ''à posteriori''.</p> <p>(Peirce, "On the Logic of Science" (1865), CE 1, 180).</p> |} We may begin the analysis of Peirce's example by making the following assignments of letters to the qualitative attributes mentioned in it: :* A = 'Wisdom', :* B = 'a certain character', :* C = 'a certain conduct', :* D = 'done by a wise man', :* E = 'a certain occasion'. Recognizing that a little more concreteness will serve as an aid to the understanding, let's augment the Spartan features of Peirce's illustration in the following way: :* B = 'Benevolence', a certain character, :* C = 'Contributes to Charity', a certain conduct, :* E = 'Earlier today', a certain occasion. The converging operation of all three reasonings is shown in Figure 2. {| align="center" cellpadding="8" style="text-align:center" | <pre> o---------------------------------------------------------------------o | | | D ("done by a wise man") | | o | | \* | | \ * | | \ * | | \ * | | \ * | | \ * | | \ * A ("a wise act") | | \ o | | \ /| * | | \ / | * | | \ / | * | | . | o B ("benevolence", a certain character) | | / \ | * | | / \ | * | | / \| * | | / o | | / * C ("contributes to charity", a certain conduct) | | / * | | / * | | / * | | / * | | / * | | /* | | o | | E ("earlier today", a certain occasion) | | | o---------------------------------------------------------------------o Figure 2. A Thrice Wise Act </pre> |} One of the styles of syntax that Aristotle uses for syllogistic propositions suggests the composite symbols that geometers have long used for labeling line intervals in a geometric figure, and it comports quite nicely with the Figure that we have just drawn. Specifically, the proposition that predicates X of the subject Y is represented by the digram 'XY' and associated with the line interval XY that descends from the point X to the point Y in the corresponding lattice diagram. In this wise we make the following observations: The common proposition that concludes each argument is AC. Introducing the symbol "&rArr;" to denote the relation of logical implication, the proposition AC can be written as C &rArr; A, and read as "C implies A". Adopting the parenthetical form of Peirce's alpha graphs, in their ''existential interpretation'', AC can be written as (C (A)), and most easily comprehended as "not C without A". In the context of the present example, all of these forms are equally good ways of expressing the same concrete proposition, namely, "contributing to charity is wise". :* Deduction could have obtained the Fact AC from the Rule AB, "benevolence is wisdom", along with the Case BC, "contributing to charity is benevolent". :* Induction could have gathered the Rule AC, after a manner of saying that "contributing to charity is exemplary of wisdom", from the Fact AE, "the act of earlier today is wise", along with the Case CE, "the act of earlier today was an instance of contributing to charity". :* Abduction could have guessed the Case AC, in a style of expression stating that "contributing to charity is explained by wisdom", from the Fact DC, "contributing to charity is done by this wise man", and the Rule DA, "everything that is wise is done by this wise man". Thus, a wise man, who happens to do all of the wise things that there are to do, may nevertheless contribute to charity for no good reason, and even be known to be charitable to a fault. But all of this notwithstanding, on seeing the wise man contribute to charity we may find it natural to conjecture, in effect, to consider it as a possibility worth examining further, that charity is indeed a mark of his wisdom, and not just the accidental trait or the immaterial peculiarity of his character &mdash; in essence, that wisdom is the ''cause'' of his contribution or the ''reason'' for his charity. As a general rule, and despite many obvious exceptions, an English word that ends in ''-ion'' denotes equivocally either a process or its result. In our present application, this means that each of the words ''abduction'', ''deduction'', ''induction'' can be used to denote either the process of inference or the product of that inference, that is, the proposition to which the inference in question leads. One of the morals of Peirce's illustration can now be drawn. It demonstrates in a very graphic fashion that the three kinds of inference are three kinds of process and not three kinds of proposition, not if one takes the word ''kind'' in its literal sense as denoting a ''genus'' of being, essence, or substance. Said another way, it means that being an abductive Case, a deductive Fact, or an inductive Rule is a category of relation, indeed, one that involves at the very least a triadic relation among propositions, and not a category of essence or substance, that is, not a property that inheres in the proposition alone. This category distinction between the absolute, essential, or monadic predicates and the more properly relative predicates constitutes a very important theme in Peirce's architectonic. There is of course a parallel application of it in the theory of sign relations, or semiotics, where the distinctions among the sign relational roles of Object, Sign, and Interpretant are distinct ways of relating to other things, modes of relation that may vary from moment to moment in the extended trajectory of a sign process, and not distinctions that mark some fixed and eternal essence of the thing in itself. In the normal course of inquiry, the elementary types of inference proceed in the order: Abduction, Deduction, Induction. However, the same building blocks can be assembled in other ways to yield different types of complex inferences. Of particular importance, reasoning by analogy can be analyzed as a combination of induction and deduction, in other words, as the abstraction and the application of a rule. Because a complicated pattern of analogical inference will be used in our example of a complete inquiry, it will help to prepare the ground if we first stop to consider an example of analogy in its simplest form. ====Abduction==== : ''Main article : [[Abductive reasoning]]'' Much of Peirce's work deals with the scientific and logical questions of [[knowledge]] and [[truth]], questions grounded in his experience as a working logician and experimental scientist, one who was a member of the international community of scientists and thinkers of his day. He made important contributions to [[deductive logic]] (see below), but was primarily interested in the logic of science and specifically in what he called [[abduction (logic)|abduction]] or "hypothesis", as opposed to [[deductive reasoning|deduction]] and [[inductive reasoning|induction]]. Abduction is the process whereby a hypothesis is generated, so that surprising facts may be explained. "There is a more familiar name for it than abduction", Peirce wrote, "for it is neither more nor less than guessing". Indeed, Peirce considered abduction to be at the heart not only of scientific research but of native human intelligence as well. In his "Illustrations of the Logic of Science" (CE 3, 325-326), Peirce gives the following example of how abduction nests with deductive and inductive reasoning. Peirce begins by positing the following three statements: :*''Rule'': "All the beans from this bag are white." :*''Case'': "These beans are from this bag." :*''Result'': "These beans are white." Now let any two of these statements be Givens (their order not mattering), and let the remaining statement be the Conclusion. The result is an ''argument'', of which three kinds are possible: {| align="center" cellpadding="4" |- ! &nbsp; !! Deduction !! Induction !! Abduction |- |- style="border-top:1px solid #999;" |- | ''Premiss'' || Rule || Case || Rule |- | ''Premiss'' || Case || Fact || Fact |- | ''Conclusion'' || Fact || Rule || Case |} ====Deduction==== : ''Main article : [[Deductive reasoning]]'' ====Induction==== : ''Main article : [[Inductive reasoning]]'' ====Analogy==== : ''Main article : [[Analogy]]'' The classic description of [[analogy]] in the syllogistic frame comes from Aristotle, who called this form of inference by the name ''paradeigma'', that is, reasoning by way of example or through the parallel comparison of cases. {| align="center" cellpadding="4" width="90%" <!--QUOTE--> | <p>We have an Example [&#960;&#945;&#961;&#945;&#948;&#949;&#953;&#947;&#956;&#945;, analogy] when the major extreme is shown to be applicable to the middle term by means of a term similar to the third. It must be known both that the middle applies to the third term and that the first applies to the term similar to the third. (Aristotle, "Prior Analytics", 2.24).</p> |} Aristotle illustrates this pattern of argument with the following sample of reasoning. The setting is a discussion, taking place in Athens, on the issue of going to war with Thebes. It is apparently accepted that a war between Thebes and Phocis is or was a bad thing, perhaps from the objectivity lent by non-involvement or perhaps as a lesson of history. {| align="center" cellpadding="4" width="90%" <!--QUOTE--> | <p>For example, let ''A'' be 'bad', ''B'' 'to make war on neighbors', ''C'' 'Athens against Thebes', and ''D'' 'Thebes against Phocis'. Then if we require to prove that war against Thebes is bad, we must be satisfied that war against neighbors is bad. Evidence of this can be drawn from similar examples, for example, that war by Thebes against Phocis is bad. Then since war against neighbors is bad, and war against Thebes is war against neighbors, it is evident that war against Thebes is bad. (Aristotle, "Prior Analytics", 2.24, with minor alterations).</p> |} Aristotle's sample of argument from analogy may be analyzed in the following way: First, a Rule is induced from the consideration of a similar Case and a relevant Fact: :* Case: D &rArr; B, Thebes vs Phocis is war against neighbors. :* Fact: D &rArr; A, Thebes vs Phocis is bad. :* Rule: B &rArr; A, War against neighbors is bad. Next, the Fact to be proved is deduced from the application of the previously induced Rule to the present Case: :* Case: C &rArr; B, Athens vs Thebes is war against neighbors. :* Rule: B &rArr; A, War against neighbors is bad. :* Fact: C &rArr; A, Athens vs Thebes is bad. In practice, of course, it would probably take a mass of comparable cases to establish a rule. As far as the logical structure goes, however, this quantitative confirmation only amounts to 'gilding the lily'. Perfectly valid rules can be guessed on the first try, abstracted from a single experience or adopted vicariously with no personal experience. Numerical factors only modify the degree of confidence and the strength of habit that govern the application of previously learned rules. Figure 3 gives a graphical illustration of Aristotle's example of 'Example', that is, the form of reasoning that proceeds by Analogy or according to a Paradigm. {| align="center" cellpadding="8" style="text-align:center" | <pre> o-----------------------------------------------------------o | | | A | | o | | /*\ | | / * \ | | / * \ | | / * \ | | / * \ | | / * \ | | / R u l e \ | | / * \ | | / * \ | | / * \ | | / * \ | | F a c t o F a c t | | / * B * \ | | / * * \ | | / * * \ | | / * * \ | | / C a s e C a s e \ | | / * * \ | | / * * \ | | / * * \ | | / * * \ | | / * * \ | | o o | | C D | | | | A = Atrocious, Adverse to All, A bad thing | | B = Belligerent Battle Between Brethren | | C = Contest of Athens against Thebes | | D = Debacle of Thebes against Phocis | | | | A is a major term | | B is a middle term | | C is a minor term | | D is a minor term, similar to C | | | o-----------------------------------------------------------o Figure 3. Aristotle's "War Against Neighbors" Example </pre> |} In this analysis of reasoning by Analogy, it is a complex or a mixed form of inference that can be seen as taking place in two steps: :* The first step is an Induction that abstracts a Rule from a Case and a Fact. :: Case: D &rArr; B, Thebes vs Phocis is a battle between neighbors. :: Fact: D &rArr; A, Thebes vs Phocis is adverse to all. :: Rule: B &rArr; A, A battle between neighbors is adverse to all. :* The final step is a Deduction that applies this Rule to a Case to arrive at a Fact. :: Case: C &rArr; B, Athens vs Thebes is a battle between neighbors. :: Rule: B &rArr; A, A battle between neighbors is adverse to all. :: Fact: C &rArr; A, Athens vs Thebes is adverse to all. As we see, Aristotle analyzed analogical reasoning into a phase of inductive reasoning followed by a phase of deductive reasoning. Peirce would pick up the story at this juncture and eventually parse analogy in a couple of different ways, both of them involving all three types of inference: abductive, deductive, and inductive. ==Example of inquiry== Examples of inquiry, that illustrate the full cycle of its abductive, deductive, and inductive phases, and yet are both concrete and simple enough to be suitable for a first (or zeroth) exposition, are somewhat rare in Peirce's writings, and so let us draw one from the work of fellow pragmatician [[John Dewey]], analyzing it according to the model of zeroth-order inquiry that we developed above. {| align="center" cellpadding="4" width="90%" <!--QUOTE--> | <p>A man is walking on a warm day. The sky was clear the last time he observed it; but presently he notes, while occupied primarily with other things, that the air is cooler. It occurs to him that it is probably going to rain; looking up, he sees a dark cloud between him and the sun, and he then quickens his steps. What, if anything, in such a situation can be called thought? Neither the act of walking nor the noting of the cold is a thought. Walking is one direction of activity; looking and noting are other modes of activity. The likelihood that it will rain is, however, something ''suggested''. The pedestrian ''feels'' the cold; he ''thinks of'' clouds and a coming shower. (John Dewey, ''How We Think'', pp. 6&ndash;7).</p> |} ===Once over quickly=== Let's first give Dewey's elegant example of inquiry in everyday life the quick once over, hitting just the high points of its analysis into Peirce's three kinds of reasoning. ====Abductive phase==== In Dewey's 'Rainy Day' or 'Sign of Rain' story, we find our peripatetic hero presented with a surprising Fact: :* Fact: C &rArr; A, In the Current situation the Air is cool. Responding to an intellectual reflex of puzzlement about the situation, his resource of common knowledge about the world is impelled to seize on an approximate Rule: :* Rule: B &rArr; A, Just Before it rains, the Air is cool. This Rule can be recognized as having a potential relevance to the situation because it matches the surprising Fact, C &rArr; A, in its consequential feature A. All of this suggests that the present Case may be one in which it is just about to rain: :* Case: C &rArr; B, The Current situation is just Before it rains. The whole mental performance, however automatic and semi-conscious it may be, that leads up from a problematic Fact and a previously settled knowledge base of Rules to the plausible suggestion of a Case description, is what we are calling an [[abductive inference]]. ====Deductive phase==== The next phase of inquiry uses deductive inference to expand the implied consequences of the abductive hypothesis, with the aim of testing its truth. For this purpose, the inquirer needs to think of other things that would follow from the consequence of his precipitate explanation. Thus, he now reflects on the Case just assumed: :* Case: C &rArr; B, The Current situation is just Before it rains. He looks up to scan the sky, perhaps in a random search for further information, but since the sky is a logical place to look for details of an imminent rainstorm, symbolized in our story by the letter B, we may safely suppose that our reasoner has already detached the consequence of the abduced Case, C &rArr; B, and has begun to expand on its further implications. So let us imagine that our up-looker has a more deliberate purpose in mind, and that his search for additional data is driven by the new-found, determinate Rule: :* Rule: B &rArr; D, Just Before it rains, Dark clouds appear. Contemplating the assumed Case in combination with this new Rule leads him by an immediate deduction to predict an additional Fact: :* Fact: C &rArr; D, In the Current situation Dark clouds appear. The reconstructed picture of reasoning assembled in this second phase of inquiry is true to the pattern of [[deductive inference]]. ====Inductive phase==== Whatever the case, our subject observes a Dark cloud, just as he would expect on the basis of the new hypothesis. The explanation of imminent rain removes the discrepancy between observations and expectations and thereby reduces the shock of surprise that made this process of inquiry necessary. ===Looking more closely=== ====Seeding hypotheses==== Figure 4 gives a graphical illustration of Dewey's example of inquiry, isolating for the purposes of the present analysis the first two steps in the more extended proceedings that go to make up the whole inquiry. {| align="center" cellpadding="8" style="text-align:center" | <pre> o-----------------------------------------------------------o | | | A D | | o o | | \ * * / | | \ * * / | | \ * * / | | \ * * / | | \ * * / | | \ R u l e R u l e / | | \ * * / | | \ * * / | | \ * * / | | \ * B * / | | F a c t o F a c t | | \ * / | | \ * / | | \ * / | | \ * / | | \ C a s e / | | \ * / | | \ * / | | \ * / | | \ * / | | \ * / | | \*/ | | o | | C | | | | A = the Air is cool | | B = just Before it rains | | C = the Current situation | | D = a Dark cloud appears | | | | A is a major term | | B is a middle term | | C is a minor term | | D is a major term, associated with A | | | o-----------------------------------------------------------o Figure 4. Dewey's "Rainy Day" Inquiry </pre> |} In this analysis of the first steps of Inquiry, we have a complex or a mixed form of inference that can be seen as taking place in two steps: :* The first step is an Abduction that abstracts a Case from the consideration of a Fact and a Rule. :: Fact: C &rArr; A, In the Current situation the Air is cool. :: Rule: B &rArr; A, Just Before it rains, the Air is cool. :: Case: C &rArr; B, The Current situation is just Before it rains. :* The final step is a Deduction that admits this Case to another Rule and so arrives at a novel Fact. :: Case: C &rArr; B, The Current situation is just Before it rains. :: Rule: B &rArr; D, Just Before it rains, a Dark cloud will appear. :: Fact: C &rArr; D, In the Current situation, a Dark cloud will appear. This is nowhere near a complete analysis of the Rainy Day inquiry, even insofar as it might be carried out within the constraints of the syllogistic framework, and it covers only the first two steps of the relevant inquiry process, but maybe it will do for a start. One other thing needs to be noticed here, the formal [[duality]] between this expansion phase of inquiry and the argument from [[analogy]]. This can be seen most clearly in the propositional [[lattice]] diagrams shown in Figures 3 and 4, where analogy exhibits a rough "A" shape and the first two steps of inquiry exhibit a rough "V" shape, respectively. Since we find ourselves repeatedly referring to this expansion phase of inquiry as a unit, let's give it a name that suggests its duality with [[analogical reasoning|analogy]] — '[[catalogical reasoning|catalogy]]' will do for the moment. This usage is apt enough if one thinks of a catalogue entry for an item as a text that lists its salient features. Notice that [[analogical reasoning|analogy]] has to do with the examples of a given quality, while [[catalogical reasoning|catalogy]] has to do with the qualities of a given example. Peirce noted similar forms of duality in many of his early writings, leading to the consummate treatment in his 1867 paper [http://members.door.net/arisbe/menu/library/bycsp/newlist/nl-frame.htm "On a New List of Categories"] (CP 1.545-559, CE 2, 49-59). ====Weeding hypotheses==== In order to comprehend the bearing of [[inductive reasoning]] on the closing phases of inquiry there are a couple of observations that we need to make: :* First, we need to recognize that smaller inquiries are typically woven into larger inquiries, whether we view the whole pattern of inquiry as carried on by a single agent or by a complex community. :* Further, we need to consider the different ways in which the particular instances of inquiry can be related to ongoing inquiries at larger scales. Three modes of inductive interaction between the micro-inquiries and the macro-inquiries that are salient here can be described under the headings of the 'Learning', the 'Transfer', and the 'Testing' of rules. ====Analogy of experience==== Throughout inquiry the reasoner makes use of rules that have to be transported across intervals of experience, from the masses of experience where they are learned to the moments of experience where they are applied. Inductive reasoning is involved in the learning and the transfer of these rules, both in accumulating a knowledge base and in carrying it through the times between acquisition and application. :* Learning. The principal way that induction contributes to an ongoing inquiry is through the learning of rules, that is, by creating each of the rules that goes into the knowledge base, or ever gets used along the way. :* Transfer. The continuing way that induction contributes to an ongoing inquiry is through the exploit of analogy, a two-step combination of induction and deduction that serves to transfer rules from one context to another. :* Testing. Finally, every inquiry that makes use of a knowledge base constitutes a 'field test' of its accumulated contents. If the knowledge base fails to serve any live inquiry in a satisfactory manner, then there is a prima facie reason to reconsider and possibly to amend some of its rules. Let's now consider how these principles of learning, transfer, and testing apply to John Dewey's 'Sign of Rain' example. =====Learning===== Rules in a knowledge base, as far as their effective content goes, can be obtained by any mode of inference. For example, a rule like: :* Rule: B &rArr; A, Just Before it rains, the Air is cool, is usually induced from a consideration of many past events, in a manner that can be rationally reconstructed as follows: :* Case: C &rArr; B, In Certain events, it is just Before it rains, :* Fact: C &rArr; A, In Certain events, the Air is cool, : ------------------------------------------------------------------------------------------ :* Rule: B &rArr; A, Just Before it rains, the Air is cool. However, the very same proposition could also be abduced as an explanation of a singular occurrence or deduced as a conclusion of a presumptive theory. =====Transfer===== What is it that gives a distinctively inductive character to the acquisition of a knowledge base? It is evidently the 'analogy of experience' that underlies its useful application. Whenever we find ourselves prefacing an argument with the phrase 'If past experience is any guide …' then we can be sure that this principle has come into play. We are invoking an analogy between past experience, considered as a totality, and present experience, considered as a point of application. What we mean in practice is this: 'If past experience is a fair sample of possible experience, then the knowledge gained in it applies to present experience'. This is the mechanism that allows a knowledge base to be carried across gulfs of experience that are indifferent to the effective contents of its rules. Here are the details of how this notion of transfer works out in the case of the 'Sign of Rain' example: Let K(pres) be a portion of the reasoner's knowledge base that is logically equivalent to the conjunction of two rules, as follows: :* K(pres) = (B &rArr; A) and (B &rArr; D). K(pres) is the present knowledge base, expressed in the form of a logical constraint on the present universe of discourse. It is convenient to have the option of expressing all logical statements in terms of their [[logical model]]s, that is, in terms of the primitive circumstances or the elements of experience over which they hold true. :* Let E(past) be the chosen set of experiences, or the circumstances that we have in mind when we refer to 'past experience'. :* Let E(poss) be the collective set of experiences, or the projective total of possible circumstances. :* Let E(pres) be the present experience, or the circumstances that are present to the reasoner at the current moment. If we think of the knowledge base K(pres) as referring to the 'regime of experience' over which it is valid, then all of these sets of models can be compared by the simple relations of [[set inclusion]] or [[logical implication]]. Figure 5 schematizes this way of viewing the 'analogy of experience'. {| align="center" cellpadding="8" style="text-align:center" | <pre> o-----------------------------------------------------------o | | | K(pres) | | o | | /|\ | | / | \ | | / | \ | | / | \ | | / Rule \ | | / | \ | | / | \ | | / | \ | | / E(poss) \ | | Fact / o \ Fact | | / * * \ | | / * * \ | | / * * \ | | / * * \ | | / * * \ | | / * Case Case * \ | | / * * \ | | / * * \ | | /* *\ | | o<<<---------------<<<---------------<<<o | | E(past) Analogy Morphism E(pres) | | More Known Less Known | | | o-----------------------------------------------------------o Figure 5. Analogy of Experience </pre> |} In these terms, the ''analogy of experience'' proceeds by inducing a Rule about the validity of a current knowledge base and then deducing a Fact, its applicability to a current experience, as in the following sequence: Inductive Phase: :* Given Case: E(past) &rArr; E(poss), Chosen events fairly sample Collective events. :* Given Fact: E(past) &rArr; K(pres), Chosen events support the Knowledge regime. : ----------------------------------------------------------------------------------------------------------------------------- :* Induce Rule: E(poss) &rArr; K(pres), Collective events support the Knowledge regime. Deductive Phase: :* Given Case: E(pres) &rArr; E(poss), Current events fairly sample Collective events. :* Given Rule: E(poss) &rArr; K(pres), Collective events support the Knowledge regime. : -------------------------------------------------------------------------------------------------------------------------------- :* Deduce Fact: E(pres) &rArr; K(pres), Current events support the Knowledge regime. =====Testing===== If the observer looks up and does not see dark clouds, or if he runs for shelter but it does not rain, then there is fresh occasion to question the utility or the validity of his knowledge base. But we must leave our foulweather friend for now and defer the logical analysis of this testing phase to another occasion. ==References== * [[Dana Angluin|Angluin, Dana]] (1989), "Learning with Hints", pp. 167&ndash;181 in David Haussler and Leonard Pitt (eds.), ''Proceedings of the 1988 Workshop on Computational Learning Theory'', MIT, 3&ndash;5 August 1988, Morgan Kaufmann, San Mateo, CA, 1989. * [[Aristotle]], "[[Prior Analytics]]", [[Hugh Tredennick]] (trans.), pp. 181&ndash;531 in ''Aristotle, Volume 1'', [[Loeb Classical Library]], [[Heinemann (book publisher)|William Heinemann]], London, UK, 1938. * Awbrey, S.M., and Awbrey, J.L. (May 2001), "Conceptual Barriers to Creating Integrative Universities", ''Organization : The Interdisciplinary Journal of Organization, Theory, and Society'' 8(2), Sage Publications, London, UK, pp. 269&ndash;284. [http://org.sagepub.com/cgi/content/abstract/8/2/269 Abstract]. * Awbrey, S.M., and Awbrey, J.L. (September 18, 1999), "Organizations of Learning or Learning Organizations : The Challenge of Creating Integrative Universities for the Next Century", ''Second International Conference of the Journal ''Organization'' '', ''Re-Organizing Knowledge, Trans-Forming Institutions : Knowing, Knowledge, and the University in the 21st Century'', University of Massachusetts, Amherst, MA. [http://www.cspeirce.com/menu/library/aboutcsp/awbrey/integrat.htm Online]. * Awbrey, J.L., and Awbrey, S.M. (Autumn 1995), "Interpretation as Action : The Risk of Inquiry", ''Inquiry : Critical Thinking Across the Disciplines'' 15(1), pp. 40&ndash;52. [http://www.chss.montclair.edu/inquiry/fall95/awbrey.html Online]. * Awbrey, J.L., and Awbrey, S.M. (June 1992), "Interpretation as Action : The Risk of Inquiry", ''The Eleventh International Human Science Research Conference'', Oakland University, Rochester, Michigan. * Awbrey, S.M., and Awbrey, J.L. (May 1991), "An Architecture for Inquiry : Building Computer Platforms for Discovery", ''Proceedings of the Eighth International Conference on Technology and Education'', Toronto, Canada, pp. 874&ndash;875. [http://home.m04.itscom.net/hhomey/tmp-a.html Online]. * Awbrey, J.L., and Awbrey, S.M. (January 1991), "Exploring Research Data Interactively : Developing a Computer Architecture for Inquiry", Poster presented at the ''Annual Sigma Xi Research Forum'', University of Texas Medical Branch, Galveston, TX. * Awbrey, J.L., and Awbrey, S.M. (August 1990), "Exploring Research Data Interactively. Theme One : A Program of Inquiry", ''Proceedings of the Sixth Annual Conference on Applications of Artificial Intelligence and CD-ROM in Education and Training'', Society for Applied Learning Technology, Washington, DC, pp. 9&ndash;15. * [[Cornelius F. Delaney|Delaney, C.F.]] (1993), ''Science, Knowledge, and Mind: A Study in the Philosophy of C.S. Peirce'', University of Notre Dame Press, Notre Dame, IN. * [[John Dewey|Dewey, John]] (1910), ''How We Think'', [[D.C. Heath]], Lexington, MA, 1910. Reprinted, Prometheus Books, Buffalo, NY, 1991. * Dewey, John (1938), ''Logic: The Theory of Inquiry'', Henry Holt and Company, New York, NY, 1938. Reprinted as pp. 1–527 in ''John Dewey, The Later Works, 1925&ndash;1953, Volume 12 : 1938'', Jo Ann Boydston (ed.), Kathleen Poulos (text. ed.), [[Ernest Nagel]] (intro.), Southern Illinois University Press, Carbondale and Edwardsville, IL, 1986. * [[Susan Haack|Haack, Susan]] (1993), ''Evidence and Inquiry&nbsp;: Towards Reconstruction in Epistemology'', Blackwell Publishers, Oxford, UK. * [[Norwood Russell Hanson|Hanson, Norwood Russell]] (1958), ''Patterns of Discovery, An Inquiry into the Conceptual Foundations of Science'', Cambridge University Press, Cambridge, UK. * [[Vincent F. Hendricks|Hendricks, Vincent F.]] (2005), ''Thought 2 Talk&nbsp;: A Crash Course in Reflection and Expression'', Automatic Press, New York, NY. * [[Cheryl J. Misak|Misak, Cheryl J.]] (1991), ''Truth and the End of Inquiry, A Peircean Account of Truth'', Oxford University Press, Oxford, UK. * [[Charles Peirce (Bibliography)|Peirce, C.S., Bibliography]]. * [[Charles Sanders Peirce|Peirce, C.S.]], (1931&ndash;1935, 1958), ''Collected Papers of Charles Sanders Peirce'', vols. 1&ndash;6, [[Charles Hartshorne]] and [[Paul Weiss (philosopher)|Paul Weiss]] (eds.), vols. 7&ndash;8, [[Arthur W. Burks]] (ed.), Harvard University Press, Cambridge, MA. Cited as CP&nbsp;volume.paragraph. * [[Robert C. Stalnaker|Stalnaker, Robert C.]] (1984), ''Inquiry'', MIT Press, Cambridge, MA. ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://mywikibiz.com/Inquiry Inquiry @ MyWikiBiz] * [http://mathweb.org/wiki/Inquiry Inquiry @ MathWeb Wiki] * [http://netknowledge.org/wiki/Inquiry Inquiry @ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Inquiry Inquiry @ OER Commons] {{col-break}} * [http://p2pfoundation.net/Inquiry Inquiry @ P2P Foundation] * [http://semanticweb.org/wiki/Inquiry Inquiry @ SemanticWeb] * [http://ref.subwiki.org/wiki/Inquiry Inquiry @ Subject Wikis] * [http://beta.wikiversity.org/wiki/Inquiry Inquiry @ Wikiversity Beta] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;] * [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;] ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Inquiry Inquiry], [http://mywikibiz.com/ MyWikiBiz] * [http://mathweb.org/wiki/Inquiry Inquiry], [http://mathweb.org/ MathWeb Wiki] * [http://netknowledge.org/wiki/Inquiry Inquiry], [http://netknowledge.org/ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Inquiry Inquiry], [http://wiki.oercommons.org/ OER Commons] {{col-break}} * [http://p2pfoundation.net/Inquiry Inquiry], [http://p2pfoundation.net/ P2P Foundation] * [http://semanticweb.org/wiki/Inquiry Inquiry], [http://semanticweb.org/ SemanticWeb] * [http://beta.wikiversity.org/wiki/Inquiry Inquiry], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://getwiki.net/-Inquiry Inquiry], [http://getwiki.net/ GetWiki] {{col-break}} * [http://wikinfo.org/index.php/Inquiry Inquiry], [http://wikinfo.org/ Wikinfo] * [http://textop.org/wiki/index.php?title=Inquiry Inquiry], [http://textop.org/wiki/ Textop Wiki] * [http://en.wikipedia.org/w/index.php?title=Inquiry&oldid=71880922 Inquiry], [http://en.wikipedia.org/ Wikipedia] * [http://forum.wolframscience.com/showthread.php?threadid=595 Inquiry], [http://forum.wolframscience.com/ NKS Forum] {{col-end}} [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Artificial Intelligence]] [[Category:Computer Science]] [[Category:Critical Thinking]] [[Category:Cybernetics]] [[Category:Education]] [[Category:Hermeneutics]] [[Category:Information Systems]] [[Category:Information Theory]] [[Category:Inquiry Driven Systems]] [[Category:Intelligence Amplification]] [[Category:Learning Organizations]] [[Category:Knowledge Representation]] [[Category:Logic]] [[Category:Philosophy]] [[Category:Pragmatics]] [[Category:Semantics]] [[Category:Semiotics]] [[Category:Systems Science]] d7187316f79163ca3f03048715e33298fd61a861 0 343 565 2010-06-23T17:16:04Z Jon Awbrey 3 + article wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. <blockquote> Every mind which passes from doubt to belief must have ideas which follow after one another in time. Every mind which reasons must have ideas which not only follow after others but are caused by them. Every mind which is capable of logical criticism of its inferences, must be aware of this determination of its ideas by previous ideas. (Peirce, "On Time and Thought", CE&nbsp;3, 68&ndash;69.) </blockquote> All through the 1860s the young Charles Peirce was busy establishing a conceptual base-camp and a technical supply line for the intellectual adventures of a lifetime. Taking the long view of this activity and trying to choose the best titles for the story, it all seems to have something to do with the dynamics of inquiry. This broad subject area has a part that is given by nature and a part that is ruled by nurture. On first approach, it is possible to see a question of articulation and a question of explanation: :* What is needed to articulate the workings of the active form of representation that is known as ''conscious experience''? :* What is needed to account for the workings of the reflective discipline of inquiry that is known as ''science''? The pursuit of answers to these questions finds them to be so entangled with each other that it's ultimately impossible to comprehend them apart from each other, but for the sake of exposition it's convenient to organize our study of Peirce's assault on the ''summa'' by following first the trails of thought that led him to develop a ''[[theory of signs]]'', one that has come to be known as '[[semiotic]]', and tracking next the ways of thinking that led him to develop a ''[[theory of inquiry]]'', one that would be up to the task of saying 'how science works'. Opportune points of departure for exploring the dynamics of representation, such as led to Peirce's theories of [[inference]] and [[information]], inquiry and signs, are those that he took for his own springboards. Perhaps the most significant influences radiate from points on parallel lines of inquiry in [[Aristotle]]'s work, points where the intellectual forerunner focused on many of the same issues and even came to strikingly similar conclusions, at least about the best ways to begin. Staying within the bounds of what will give us a more solid basis for understanding Peirce, it serves to consider the following ''loci'' in Aristotle: :* The basic terminology of [[psychology]], in ''[[On the Soul]]''. :* The founding description of [[sign relations]], in ''[[On Interpretation]]''; :* The differentiation of the genus of reasoning into three species of [[inference]] that are commonly translated into English as ''[[Abductive reasoning|abduction]]'', ''[[Deductive reasoning|deduction]]'', and ''[[Inductive reasoning|induction]]'', in the ''[[Prior Analytics]]''. In addition to the three elements of inference, that Peirce would assay to be [[irreducible]], [[Aristotle]] analyzed several types of [[compound inference]], most importantly the type known as 'reasoning by [[analogy]]' or 'reasoning from [[example]]', employing for the latter description the Greek word 'paradeigma', from which we get our word '[[paradigm]]'. Inquiry is a form of reasoning process, in effect, a particular way of conducting thought, and thus it can be said to institute a specialized manner, style, or turn of thinking. Philosophers of the school that is commonly called 'pragmatic' hold that all thought takes place in signs, where 'sign' is the word they use for the broadest conceivable variety of characters, expressions, formulas, messages, signals, texts, and so on up the line, that might be imagined. Even intellectual concepts and mental ideas are held to be a special class of signs, corresponding to internal states of the thinking agent that both issue in and result from the interpretation of external signs. The subsumption of inquiry within reasoning in general and the inclusion of thinking within the class of sign processes allows us to approach the subject of inquiry from two different perspectives: :* The ''[[syllogistic]]'' approach treats inquiry as a species of logical process, and is limited to those of its aspects that can be related to the most basic laws of inference. :* The ''[[sign-theoretic]]'' approach views inquiry as a genus of ''[[semiosis]]'', an activity taking place within the more general setting of [[sign relation]]s and [[sign process]]es. The distinction between signs denoting and objects denoted is critical to the discussion of Peirce's theory of signs. Wherever needed in the rest of this article, therefore, in order to mark this distinction a little more emphatically than usual, double quotation marks placed around a given sign, for example, a string of zero or more characters, will be used to create a new sign that denotes the given sign as its object. ===Semeiotic : Peirce's theory of signs=== Peirce referred to his general study of signs, based on the concept of a [[triadic relation|triadic]] [[sign relation]], as ''[[semeiotic]]'' or ''[[semiotic]]'', either of which terms are currently used in both singular of plural forms. Peirce began writing on semeiotic in the 1860s, around the time that he devised his system of three categories. He eventually defined ''[[semiosis]]'' as an "action, or influence, which is, or involves, a cooperation of ''three'' subjects, such as a sign, its object, and its interpretant, this tri-relative influence not being in any way resolvable into actions between pairs". (Houser 1998: 411, written 1907). This triadic relation grounds the semeiotic. In order to understand what a ''sign'' is we need to understand what a ''[[sign relation]]'' is, for signhood is a way of being in relation, not a way of being in itself. In order to understand what a sign relation is we need to understand what a ''[[triadic relation]]'' is, for the role of a sign is constituted as one among three, where roles in general are distinct even when the things that fill them are not. In order to understand what a triadic relation is we need to understand what a ''[[relation (mathematics)|relation]]'' is, and here there are traditionally two ways of understanding what a relation is, both of which are necessary if not sufficient to complete understanding, namely, the way of ''[[extension (semantics)|extension]]'' and the way of ''[[intension]]''. To these traditional approximations, Peirce adds a third way, the way of ''[[semiotic information theory|information]]'', that integrates the other two approaches in a unified whole. ====Sign relations==== : ''Main article'' : [[Sign relation]] With that hasty map of relations and relatives sketched above, we may now trek into the terrain of ''sign relations'', the main subject matter of Peirce's ''semeiotic'', or theory of signs. ====Types of signs==== Peirce proposes several typologies and definitions of the signs. More than 76 definitions of what a sign is have been collected throughout Peirce's work. Some canonical typologies can nonetheless be observed, one crucial one being the distinction between "icons", "indices" and "symbols" (CP 2.228, CP 2.229 and CP 5.473). This typology emphasizes the different ways in which the ''representamen'' (or its ''ground'') addresses or refers to its ''object'', through a particular mobilisation of an ''interpretant'' (but Peirce proposes also other typologies based on other criteria). * An '''icon''' is a sign that denotes its objects by virtue of a quality that it shares with them. The sign is perceived as resembling or imitating the object it refers to (e.g. fork on a sign by the road indicating a rest stop). In other words, an icon thus "resembles" to its object. It shares a character or an aspect with it, which allows for it to be interpreted as a sign even if the object does not exist. It signifies essentially on the basis of its "ground". * An '''index''' is a sign that denotes its objects by virtue of an existential connection that it has with them. For an index to signify, the relation to the object is crucial. The ''representamen'' is directly connected in some way (physically or casually) to the object it denotes (e.g. smoke coming from a building is an index of fire). Hence, an index refers to the object because it is really affected or modified by it, and thus may stand as a trace of the existence of the object. * A '''symbol''' is a sign that denotes its objects solely by virtue of the fact that it is interpreted to do so. The ''representamen'' does not resemble the object signified but is fundamentally conventional, so that the signifying relationship must be learned and agreed upon (e.g. the word "cat"). A symbol thus denotes, primarily, by virtue of its ''interpretant''. Its action (''semeiosis'') is ruled by a convention, a more or less systematic set of associations that guarantees its interpretation, independently of any resemblance or any material relation with its object. Note that these definitions are specific to Peirce's theory of signs and are not exactly equivalent to general uses of the notion of "[[icon]]", "[[symbol]]" or "[[index]]". ===Theory of inquiry=== : ''Main article'' : [[Inquiry]] : Upon this first, and in one sense this sole, rule of reason, that in order to learn you must desire to learn, and in so desiring not be satisfied with what you already incline to think, there follows one corollary which itself deserves to be inscribed upon every wall of the city of philosophy: <center> Do not block the way of inquiry.</center> : Although it is better to be methodical in our investigations, and to consider the economics of research, yet there is no positive sin against logic in ''trying'' any theory which may come into our heads, so long as it is adopted in such a sense as to permit the investigation to go on unimpeded and undiscouraged. On the other hand, to set up a philosophy which barricades the road of further advance toward the truth is the one unpardonable offence in reasoning, as it is also the one to which metaphysicians have in all ages shown themselves the most addicted. (Peirce, "F.R.L." (c. 1899), CP 1.135&ndash;136.) Peirce extracted the pragmatic model or [[theory]] of [[inquiry]] from its raw materials in classical logic and refined it in parallel with the early development of symbolic logic to address problems about the nature of scientific reasoning. Borrowing a brace of concepts from [[Aristotle]], Peirce examined three fundamental modes of reasoning that play a role in inquiry, processes that are currently known as ''[[abductive]]'', ''[[deductive]]'', and ''[[inductive]]'' [[inference]]. In the roughest terms, [[abductive reasoning|abduction]] is what we use to generate a likely [[hypothesis]] or an initial [[diagnosis]] in response to a [[phenomenon]] of interest or a [[problem]] of concern, while [[deductive reasoning|deduction]] is used to clarify, to derive, and to explicate the relevant consequences of the selected [[hypothesis]], and [[inductive reasoning|induction]] is used to test the sum of the predictions against the sum of the data. These three processes typically operate in a cyclic fashion, systematically operating to reduce the uncertainties and the difficulties that initiated the inquiry in question, and in this way, to the extent that inquiry is successful, leading to an increase in the [[knowledge]] or [[skills]], in other words, an [[augmentation]] in the [[competence]] or [[performance]], of the agent or community engaged in the inquiry. In the pragmatic way of thinking every thing has a purpose, and the purpose of any thing is the first thing that we should try to note about it. The purpose of inquiry is to reduce doubt and lead to a state of belief, which a person in that state will usually call ''knowledge'' or ''certainty''. It needs to be appreciated that the three kinds of inference, insofar as they contribute to the end of inquiry, describe a cycle that can be understood only as a whole, and none of the three makes complete sense in isolation from the others. For instance, the purpose of [[abductive reasoning|abduction]] is to generate guesses of a kind that [[deductive reasoning|deduction]] can explicate and that [[inductive reasoning|induction]] can evaluate. This places a mild but meaningful [[constraint]] on the production of hypotheses, since it is not just any wild guess at [[explanation]] that submits itself to reason and bows out when defeated in a match with reality. In a similar fashion, each of the other types of [[inference]] realizes its purpose only in accord with its proper role in the whole [[cycle of inquiry]]. No matter how much it may be necessary to study these processes in abstraction from each other, the [[integrity]] of inquiry places strong limitations on the effective modularity of its principal components. If we then think to inquire, "What sort of [[constraint]], exactly, does pragmatic thinking place on our guesses?", we have asked the question that is generally recognized as the problem of "giving a rule to abduction". Peirce's way of answering it is given in terms of the so-called ''[[pragmatic maxim]]'', and this in turn gives us a clue as to the central role of abductive reasoning in Peirce's pragmatic philosophy. ===Logic of information=== : ''Main article'' : [[Logic of information]] <blockquote> Let us now return to the information. The information of a term is the measure of its superfluous comprehension. That is to say that the proper office of the comprehension is to determine the extension of the term. For instance, you and I are men because we possess those attributes &mdash; having two legs, being rational, &tc. &mdash; which make up the comprehension of ''man''. Every addition to the comprehension of a term lessens its extension up to a certain point, after that further additions increase the information instead. (C.S. Peirce, "The Logic of Science, or, Induction and Hypothesis" (1866), CE 1, 467.) </blockquote> ==Source materials== [[C.S. Peirce]], &ldquo;On Time and Thought&rdquo;, MS 215, 8 March 1873. <blockquote> Every mind which passes from doubt to belief must have ideas which follow after one another in time. Every mind which reasons must have ideas which not only follow after others but are caused by them. Every mind which is capable of logical criticism of its inferences, must be aware of this determination of its ideas by previous ideas. But is it pre-supposed in the conception of a logical mind, that the temporal succession in its ideas is continuous, and not by discrete steps? A continuum such as we suppose time and space to be, is defined as something any part of which itself has parts of the same kind. So that the point of time or the point of space is nothing but the ideal limit towards which we approach, but which we can never reach in dividing time or space; and consequently nothing is true of a point which is not true of a space or a time. A discrete quantum, on the other hand, has ultimate parts which differ from any other part of the quantum in their absolute separation from one another. If the succession of images in the mind is by discrete steps, time for that mind will be made up of indivisible instants. Any one idea will be absolutely distinguished from every other idea by its being present only in the passing moment. And the same idea can not exist in two different moments, however similar the ideas felt in the two different moments may, for the sake of argument, be allowed to be. Now an idea exists only so far as the mind thinks it; and only when it is present to the mind. An idea therefore has no characters or qualities but what the mind thinks of it at the time when it is present to the mind. It follows from this that if the succession of time were by separate steps, no idea could resemble another; for these ideas if they are distinct, are present to the mind at different times. Therefore at no time when one is present to the mind, is the other present. Consequently the mind never compares them nor thinks them to be alike; and consequently they are not alike; since they are only what they are thought to be at the time when they are present. It may be objected that though the mind does not directly think them to be alike; yet it may think together reproductions of them, and thus think them to be alike. This would be a valid objection were it not necessary, in the first place, in order that one idea should be the representative of another, that it should resemble that idea, which it could only do by means of some representation of it again, and so on to infinity; the link which is to bind the first two together which are to be pronounced alike, never being found. In short the resemblance of ideas implies that some two ideas are to be thought together which are present to the mind at different times. And this never can be, if instants are separated from one another by absolute steps. This conception is therefore to be abandoned, and it must be acknowledged to be already presupposed in the conception of a logical mind that the flow of time should be continuous. Let us consider then how we are to conceive what is present to the mind. We are accustomed to say that nothing is present but a fleeting instant, a point of time. But this is a wrong view of the matter because a point differs in no respect from a space of time, except that it is the ideal limit which, in the division of time, we never reach. It can not therefore be that it differs from an interval of time in this respect that what is present is only in a fleeting instant, and does not occupy a whole interval of time, unless what is present be an ideal something which can never be reached, and not something real. The true conception is, that ideas which succeed one another during an interval of time, become present to the mind through the successive presence of the ideas which occupy the parts of that time. So that the ideas which are present in each of these parts are more immediately present, or rather less mediately present than those of the whole time. And this division may be carried to any extent. But you never reach an idea which is quite immediately present to the mind, and is not made present by the ideas which occupy the parts of the time that it occupies. Accordingly, it takes time for ideas to be present to the mind. They are present during a time. And they are present by means of the presence of the ideas which are in the parts of that time. Nothing is therefore present to the mind in an instant, but only during a time. The events of a day are less mediately present to the mind than the events of a year; the events of a second less mediately present than the events of a day. (C.S. Peirce, CE 3, pp. 68&ndash;70). </blockquote> Charles Sanders Peirce, MS 215, 1873, [&ldquo;On Time and Thought&rdquo;], pp. 68&ndash;71 in ''Writings of Charles S. Peirce : A Chronological Edition, Volume 3, 1872&ndash;1878'', Peirce Edition Project, Indiana University Press, Bloomington, IN, 1986. ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://mywikibiz.com/Dynamics_of_inquiry Dynamics of Inquiry @ MyWikiBiz] * [http://mathweb.org/wiki/Dynamics_of_inquiry Dynamics of Inquiry @ MathWeb Wiki] * [http://netknowledge.org/wiki/Dynamics_of_inquiry Dynamics of Inquiry @ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Dynamics_of_inquiry Dynamics of Inquiry @ OER Commons] {{col-break}} * [http://p2pfoundation.net/Dynamics_of_Inquiry Dynamics of Inquiry @ P2P Foundation] * [http://semanticweb.org/wiki/Dynamics_of_inquiry Dynamics of Inquiry @ SemanticWeb] * [http://ref.subwiki.org/wiki/Dynamics_of_inquiry Dynamics of Inquiry @ Subject Wikis] * [http://beta.wikiversity.org/wiki/Dynamics_of_inquiry Dynamics of Inquiry @ Wikiversity Beta] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;] * [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;] ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Dynamics_of_inquiry Dynamics of Inquiry], [http://mywikibiz.com/ MyWikiBiz] * [http://mathweb.org/wiki/Dynamics_of_inquiry Dynamics of Inquiry], [http://mathweb.org/ MathWeb Wiki] * [http://netknowledge.org/wiki/Dynamics_of_inquiry Dynamics of Inquiry], [http://netknowledge.org/ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Dynamics_of_inquiry Dynamics of Inquiry], [http://wiki.oercommons.org/ OER Commons] * [http://p2pfoundation.net/Dynamics_of_Inquiry Dynamics of Inquiry], [http://p2pfoundation.net/ P2P Foundation] {{col-break}} * [http://semanticweb.org/wiki/Dynamics_of_inquiry Dynamics of Inquiry], [http://semanticweb.org/ SemanticWeb] * [http://beta.wikiversity.org/wiki/Dynamics_of_inquiry Dynamics of Inquiry], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://www.getwiki.net/-Dynamics_of_Inquiry_(C.S._Peirce) Dynamics of Inquiry], [http://www.getwiki.net/ GetWiki] * [http://www.wikinfo.org/index.php/Dynamics_of_inquiry Dynamics of Inquiry], [http://www.wikinfo.org/ Wikinfo] * [http://en.wikipedia.org/w/index.php?title=Charles_Sanders_Peirce&oldid=111891138#Dynamics_of_inquiry Dynamics of Inquiry], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Artificial Intelligence]] [[Category:Computer Science]] [[Category:Critical Thinking]] [[Category:Cybernetics]] [[Category:Education]] [[Category:Hermeneutics]] [[Category:Information Systems]] [[Category:Information Theory]] [[Category:Inquiry Driven Systems]] [[Category:Intelligence Amplification]] [[Category:Learning Organizations]] [[Category:Knowledge Representation]] [[Category:Logic]] [[Category:Philosophy]] [[Category:Pragmatics]] [[Category:Semantics]] [[Category:Semiotics]] [[Category:Systems Science]] 8db7cc43b9911d47c9f90c3c8d5588ab509afe42 0 344 566 2010-06-23T18:58:19Z Jon Awbrey 3 + article wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. '''''Semeiotic''''' is one of the terms that [[Charles Sanders Peirce]] used to describe his theory of [[triadic relation|triadic]] [[sign relations]], along with ''semiotic'' and the plural variants of both terms. The form ''semeiotic'' is often used to distinguish Peirce's theory, since it is less often used by other writers to denote their particular approaches to the subject. ==Types of signs== There are three principal ways that a sign can denote its objects. These are usually described as ''kinds'', ''species'', or ''types'' of signs, but it is important to recognize that these are not ontological species, that is, they are not mutually exclusive features of description, since the same thing can be a sign in several different ways. Beginning very roughly, the three main ways of being a sign can be described as follows: :* An ''icon'' is a sign that denotes its objects by virtue of a quality that it shares with its objects. :* An ''index'' is a sign that denotes its objects by virtue of an existential connection that it has with its objects. :* A ''symbol'' is a sign that denotes its objects solely by virtue of the fact that it is interpreted to do so. One of Peirce's early delineations of the three types of signs is still quite useful as a first approach to understanding their differences and their relationships to each other: {| align="center" cellpadding="8" width="90%" | <p>In the first place there are likenesses or copies &mdash; such as ''statues'', ''pictures'', ''emblems'', ''hieroglyphics'', and the like. Such representations stand for their objects only so far as they have an actual resemblance to them &mdash; that is agree with them in some characters. The peculiarity of such representations is that they do not determine their objects &mdash; they stand for anything more or less; for they stand for whatever they resemble and they resemble everything more or less.</p> <p>The second kind of representations are such as are set up by a convention of men or a decree of God. Such are ''tallies'', ''proper names'', &c. The peculiarity of these ''conventional signs'' is that they represent no character of their objects. Likenesses denote nothing in particular; ''conventional signs'' connote nothing in particular.</p> <p>The third and last kind of representations are ''symbols'' or general representations. They connote attributes and so connote them as to determine what they denote. To this class belong all ''words'' and all ''conceptions''. Most combinations of words are also symbols. A proposition, an argument, even a whole book may be, and should be, a single symbol. (Peirce 1866, "Lowell Lecture 7", CE 1, 467&ndash;468).</p> |} ==References== * [[Charles Sanders Peirce|Peirce, C.S.]], [[Charles Sanders Peirce (Bibliography)|Bibliography]]. * Peirce, C.S., ''Writings of Charles S. Peirce : A Chronological Edition, Volume&nbsp;1, 1857&ndash;1866'', Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1982. Cited as CE&nbsp;1. * Peirce, C.S. (1865), "On the Logic of Science", Harvard University Lectures, CE&nbsp;1, 161&ndash;302. * Peirce, C.S. (1866), "The Logic of Science, or, Induction and Hypothesis", Lowell Institute Lectures, CE&nbsp;1, 357&ndash;504. ==Readings== * Awbrey, Jon, and Awbrey, Susan (1995), &ldquo;Interpretation as Action : The Risk of Inquiry&rdquo;, ''Inquiry : Critical Thinking Across the Disciplines'' 15, 40&ndash;52. [http://www.chss.montclair.edu/inquiry/fall95/awbrey.html Online]. ==Resources== * [http://vectors.usc.edu/thoughtmesh/publish/142.php Semeiotic &rarr; ThoughtMesh] * Bergman & Paavola (eds.), ''Commens Dictionary of Peirce's Terms'', [http://www.helsinki.fi/science/commens/dictionary.html Webpage] ** ''[http://www.helsinki.fi/science/commens/terms/semeiotic.html Semeiotic]'' ** ''[http://www.helsinki.fi/science/commens/terms/icon.html Icon]'' ** ''[http://www.helsinki.fi/science/commens/terms/index2.html Index]'' ** ''[http://www.helsinki.fi/science/commens/terms/symbol.html Symbol]'' ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://mywikibiz.com/Semeiotic Semeiotic @ MyWikiBiz] * [http://mathweb.org/wiki/Semeiotic Semeiotic @ MathWeb Wiki] * [http://netknowledge.org/wiki/Semeiotic Semeiotic @ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Semeiotic Semeiotic @ OER Commons] {{col-break}} * [http://p2pfoundation.net/Semeiotic Semeiotic @ P2P Foundation] * [http://semanticweb.org/wiki/Semeiotic Semeiotic @ SemanticWeb] * [http://ref.subwiki.org/wiki/Semeiotic Semeiotic @ Subject Wikis] * [http://beta.wikiversity.org/wiki/Semeiotic Semeiotic @ Wikiversity Beta] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;] * [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;] ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Semeiotic Semeiotic], [http://mywikibiz.com/ MyWikiBiz] * [http://mathweb.org/wiki/Semeiotic Semeiotic], [http://mathweb.org/wiki/ MathWeb Wiki] * [http://netknowledge.org/wiki/Semeiotic Semeiotic], [http://netknowledge.org/ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Semeiotic Semeiotic], [http://wiki.oercommons.org/ OER Commons] {{col-break}} * [http://p2pfoundation.net/Semeiotic Semeiotic], [http://p2pfoundation.net/ P2P Foundation] * [http://semanticweb.org/wiki/Semeiotic Semeiotic], [http://semanticweb.org/ Semantic Web] * [http://knol.google.com/k/jon-awbrey/semeiotic/3fkwvf69kridz/4 Semeiotic], [http://knol.google.com/ Google Knol] * [http://vectors.usc.edu/thoughtmesh/publish/142.php Semeiotic], [http://vectors.usc.edu/thoughtmesh/ ThoughtMesh] {{col-break}} * [http://getwiki.net/-Semeiotic Semeiotic], [http://getwiki.net/ GetWiki] * [http://wikinfo.org/index.php/Semeiotic Semeiotic], [http://wikinfo.org/ Wikinfo] * [http://textop.org/wiki/index.php?title=Semeiotic Semeiotic], [http://textop.org/wiki/ Textop Wiki] * [http://en.wikipedia.org/w/index.php?title=Semeiotic&oldid=246563989 Semeiotic], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Artificial Intelligence]] [[Category:Critical Thinking]] [[Category:Cybernetics]] [[Category:Education]] [[Category:Hermeneutics]] [[Category:Information Systems]] [[Category:Intelligence Amplification]] [[Category:Learning Organizations]] [[Category:Knowledge Representation]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Philosophy]] [[Category:Pragmatics]] [[Category:Semantics]] [[Category:Semiotics]] [[Category:Systems Science]] a09d5689a54dfcaf942a8a36e7078ecf4aab0694 0 346 568 2010-06-23T19:54:46Z Jon Awbrey 3 + article wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. The '''logic of information''', or the ''logical theory of information'', considers the information content of logical [[semiotics|signs]] &mdash; everything from bits to books and beyond &mdash; along the lines initially developed by [[Charles Sanders Peirce]]. In this line of development the concept of information serves to integrate the aspects of logical signs that are separately covered by the concepts of [[denotation]] and [[connotation]], or, in roughly equivalent terms, by the concepts of [[extension (logic)|extension]] and [[comprehension (logic)|comprehension]]. Peirce began to develop these ideas in his lectures "On the Logic of Science" at Harvard University (1865) and the Lowell Institute (1866). Here is one of the starting points: {| align="center" cellpadding="8" width="90%" | <p>Let us now return to the information. The information of a term is the measure of its superfluous comprehension. That is to say that the proper office of the comprehension is to determine the extension of the term. For instance, you and I are men because we possess those attributes &mdash; having two legs, being rational, &tc. &mdash; which make up the comprehension of ''man''. Every addition to the comprehension of a term lessens its extension up to a certain point, after that further additions increase the information instead.</p> <p>Thus, let us commence with the term ''colour''; add to the comprehension of this term, that of ''red''. ''Red colour'' has considerably less extension than ''colour''; add to this the comprehension of ''dark''; ''dark red colour'' has still less [extension]. Add to this the comprehension of ''non-blue'' &mdash; ''non-blue dark red colour'' has the same extension as ''dark red colour'', so that the ''non-blue'' here performs a work of supererogation; it tells us that no ''dark red colour'' is blue, but does none of the proper business of connotation, that of diminishing the extension at all.</p> <p>Thus information measures the superfluous comprehension. And, hence, whenever we make a symbol to express any thing or any attribute we cannot make it so empty that it shall have no superfluous comprehension. I am going, next, to show that inference is symbolization and that the puzzle of the validity of scientific inference lies merely in this superfluous comprehension and is therefore entirely removed by a consideration of the laws of ''information''. (C.S. Peirce, "The Logic of Science, or, Induction and Hypothesis" (1866), CE 1, 467).</p> |} ==References== * [[Charles Sanders Peirce (Bibliography)|Peirce, C.S., Bibliography]]. * De Tienne, André (2006), "Peirce's Logic of Information", Seminario del Grupo de Estudios Peirceanos, Universidad de Navarra, 28 Sep 2006. [http://www.unav.es/gep/SeminariodeTienne.html Online]. * Peirce, C.S. (1867), "Upon Logical Comprehension and Extension", [http://www.iupui.edu/~peirce/writings/v2/w2/w2_06/v2_06.htm Online]. ==Resources== * [http://vectors.usc.edu/thoughtmesh/publish/145.php Logic of Information &rarr; ThoughtMesh] ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://mywikibiz.com/Logic_of_information Logic of Information @ MyWikiBiz] * [http://mathweb.org/wiki/Logic_of_information Logic of Information @ MathWeb Wiki] * [http://netknowledge.org/wiki/Logic_of_information Logic of Information @ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Logic_of_information Logic of Information @ OER Commons] {{col-break}} * [http://p2pfoundation.net/Logic_of_Information Logic of Information @ P2P Foundation] * [http://semanticweb.org/wiki/Logic_of_information Logic of Information @ SemanticWeb] * [http://ref.subwiki.org/wiki/Logic_of_information Logic of Information @ Subject Wikis] * [http://beta.wikiversity.org/wiki/Logic_of_information Logic of Information @ Wikiversity Beta] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;] * [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;] ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Logic_of_information Logic of Information], [http://mywikibiz.com/ MyWikiBiz] * [http://mathweb.org/wiki/Logic_of_information Logic of Information], [http://mathweb.org/wiki/ MathWeb Wiki] * [http://netknowledge.org/wiki/Logic_of_information Logic of Information], [http://netknowledge.org/ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Logic_of_information Logic of Information], [http://wiki.oercommons.org/ OER Commons] * [http://p2pfoundation.net/Logic_of_Information Logic of Information], [http://p2pfoundation.net/ P2P Foundation] * [http://semanticweb.org/wiki/Logic_of_information Logic of Information], [http://semanticweb.org/ Semantic Web] {{col-break}} * [http://beta.wikiversity.org/wiki/Logic_of_information Logic of Information], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://vectors.usc.edu/thoughtmesh/publish/145.php Logic of Information], [http://vectors.usc.edu/thoughtmesh/ ThoughtMesh] * [http://getwiki.net/-Logic_of_Information_(C.S._Peirce) Logic of Information : 1], [http://getwiki.net/ GetWiki] * [http://getwiki.net/-Logic_of_Information_(Jon_Awbrey) Logic of Information : 2], [http://getwiki.net/ GetWiki] * [http://wikinfo.org/index.php/Logic_of_information Logic of Information], [http://wikinfo.org/ Wikinfo] * [http://en.wikipedia.org/w/index.php?title=Logic_of_information&oldid=67770000 Logic of Information], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Artificial Intelligence]] [[Category:Cybernetics]] [[Category:Hermeneutics]] [[Category:Information Systems]] [[Category:Information Theory]] [[Category:Intelligence Amplification]] [[Category:Knowledge Representation]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Philosophy]] [[Category:Pragmatics]] [[Category:Science]] [[Category:Semantics]] [[Category:Semiotics]] [[Category:Systems Science]] e61b6d533f5c3cba52ef3691ee2f6b3096c5e25f 0 347 569 2010-06-24T17:12:07Z Jon Awbrey 3 + article wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. The '''pragmatic maxim''', also known as the ''maxim of [[pragmatism]]'' or the ''maxim of [[pragmaticism]]'', is a maxim of logic formulated by [[Charles Sanders Peirce]]. Serving as a normative recommendation or a regulative principle in the [[normative science]] of logic, its function is to guide the conduct of thought toward the achievement of its purpose, advising the addressee on an optimal way of "attaining clearness of apprehension". ==Seven ways of looking at a pragmatic maxim== Peirce stated the pragmatic maxim in many different ways over the years, each of which adds its own bit of clarity or correction to their collective corpus. * The first excerpt appears in the form of a dictionary entry, intended as a definition of ''pragmatism''. <blockquote> Pragmatism. The opinion that metaphysics is to be largely cleared up by the application of the following maxim for attaining clearness of apprehension: "Consider what effects, that might conceivably have practical bearings, we conceive the object of our conception to have. Then, our conception of these effects is the whole of our conception of the object. (Peirce, CP 5.2, 1878/1902). </blockquote> * The second excerpt presents another version of the pragmatic maxim, a recommendation about a way of clarifying meaning that can be taken to stake out the general philosophy of pragmatism. <blockquote> Pragmaticism was originally enounced in the form of a maxim, as follows: Consider what effects that might ''conceivably'' have practical bearings you ''conceive'' the objects of your ''conception'' to have. Then, your ''conception'' of those effects is the whole of your ''conception'' of the object. (Peirce, CP 5.438, 1878/1905). </blockquote> * The third excerpt puts a gloss on the meaning of a ''practical bearing'' and provides an alternative statement of the maxim. <blockquote> <p>Such reasonings and all reasonings turn upon the idea that if one exerts certain kinds of volition, one will undergo in return certain compulsory perceptions. Now this sort of consideration, namely, that certain lines of conduct will entail certain kinds of inevitable experiences is what is called a "practical consideration". Hence is justified the maxim, belief in which constitutes pragmatism; namely:</p> <p>''In order to ascertain the meaning of an intellectual conception one should consider what practical consequences might conceivably result by necessity from the truth of that conception; and the sum of these consequences will constitute the entire meaning of the conception.'' (Peirce, CP 5.9, 1905).</p> </blockquote> * The fourth excerpt illustrates one of Peirce's many attempts to get the sense of the pragmatic philosophy across by rephrasing the pragmatic maxim in an alternative way. In introducing this version, he addresses an order of prospective critics who do not deem a simple heuristic maxim, much less one that concerns itself with a routine matter of logical procedure, as forming a sufficient basis for a whole philosophy. <blockquote> <p>On their side, one of the faults that I think they might find with me is that I make pragmatism to be a mere maxim of logic instead of a sublime principle of speculative philosophy. In order to be admitted to better philosophical standing I have endeavored to put pragmatism as I understand it into the same form of a philosophical theorem. I have not succeeded any better than this:</p> <p>Pragmatism is the principle that every theoretical judgment expressible in a sentence in the [[indicative mood]] is a confused form of thought whose only meaning, if it has any, lies in its tendency to enforce a corresponding practical maxim expressible as a [[conditional sentence]] having its [[apodosis]] in the [[imperative mood]]. (Peirce, CP 5.18, 1903).</p> </blockquote> * The fifth excerpt is useful by way of additional clarification, and was aimed to correct a variety of historical misunderstandings that arose over time with regard to the intended meaning of the pragmatic maxim. <blockquote> The doctrine appears to assume that the end of man is action &mdash; a [[Stoics|stoical]] axiom which, to the present writer at the age of sixty, does not recommend itself so forcibly as it did at thirty. If it be admitted, on the contrary, that action wants an end, and that that end must be something of a general description, then the spirit of the maxim itself, which is that we must look to the upshot of our concepts in order rightly to apprehend them, would direct us towards something different from practical facts, namely, to general ideas, as the true interpreters of our thought. (Peirce, CP 5.3, 1902). </blockquote> * A sixth excerpt is useful in stating the bearing of the pragmatic maxim on the topic of reflection, namely, that it makes all of pragmatism boil down to nothing more or less than a method of reflection. <blockquote> The study of philosophy consists, therefore, in reflexion, and ''pragmatism'' is that method of reflexion which is guided by constantly holding in view its purpose and the purpose of the ideas it analyzes, whether these ends be of the nature and uses of action or of thought. &hellip; It will be seen that ''pragmatism'' is not a ''Weltanschauung'' but is a method of reflexion having for its purpose to render ideas clear. (Peirce, CP 5.13 note 1, 1902). </blockquote> * The seventh excerpt is a late reflection on the reception of pragmatism. With a sense of exasperation that is almost palpable, Peirce tries to justify the maxim of pragmatism and to correct its misreadings by pinpointing a number of false impressions that the intervening years have piled on it, and he attempts once more to prescribe against the deleterious effects of these mistakes. Recalling the very conception and birth of pragmatism, he reviews its initial promise and its intended lot in the light of its subsequent vicissitudes and its apparent fate. Adopting the style of a ''post mortem'' analysis, he presents a veritable autopsy of the ways that the main idea of pragmatism, for all its practicality, can be murdered by a host of misdissecting disciplinarians, by what are ostensibly its most devoted followers. <blockquote> This employment five times over of derivates of ''concipere'' must then have had a purpose. In point of fact it had two. One was to show that I was speaking of meaning in no other sense than that of intellectual purport. The other was to avoid all danger of being understood as attempting to explain a concept by percepts, images, schemata, or by anything but concepts. I did not, therefore, mean to say that acts, which are more strictly singular than anything, could constitute the purport, or adequate proper interpretation, of any symbol. I compared action to the finale of the symphony of thought, belief being a demicadence. Nobody conceives that the few bars at the end of a musical movement are the purpose of the movement. They may be called its upshot. (Peirce, CP 5.402 note 3, 1906). </blockquote> ==References== * Peirce, C.S., ''Collected Papers of Charles Sanders Peirce'', vols. 1&ndash;6, [[Charles Hartshorne]] and [[Paul Weiss]] (eds.), vols. 7&ndash;8, [[Arthur W. Burks]] (ed.), Harvard University Press, Cambridge, MA, 1931&ndash;1935, 1958. (Cited as CP&nbsp;''n''.''m'' for volume&nbsp;''n'', paragraph&nbsp;''m''). ==Resources== * [http://vectors.usc.edu/thoughtmesh/publish/141.php Pragmatic Maxim &rarr; ThoughtMesh] ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://mywikibiz.com/Pragmatic_maxim Pragmatic Maxim @ MyWikiBiz] * [http://mathweb.org/wiki/Pragmatic_maxim Pragmatic Maxim @ MathWeb Wiki] * [http://netknowledge.org/wiki/Pragmatic_maxim Pragmatic Maxim @ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Pragmatic_maxim Pragmatic Maxim @ OER Commons] {{col-break}} * [http://p2pfoundation.net/Pragmatic_Maxim Pragmatic Maxim @ P2P Foundation] * [http://semanticweb.org/wiki/Pragmatic_maxim Pragmatic Maxim @ SemanticWeb] * [http://ref.subwiki.org/wiki/Pragmatic_maxim Pragmatic Maxim @ Subject Wikis] * [http://beta.wikiversity.org/wiki/Pragmatic_maxim Pragmatic Maxim @ Wikiversity Beta] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;] * [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;] ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Pragmatic_maxim Pragmatic Maxim], [http://mywikibiz.com/ MyWikiBiz] * [http://mathweb.org/wiki/Pragmatic_maxim Pragmatic Maxim], [http://mathweb.org/wiki/ MathWeb Wiki] * [http://netknowledge.org/wiki/Pragmatic_maxim Pragmatic Maxim], [http://netknowledge.org/ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Pragmatic_maxim Pragmatic Maxim], [http://wiki.oercommons.org/ OER Commons] * [http://p2pfoundation.net/Pragmatic_Maxim Pragmatic Maxim], [http://p2pfoundation.net/ P2P Foundation] * [http://semanticweb.org/wiki/Pragmatic_maxim Pragmatic Maxim], [http://semanticweb.org/ Semantic Web] * [http://beta.wikiversity.org/wiki/Pragmatic_maxim Pragmatic Maxim], [http://beta.wikiversity.org/ Wikiversity Beta] {{col-break}} * [http://knol.google.com/k/jon-awbrey/pragmatic-maxim/3fkwvf69kridz/6 Pragmatic Maxim], [http://knol.google.com/ Google Knol] * [http://vectors.usc.edu/thoughtmesh/publish/141.php Pragmatic Maxim], [http://vectors.usc.edu/thoughtmesh/ ThoughtMesh] * [http://getwiki.net/-Maxim_of_Pragmatism Pragmatic Maxim], [http://getwiki.net/ GetWiki] * [http://wikinfo.org/index.php/Pragmatic_maxim Pragmatic Maxim], [http://wikinfo.org/ Wikinfo] * [http://textop.org/wiki/index.php?title=Pragmatic_maxim Pragmatic Maxim], [http://textop.org/wiki/ Textop Wiki] * [http://en.wikipedia.org/w/index.php?title=Pragmatic_maxim&oldid=45528828 Pragmatic Maxim], [http://en.wikipedia.org/ Wikipedia] * [http://suo.ieee.org/ontology/msg05407.html Pragmatic Maxim], [http://suo.ieee.org/ontology/mail1.html Ontology List] {{col-end}} [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Artificial Intelligence]] [[Category:Critical Thinking]] [[Category:Cybernetics]] [[Category:Education]] [[Category:Hermeneutics]] [[Category:Information Systems]] [[Category:Intelligence Amplification]] [[Category:Learning Organizations]] [[Category:Knowledge Representation]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Normative Sciences]] [[Category:Philosophy]] [[Category:Pragmatics]] [[Category:Semantics]] [[Category:Semiotics]] [[Category:Systems Science]] e5bd1c99180fe389a31b76665ad746e4ff966d5e 0 348 570 2010-06-24T18:02:19Z Jon Awbrey 3 + article wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''truth theory''' or a '''theory of truth''' is a conceptual framework that underlies a particular conception of truth, such as those used in art, ethics, logic, mathematics, philosophy, the sciences, or any discussion that either mentions or makes use of a notion of truth. A truth theory can be anything from an informal theory, based on implicit or tacit ideas, to a formal theory, constructed from explicit axioms and definitions and developed by means of definite rules of inference. The scope of a truth theory can be restricted to tightly-controlled and well-bounded universes of discourse or its horizon may extend to the limits of the human imagination. ==Truth in perspective== Notions of truth are notoriously difficult to disentangle from many of our most basic concepts &mdash; meaning, reality, and values in general, to mention just a few. The subjects of meaning and truth are commonly treated together, the idea being that a thing must be meaningful before it can be true or false. This association is found in ancient times, and has become standard in modern times under the heading of ''semantics'', especially ''formal semantics'' and ''model theory''. Another association of longstanding interest is the relation between truth and ''logical validity'', "because the fundamental notion of logic is validity and this is definable in terms of truth and falsehood" (Kneale and Kneale, 16). Though not the main subjects of this article, meaning and validity are truth's neighbors, and incidental inquiries of them can serve to cast light on truth's character. Beyond this minor note of accord, hardly universal, suggesting that meaning is necessary to truth, reflectors on the idea of truth just as quickly disperse into schools of thought that barely comprehend each other's thinking. A few of the more notable points of departure are these: # One of the first partings of the ways occurs at the watershed between literal and symbolic meanings, leading to a corresponding division in truths. People often speak of truth in art, truth in drama, truth in fiction, human truth, moral, religious, and spiritual truth, along with the difference between truth in principle and truth in practice. These topics demand a perspective on meaning, reality, and truth that looks beyond the bounds of literal truth and the branches of philosophy that are limited to it. # Merely resolving that meaning precedes truth, logically speaking, only brings up a host of new questions, since the meaning of the word ''meaning'' is notoriously hard to pin down. There are just to start at least two different dimensions of meaning that are commonly recognized, namely, ''[[connotation|connotative meaning]]'' and ''[[denotation|denotative meaning]]''. In one classical formulation, truth is defined as the good of [[logic]], where logic is treated as a [[normative science]], that is, an [[inquiry]] into a ''good'' or a ''value'' that seeks knowledge of it and the means to achieve it. In this scheme of ideas, truth is the positive quality of a sign that indicates the right course of action for reaching a value that we value for its own sake. As such, truth takes its place among justice and beauty, whose normative sciences are [[ethics]] and [[aesthetics]], respectively. Viewed in this light, it is pointless to discuss truth in isolation from a frame of reference that encompasses the topics of inquiry, knowledge, logic, meaning, practice, and value, all very broadly conceived. In contexts bounded by formal linguistic analysis, a '''truth theory''' is defined as "a theory providing the truth definition for a language" (Blackburn, 382). A ''truth definition'' is in turn defined as "a definition of the predicate "__is&nbsp;true" for a language that satisfies ''convention T'', the material adequacy condition laid down by [[Tarski]]" (Blackburn, 382). ==Historical overview== In an ancient fragment of text called the ''Dissoi Logoi'', a writer is evidently trying to prove the impossibility of speaking consistently about truth and falsehood. One of the conundrums put forward to confound the reader cites the case of the verbal form, "I am an initiate", which is true when ''A'' says it but false when ''B'' says it. Escape from befuddlement seems easy enough if one observes that it is not the verbal expression, the sentence, to which the predicates of truth and falsity apply but what the sentence expresses, the proposition that it states. (Cf. Kneale and Kneale, 16). This same tension between strings of characters and their meanings remains with us to this day. In his early work &#928;&#949;&#961;&#953; &#917;&#961;&#956;&#951;&#957;&#949;&#953;&#945;s (''Peri Hermeneias'' or ''[[On Interpretation]]'') [[Aristotle]] strikes a chord that not only sets the key for a number of philosophical movements down through the ages but supplies the initial motif for many themes in the logic of meaning and truth that are still undergoing active development in our time. <blockquote> Words spoken (phoné) are symbols or signs (symbola) of affections or impressions (pathemata) of the soul (psyche); written words (graphomena) are the signs of words spoken. As writing, so also is speech not the same for all races of men. But the mental affections themselves, of which these words are primarily signs (semeia), are the same for the whole of mankind, as are also the objects (pragmata) of which those affections are representations or likenesses, images, copies (homoiomata). (Aristotle, ''On Interpretation'', 1.16^a4). </blockquote> Some of the points to be noted in this passage are these: # Aristotle employs a distinction in Greek that is drawn between natural or physical signs (semeia) and artificial or cultural signs (symbola). # The passage mentions three principal domains of elements, namely, the ''objects'' (pragmata), the ''signs'' (semeia, symbola), and the psychological elements (pathemata). The last domain extends over the full range of a human being's affective and cognitive experiences, for brevity summed up as ''ideas'' and ''impressions'', where these words are taken in their broadest conceivable senses. # This means that the phenomena under investigation have to do with the types of [[three-place relation]]s that conceivably exist among three domains of this sort. As a general rule, three-place relations can be very complex, and a commonly-tried strategy for approaching their complexity is to consider the [[two-place relation]]s that are left when the presence of a selected domain is simply ignored. # There are two types of two-place relation on the face of the overall three-place relation that Aristotle takes the trouble to mention, namely these:<p>Sign <math>\longrightarrow</math> Idea. Words spoken are signs or symbols of pathemata.</p><p>Idea <math>\longrightarrow</math> Object. Pathemata are icons (homoiomata) of pragmata.</p> # More incidentally, but still bearing heavily on many later discussions, Aristotle holds that the relation between writing and speech is analogous to the relation between speech and the realm of experiences, feelings, and thoughts.<p>Writing <math>\longrightarrow</math> Speech. Written words are symbols of spoken words.</p><p> Speech <math>\longrightarrow</math> Ideation. Spoken words are symbols of impressions.</p> ==Elements of theory== It is customary in philosophy to refer to a distinctive treatment of a particular subject matter, frequently summed up in a succinctly stated thesis, as a ''theory'', whether or not it qualifies as a theory by strict empirical or logical standards. When there is any risk of confusion, an informal thesis of this kind may be referred to as an ''account'', a ''perspective'', a ''treatment'', or so on, reserving the term ''theory'' for the type of [[formal system]] that serves in logic and science. Theories of truth can be classified according to the following features: * Primary subjects. What kinds of things are potentially meaningful enough to be asserted or not, believed or not, or considered true or false? * Relevant objects. What kinds of things, in addition to primary subjects, are pertinent to deciding whether to assert them or not, believe them or not, or consider them true or false? * Value predicates. What kinds of things are legitimate to say about primary subjects, either in themselves, or in relation to relevant objects? In some discussions of meaning and truth that consider forms of expression well beyond the limits of literally-interpreted linguistic forms, potentially meaningful elements are called ''[[representation]]s'', or ''[[sign (semiotics)|signs]]'' for short, taking these words in the broadest conceivable senses. Most treatments of truth draw an important distinction at this point, though the language in which they draw it may vary. On the one hand there is a type of incomplete sign that is nevertheless said to be true or false of various objects. For example, in logic there are ''terms'' such as "man" or "woman" that are true of some things and false of others, and there are ''predicates'' such as "__is a man" or "__is a woman" that are true or false in the same way. On the other hand there is a type of complete sign that expresses what grammarians traditionally call a ''complete thought''. Here one speaks of ''sentences'' and ''propositions''. Some considerations of truth admit both types of signs, ''terms'' and ''sentences'', while others admit only the bearers of complete thoughts into the arena of judgment. In a number of recent discussions that focus on linguistic analysis, the vehicles of complete thoughts are described as ''truthbearers'', with no intention of prejudging whether they bear truth or falsehood. The things that can be said about any of these representations, signs, or truthbearers are expressed in what most truth theorists describe as ''truth predicates''. Most inquiries into the character of truth begin with a notion of an informative, meaningful, or significant element, the truth of whose information, meaning, or significance may be put into question and needs to be evaluated. Depending on the context, this element might be called an ''artefact'', ''expression'', ''image'', ''impression'', ''lyric'', ''mark'', ''performance'', ''picture'', ''sentence'', ''sign'', ''string'', ''symbol'', ''text'', ''thought'', ''token'', ''utterance'', ''word'', ''work'', and so on. For the sake of brevity, it is convenient to use the term ''sign'' for any one of these elements. Whatever the case, one has the task of judging whether the bearers of information, meaning, or significance are indeed ''truth-bearers''. This judgment is typically expressed in the form of a specific ''truth predicate'', whose positive application to a sign asserts that the sign is true. Considered within the broadest horizon, there is little reason to imagine that the process of judging a ''work'', that leads to a predication of false or true, is necessarily amenable to formalization, and it may always remain what is commonly called a ''judgment call''. But there are indeed many well-circumscribed domains where it is useful to consider disciplined forms of evaluation, and the observation of these limits allows for the institution of what is called a ''[[method]]'' of judging truth and falsity. One of the first questions that can be asked in this setting is about the relationship between the significant performance and its reflective critique. If one expresses oneself in a particular fashion, and someone says "that's true", is there anything useful at all that can be said in general terms about the relationship between these two acts? For instance, does the critique add value to the expression criticized, does it say something significant in its own right, or is it just an insubstantial echo of the original sign? Theories of truth may be described according to several dimensions of description that affect the character of the predicate "true". The truth predicates that are used in different theories may be classified by the number of things that have to be mentioned in order to assess the truth of a sign, counting the sign itself as the first thing. In formal logic, this number is called the ''[[arity]]'' of the predicate. The kinds of truth predicates may then be subdivided according to any number of more specific characters that various theorists recognize as important. # A ''monadic'' truth predicate is one that applies to its main subject ? typically a concrete representation or its abstract content ? independently of reference to anything else. In this case one can say that a truth bearer is true in and of itself. # A ''dyadic'' truth predicate is one that applies to its main subject only in reference to something else, a second subject. Most commonly, the auxiliary subject is either an ''object'', an ''interpreter'', or a ''language'' to which the representation bears some [[relation (mathematics)|relation]]. # A ''triadic'' truth predicate is one that applies to its main subject only in reference to a second and a third subject. For example, in a pragmatic theory of truth, one has to specify both the object of the sign, and either its interpreter or another sign called the ''interpretant'' before one can say that the sign is true ''of'' its object ''to'' its interpreting agent or sign. Several qualifications must be kept in mind with respect to any such radically simple scheme of classification, as real practice seldom presents any pure types, and there are settings in which it is useful to speak of a theory of truth that is "almost" ''k''-adic, or that "would be" ''k''-adic if certain details can be abstracted away and neglected in a particular context of discussion. That said, given the generic division of truth predicates according to their arity, further species can be differentiated within each genus according to a number of more refined features. The truth predicate of interest in a typical [[correspondence theory of truth]] tells of a relation between representations and objective states of affairs, and is therefore expressed, for the most part, by a dyadic predicate. In general terms, one says that a representation is ''true of'' an objective situation, more briefly, that a sign is true of an object. The nature of the correspondence may vary from theory to theory in this family. The correspondence can be fairly arbitrary or it can take on the character of an ''[[analogy]]'', an ''[[icon]]'', or a ''[[morphism]]'', whereby a representation is rendered true of its object by the existence of corresponding elements and a similar structure. ===Signs=== In some branches of philosophy and fields of science the domain of potentially meaningful entities may include almost any kind of informative or significant element. The generic terms ''sign'' or ''representation'' suffice for these, with the qualification that the terms are used equivocally up and down a full spectrum from the more abstract ''[[type]]s'' to the more concrete ''[[token]]s'' that are associated with each other. More specifically, the [[linguistic turn]] in [[analytic philosophy]] begins with a focus on the syntactic character of the ''sentence'', from which is abstracted its meaningful content, referred to as the corresponding ''proposition''. A proposition is the content expressed by a sentence, held in a belief, or affirmed in an assertion or judgment. ''[[Truthbearer]]'' is used by a number of writers to refer to any entity that can be judged true or false. The term ''truthbearer'' may be applied to [[proposition]]s, [[sentence (linguistics)|sentence]]s, [[statement]]s, [[idea]]s, [[belief]]s, and [[judgment]]s. Some writers exclude one or more of these categories, or argue that some of them are true (or false) only in a derivative sense. Other writers may add additional entities to the list. Truthbearers typically have two possible values, true or false. Fictional forms of expression are usually regarded as false if interpreted literally, but may be said to bear a species of truth if interpreted suitably. Still other truthbearers may be judged true or false to a greater or lesser degree. ===Higher order signs=== As ''predicate terms'', most discussions of truth allow for a number of phrases that are used to say in what ways signs or sentences or their abstract senses are regarded as true, either by themselves or in relation to other things. Theorists who admit the term call these phrases ''[[truth predicate]]s''. A truth predicate that is used to ascribe truth to something, in and of itself, in effect treating truth as an [[intrinsic property (philosophy)|intrinsic property]] of the thing, is called a ''one-place'' or ''monadic'' truth predicate. Other forms of truth predicates may be used to say that something is true in relation to specified numbers and types of other things. These are called ''many-place'' or ''polyadic'' truth predicates. In ordinary parlance, the things that one says about a subject are expressed in predicates. If one says that a sentence is true, then one is predicating truth of that sentence. Is this the same thing as asserting the sentence? This question serves as useful touchstone for sorting out some of the theories of truth. ===Propositional attitudes=== <blockquote> What sort of name shall we give to verbs like 'believe' and 'wish' and so forth? I should be inclined to call them 'propositional verbs'. This is merely a suggested name for convenience, because they are verbs which have the ''form'' of relating an object to a proposition. As I have been explaining, that is not what they really do, but it is convenient to call them propositional verbs. Of course you might call them 'attitudes', but I should not like that because it is a psychological term, and although all the instances in our experience are psychological, there is no reason to suppose that all the verbs I am talking of are psychological. There is never any reason to suppose that sort of thing. (Russell 1918, 227). </blockquote> What a proposition is, is one thing. How we feel about it, or how we regard it, is another. We can accept it, assert it, believe it, command it, contest it, declare it, deny it, doubt it, enjoin it, exclaim it, expect it, imagine it, intend it, know it, observe it, prove it, question it, suggest it, or wish it were so. Different attitudes toward propositions are called ''propositional attitudes'', and they are also discussed under the headings of ''intentionality'' and ''linguistic modality''. The formal properties of verbs like ''assert'', ''believe'', ''command'', ''consider'', ''deny'', ''doubt'', ''hunt'', ''imagine'', ''judge'', ''know'', ''want'', ''wish'', and a host of others, are studied under these headings by linguists and logicians alike. Many problematic situations in real life arise from the circumstance that many different propositions in many different modalities are in the air at once. In order to compare propositions of different colors and flavors, as it were, we have no basis for comparison but to examine the underlying propositions themselves. Thus we are brought back to matters of language and logic. Despite the name, propositional attitudes are not regarded as psychological attitudes proper, since the formal disciplines of linguistics and logic are concerned with nothing more concrete than what can be said in general about their formal properties and their patterns of interaction. The variety of attitudes that a proposer can bear toward a single proposition is a critical factor in evaluating its truth. One topic of central concern is the relation between the modalities of assertion and belief, especially when viewed in the light of the proposer's intentions. For example, we frequently find ourselves faced with the question of whether a person's assertions conform to his or her beliefs. Discrepancies here can occur for many reasons, but when the departure of assertion from belief is intentional, we usually call that a ''[[lie]]''. Other comparisons of multiple modalities that frequently arise are the relationships between belief and knowledge and the discrepancies that occur among observations, expectations, and intentions. Deviations of observations from expectations are commonly perceived as ''[[surprise (emotion)|surprise]]s'', phenomena that call for ''[[explanation]]s'' to reduce the shock of amazement. Deviations of observations from intentions are commonly experienced as ''[[problem]]s'', situations that call for plans of action to reduce the drive of dissatisfaction. Either type of discrepancy forms an impulse to ''[[inquiry]]'' (Awbrey & Awbrey 1995). ===Reflection and quotation=== The study of propositional attitudes is no sooner begun than it leads to the all-important philosophical distinction between (1) using a meaning-bearer to bear its meaning in an active manner and (2) mentioning a meaning-bearer in a form that keeps its meaning in a more inert or inhibited state. The reasons for doing the latter are various, but involve the need to reflect on a potential meaning, to compare and contrast it with others, to criticize and evaluate both its logical implications and its practical consequences, all before deciding whether to put its meaning into action or not. The word "quote" derives from the Latin verb ''quotare'', which refers to the practice of numbering references and referring to pieces of text by marking their numbers. There is a certain [[aesthetic distance]] involved in this practice, and it leads, if only for moments at a time, to viewing each piece of text as a string of characters that bears its own litter of meanings, but meanings to be reflected on and critically compared with others, both in and out of their litter. It is hardly an accident, then, that matters of [[Gödel number|numbering phrases]], [[quotation]], and [[reflection (computer science)|reflection]] are bound up with each other in [[mathematical logic]] and [[computation theory]]. ==Varieties of truth theory== ===Nominal truth theories=== A ''nominal truth'' theory is defined by the axiom that the concept ''truth'' is a mere name. In traditional systems of logic, a concept is always a symbol, specifically, a mental symbol, and so the word ''mere'' in the nominal axiom says that ''truth'' is nothing more than a symbol. One of the aims of nominal philosophies, generally speaking, is to clear away the conceptual clutter of excess metaphysical ideas through the searching examination of their verbal formulations. Thus the question arises whether ''truth'' is one of the essentials or one of the excesses of rational thought. One method of critical analysis that is commonly brought to bear at this juncture is based on the nominal corollary that if one can do without the word in every linguistic context, then one can do without the concept, which is after all nothing but the word. ===Real truth theories=== ===Formal truth theories=== There is a generally acknowledged distinction between merely contemplating or entertaining a proposition, and actually asserting or believing it. This does not mean that there is general agreement as to the precise nature of the distinction. Although there are many ways of talking about the distinction, words alone do not guarantee clarity, and they often lead to the problem of having to decide which descriptions say the same thing and which say something different. For example, formal logic provides symbolic operators for indicating the assertion of a sentence, or the assertion of the proposition that comes from interpreting the sentence relative to a particular context of discussion. Another way of saying something about a sentence or the corresponding proposition is by means of various semantic predicates, including truth predicates as a special case. This raises the question of how these operators and predicates are related to one another. As noted before, one of the first questions of this sort is whether asserting a proposition amounts to the same thing as predicating truth of that proposition. ===Semantic relations=== A ''denotation relation'', or a ''name relation'', is a [[relation (mathematics)|relation]] between symbols (formulas, words, phrases) and the things that they are interpreted as denoting or naming in a particular context of discussion (Church 1962). The things denoted, which may be quite literally anything that can be talked about or thought about, are called the ''objects'' of denotation. Different theories of meaning vary in their use of denotation relations and the properties that they require of them. The following are two criteria that serve to distinguish particular theories of denotation: # How many things can a symbol denote? For instance, can a symbol denote more than one thing, or must a symbol always denote at most one thing? # Is denoting the same sort of relation as ''being true of'', and thus a state of affairs that can be described by a particular type of truth predicate, or is denoting a very different sort of relation than that? ==Truth and the conduct of life== <blockquote> Again, in a ship, if a man were at liberty to do what he chose, but were devoid of mind and excellence in navigation ([[arete (excellence)|&#945;&#961;&#949;&#964;&#951;s]] &#954;&#965;&#946;&#949;&#961;&#957;&#951;&#964;&#953;&#954;&#951;s), do you perceive what must happen to him and his fellow sailors? ([[Plato]], ''[[Alcibiades]]'', 135A). </blockquote> ==References== * [[Aristotle]], "On Interpretation", [[Harold P. Cooke]] (trans.), pp. 111?179 in ''Aristotle, Volume 1'', [[Loeb Classical Library]], [[William Heinemann]], London, UK, 1938. * Awbrey, Jon, and Awbrey, Susan (1995), "Interpretation as Action: The Risk of Inquiry", ''Inquiry: Critical Thinking Across the Disciplines'', 15, 40?52. [http://www.chss.montclair.edu/inquiry/fall95/awbrey.html Eprint] * [[Simon Blackburn|Blackburn, Simon]] (1996), ''The Oxford Dictionary of Philosophy'', Oxford University Press, Oxford, UK, 1994. Paperback edition with new Chronology, 1996. * Blackburn, Simon, and [[Keith Simmons|Simmons, Keith]] (eds., 1999), ''Truth'', Oxford University Press, Oxford, UK. * [[Alonzo Church|Church, Alonzo]] (1962a), "Name Relation, or Meaning Relation", p. 204 in Dagobert D. Runes (ed.), ''Dictionary of Philosophy'', Littlefield, Adams, and Company, Totowa, NJ. * Church, Alonzo (1962b), "Truth, Semantical", p. 322 in Dagobert D. Runes (ed.), ''Dictionary of Philosophy'', Littlefield, Adams, and Company, Totowa, NJ. * [[William Kneale|Kneale, W.]], and [[Martha Kneale|Kneale, M.]] (1962), ''The Development of Logic'', Oxford University Press, London, UK, 1962. Reprinted with corrections, 1975. * [[Plato]], "Alcibiades 1", [[W.R.M. Lamb]] (trans.), pp. 93?223 in ''Plato, Volume 12'', [[Loeb Classical Library]], [[William Heinemann]], London, UK, 1927. * Russell, Bertrand (1918), "The Philosophy of Logical Atomism", ''The Monist'', 1918. Reprinted, pp. 177?281 in ''Logic and Knowledge: Essays 1901?1950'', [[Robert Charles Marsh]] (ed.), Unwin Hyman, London, UK, 1956. Reprinted, pp. 35?155 in ''The Philosophy of Logical Atomism'', [[David Pears]] (ed.), Open Court, La Salle, IL, 1985. ==Further reading== * [[Michael Beaney|Beaney, Michael]] (ed., 1997), ''The Frege Reader'', Blackwell Publishers, Oxford, UK. * [[John Dewey|Dewey, John]] (1900?1901), ''Lectures on Ethics 1900?1901'', Donald F. Koch (ed.), Southern Illinois University Press, Carbondale and Edwardsville, IL, 1991. * Dewey, John (1932), ''Theory of the Moral Life'', Part 2 of John Dewey and [[James H. Tufts]], ''Ethics'', Henry Holt and Company, New York, NY, 1908. 2nd edition, Holt, Rinehart, and Winston, 1932. Reprinted, Arnold Isenberg (ed.), Victor Kestenbaum (pref.), Irvington Publishers, New York, NY, 1980. * [[Michael Dummett|Dummett, Michael]] (1991), ''Frege and Other Philosophers'', Oxford University Press, Oxford, UK. * Dummett, Michael (1993), ''Origins of Analytical Philosophy'', Harvard University Press, Cambridge, MA. * [[Michel Foucault|Foucault, Michel]] (1997), ''Essential Works of Foucault, 1954?1984, Volume 1, Ethics: Subjectivity and Truth'', Paul Rabinow (ed.), Robert Hurley et al. (trans.), The New Press, New York, NY. * [[Hans-Georg Gadamer|Gadamer, Hans-Georg]] (1986), ''The Idea of the Good in Platonic?Aristotelian Philosophy'', P. Christopher Smith (trans.), Yale University Press, New Haven, CT. 1st published, ''Die Idee des Guten zwischen Plato und Aristoteles'', J.C.B. Mohr, Heidelberg, Germany, 1978. * Grover, Dorothy (1992), ''A Prosentential Theory of Truth'', Princeton University Press, Princeton, NJ. * [[Jürgen Habermas|Habermas, Jürgen]] (1979), ''Communication and the Evolution of Society'', Thomas McCarthy (trans.), Beacon Press, Boston, MA. * Habermas, Jürgen (1990), ''Moral Consciousness and Communicative Action'', Christian Lenhardt and Shierry Weber Nicholsen (trans.), Thomas McCarthy (intro.), MIT Press, Cambridge, MA. * Habermas, Jürgen (2003), ''Truth and Justification'', Barbara Fultner (trans.), MIT Press, Cambridge, MA. * Kirkham, Richard L. (1992), ''Theories of Truth'', MIT Press, Cambridge, MA. * [[Saul A. Kripke|Kripke, Saul A.]] (1975), "An Outline of a Theory of Truth", ''Journal of Philosophy'' 72 (1975), 690?716. * Kripke, Saul A. (1980), ''Naming and Necessity'', Harvard University Press, Cambridge, MA. * [[Clarence Irving Lewis|Lewis, C.I.]] (1946), ''An Analysis of Knowledge and Valuation'', 'The Paul Carus Lectures, Series 8', Open Court, La Salle, IL. * [[Leonard Linsky|Linsky, Leonard]] (ed., 1971), ''Reference and Modality'', Oxford University Press, London, UK. * [[Robert L. Martin|Martin, Robert L.]] (ed., 1984), ''Recent Essays on Truth and the Liar Paradox'', Oxford University Press, Oxford, UK. * [[Ernest A. Moody|Moody, Ernest A.]] (1953), ''Truth and Consequence in Mediaeval Logic'', North-Holland, Amsterdam, Netherlands, 1953. Reprinted, Greenwood Press, Westport, CT, 1976. * Nietzsche, Friedrich [1873] (1968). "Uber Wahrheit und Lüge im aussermoralischen Sinn", ("On Truth and Lying in an Extra-moral Sense"), in Jürgen Habermas (ed.), ''Erkenntnistheoretische Schriften'', Suhrkamp, Frankfurt, Germany. * [[Hilary Putnam|Putnam, Hilary]] (1981), ''Reason, Truth, and History'', Cambridge University Press, Cambridge, UK. * Quine, W.V. (1982), ''Methods of Logic'', (1st ed. 1950), (2nd ed. 1959), (3rd ed. 1972), 4th edition, Harvard University Press, Cambridge, MA. * Quine, W.V. (1992), ''Pursuit of Truth'', Harvard University Press, Cambridge, MA, 1990. Revised edition, Harvard University Press, Cambridge, MA, 1992. * Quine, W.V., and [[J.S. Ullian|Ullian, J.S.]] (1978), ''The Web of Belief'', Random House, New York, NY, 1970. 2nd edition, Random House, New York, NY, 1978. * [[John Rawls|Rawls, John]] (2000), ''Lectures on the History of Moral Philosophy'', Barbara Herman (ed.), Harvard University Press, Cambridge, MA. * Rescher, Nicholas (1973), ''The Coherence Theory of Truth'', Oxford University Press, Oxford, UK. * [[Richard Rorty|Rorty, Richard]] (1991), ''Objectivity, Relativism, and Truth: Philosophical Papers, Volume 1'', Cambridge University Press, Cambridge, UK. * [[Bertrand Russell|Russell, Bertrand]] (1913), ''Theory of Knowledge (The 1913 Manuscript)'', Elizabeth Ramsden Eames (ed.), Kenneth Blackwell (collab.), George Allen & Unwin, 1984. Reprinted, Routledge, London, UK, 1992. * Russell, Bertrand (1940), ''An Inquiry into Meaning and Truth'', 'The William James Lectures for 1940 Delivered at Harvard University', George Allen & Unwin, 1950. Reprinted, Thomas Baldwin (intro.), Routledge, London, UK, 1992. * [[Nathan Salmon|Salmon, Nathan]], and [[Scott Soames|Soames, Scott]] (eds., 1988), ''Propositions and Attitudes'', Oxford University Press, Oxford, UK. * [[Ninian Smart|Smart, Ninian]] (1969), ''The Religious Experience of Mankind'', Charles Scribner's Sons, New York, NY. * Tarski, Alfred (1944), "The Semantic Conception of Truth and the Foundations of Semantics", ''Philosophy and Phenomenological Research'' 4 (3), 341-376. * [[Anthony F.C. Wallace|Wallace, Anthony F.C.]] (1966), ''Religion, An Anthropological View'', Random House, New York, NY. * Williams, Bernard (2002), ''Truth and Truthfulness: An Essay in Genealogy'', Princeton University Press, Princeton, NJ. ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://mywikibiz.com/Truth_theory Truth Theory @ MyWikiBiz] * [http://mathweb.org/wiki/Truth_theory Truth Theory @ MathWeb Wiki] * [http://netknowledge.org/wiki/Truth_theory Truth Theory @ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Truth_theory Truth Theory @ OER Commons] {{col-break}} * [http://p2pfoundation.net/Truth_Theory Truth Theory @ P2P Foundation] * [http://semanticweb.org/wiki/Truth_theory Truth Theory @ SemanticWeb] * [http://ref.subwiki.org/wiki/Truth_theory Truth Theory @ Subject Wikis] * [http://beta.wikiversity.org/wiki/Truth_theory Truth Theory @ Wikiversity Beta] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;] * [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;] ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Truth_theory Truth Theory], [http://mywikibiz.com/ MyWikiBiz] * [http://mathweb.org/wiki/Truth_theory Truth Theory], [http://mathweb.org/wiki/ MathWeb Wiki] * [http://netknowledge.org/wiki/Truth_theory Truth Theory], [http://netknowledge.org/ NetKnowledge] {{col-break}} * [http://wiki.oercommons.org/mediawiki/index.php/Truth_theory Truth Theory], [http://wiki.oercommons.org/ OER Commons] * [http://p2pfoundation.net/Truth_Theory Truth Theory], [http://p2pfoundation.net/ P2P Foundation] * [http://semanticweb.org/wiki/Truth_theory Truth Theory], [http://semanticweb.org/ Semantic Web] {{col-break}} * [http://www.getwiki.net/-Truth_Theory Truth Theory], [http://www.getwiki.net/ GetWiki] * [http://wikinfo.org/index.php/Truth_theory Truth Theory], [http://wikinfo.org/index.php/Main_Page Wikinfo] * [http://textop.org/wiki/index.php?title=Truth_theory Truth Theory], [http://textop.org/wiki/ Textop Wiki] {{col-end}} This article contains material from an earlier version of the former [http://en.wikipedia.org/ Wikipedia] article, &ldquo;Truth Theory&rdquo;, no longer extant. The Wikipedia article was deleted by Wikipedia administrators, replaced with a redirect to the Wikipedia article &ldquo;Truth&rdquo;, and its edit history was destroyed, in violation of the GNU Free Documentation License. A record of the Wikipedia AFD (Article For Deletion) proceedings can be found at the following locations: {{col-begin}} {{col-break}} * [http://en.wikipedia.org/w/index.php?title=Wikipedia:Articles_for_deletion/Truth_theory&oldid=54630517 1st AFD proceeding] {{col-break}} * [http://en.wikipedia.org/w/index.php?title=Wikipedia:Articles_for_deletion/Truth_theory(2)&oldid=71295142 2nd AFD proceeding] {{col-break}} * [http://en.wikipedia.org/w/index.php?title=Wikipedia:Articles_for_deletion/Truth_theory_(3rd_nomination)&oldid=81774680 3rd AFD proceeding] {{col-end}} [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Artificial Intelligence]] [[Category:Critical Thinking]] [[Category:Cybernetics]] [[Category:Education]] [[Category:Hermeneutics]] [[Category:Information Systems]] [[Category:Intelligence Amplification]] [[Category:Learning Organizations]] [[Category:Knowledge Representation]] [[Category:Logic]] [[Category:Philosophy]] [[Category:Pragmatics]] [[Category:Semantics]] [[Category:Semiotics]] [[Category:Systems Science]] d3d33a0d9a956e3be9cc0bd051f4758bffd17742 0 349 571 2010-06-25T03:32:41Z Jon Awbrey 3 + article wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''normative science''' is a form of [[inquiry]], typically involving a community of inquiry and its accumulated body of provisional knowledge, that seeks to discover good ways of achieving recognized aims, ends, goals, objectives, or purposes. The three '''normative sciences''', according to traditional conceptions in philosophy, are ''[[aesthetics]]'', ''[[ethics]]'', and ''[[logic]]''. ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://mywikibiz.com/Normative_science Normative Science @ MyWikiBiz] * [http://mathweb.org/wiki/Normative_science Normative Science @ MathWeb Wiki] * [http://netknowledge.org/wiki/Normative_science Normative Science @ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Normative_science Normative Science @ OER Commons] {{col-break}} * [http://p2pfoundation.net/Normative_Science Normative Science @ P2P Foundation] * [http://semanticweb.org/wiki/Normative_science Normative Science @ SemanticWeb] * [http://ref.subwiki.org/wiki/Normative_science Normative Science @ Subject Wikis] * [http://beta.wikiversity.org/wiki/Normative_science Normative Science @ Wikiversity Beta] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;] * [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;] ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Normative_science Normative Science], [http://mywikibiz.com/ MyWikiBiz] * [http://mathweb.org/wiki/Normative_science Normative Science], [http://mathweb.org/wiki/ MathWeb Wiki] * [http://netknowledge.org/wiki/Normative_science Normative Science], [http://netknowledge.org/ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Normative_science Normative Science], [http://wiki.oercommons.org/ OER Commons] {{col-break}} * [http://p2pfoundation.net/Normative_Science Normative Science], [http://p2pfoundation.net/ P2P Foundation] * [http://semanticweb.org/wiki/Normative_science Normative Science], [http://semanticweb.org/ Semantic Web] * [http://beta.wikiversity.org/wiki/Normative_science Normative Science], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Normative_science&oldid=51993011 Normative Science], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Normative Sciences]] [[Category:Philosophy]] [[Category:Philosophy of Science]] [[Category:Science]] b46b32264ef1babc55460a51bd852a97cfa66f2f 0 350 572 2010-06-25T11:36:05Z Jon Awbrey 3 + article wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''descriptive science''', also called a '''special science''', is a form of [[inquiry]], typically involving a community of inquiry and its accumulated body of provisional knowledge, that seeks to discover what is true about a recognized domain of [[phenomena]]. ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://mywikibiz.com/Descriptive_science Descriptive Science @ MyWikiBiz] * [http://mathweb.org/wiki/Descriptive_science Descriptive Science @ MathWeb Wiki] * [http://netknowledge.org/wiki/Descriptive_science Descriptive Science @ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Descriptive_science Descriptive Science @ OER Commons] {{col-break}} * [http://p2pfoundation.net/Descriptive_Science Descriptive Science @ P2P Foundation] * [http://semanticweb.org/wiki/Descriptive_science Descriptive Science @ SemanticWeb] * [http://ref.subwiki.org/wiki/Descriptive_science Descriptive Science @ Subject Wikis] * [http://beta.wikiversity.org/wiki/Descriptive_science Descriptive Science @ Wikiversity Beta] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;] * [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;] ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Descriptive_science Descriptive Science], [http://mywikibiz.com/ MyWikiBiz] * [http://mathweb.org/wiki/Descriptive_science Descriptive Science], [http://mathweb.org/wiki/ MathWeb Wiki] * [http://netknowledge.org/wiki/Descriptive_science Descriptive Science], [http://netknowledge.org/ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Descriptive_science Descriptive Science], [http://wiki.oercommons.org/ OER Commons] {{col-break}} * [http://p2pfoundation.net/Descriptive_Science Descriptive Science], [http://p2pfoundation.net/ P2P Foundation] * [http://semanticweb.org/wiki/Descriptive_science Descriptive Science], [http://semanticweb.org/ Semantic Web] * [http://beta.wikiversity.org/wiki/Descriptive_science Descriptive Science], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Descriptive_science&oldid=51990248 Descriptive Science], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Descriptive Sciences]] [[Category:Philosophy]] [[Category:Philosophy of Science]] [[Category:Science]] 608a40930b7172ca7638aa6c5547cb8cc9dc57fc 2 163 573 394 2010-07-01T14:34:31Z Jon Awbrey 3 /* Recent Sightings */ update wikitext text/x-wiki <br> <p><font face="lucida calligraphy" size="7">Jon Awbrey</font></p> <br> __NOTOC__ ==Presently &hellip;== <center> [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Riffs_and_Rotes R&R] [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems PERS] [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Information_%3D_Comprehension_%C3%97_Extension I = C &times; E] [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Peirce%27s_1870_Logic_Of_Relatives LOR 1870] [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositions_As_Types PAT Analogy] [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Analytic_Turing_Automata DATA Project] [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Cactus_Language Cactus Language] [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Inquiry Driven Systems] Logical Graphs : [http://knol.google.com/k/jon-awbrey/logical-graphs-1/3fkwvf69kridz/3 One] [http://knol.google.com/k/jon-awbrey/logical-graphs-2/3fkwvf69kridz/8 Two] [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Differential Logic : Introduction] [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Differential Propositional Calculus] [http://www.mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] </center> ==Recent Sightings== {| align="center" style="text-align:center" width="100%" |- | [http://stderr.org/cgi-bin/mailman/listinfo/inquiry Inquiry Project] | [http://stderr.org/pipermail/inquiry/ Inquiry Archive] |- | [http://www.mywikibiz.com/User:Jon_Awbrey MyWikiBiz Page] | [http://www.mywikibiz.com/User_talk:Jon_Awbrey MyWikiBiz Talk] |- | [http://www.mywikibiz.com/Directory:Jon_Awbrey MyWikiBiz Directory] | [http://www.mywikibiz.com/Directory_talk:Jon_Awbrey MyWikiBiz Dialogue] |- | [http://planetmath.org/ PlanetMath Project] | [http://planetmath.org/?op=getuser&id=15246 PlanetMath Profile] |- | [http://planetphysics.org/ PlanetPhysics Project] | [http://planetphysics.org/?op=getuser&id=513 PlanetPhysics Profile] |- | [http://knol.google.com/ Google Knol Project] | [http://knol.google.com/k/jon-awbrey/jon-awbrey/3fkwvf69kridz/1 Google Knol Profile] |- | [http://mathforum.org/kb/ Math Forum Project] | [http://mathforum.org/kb/accountView.jspa?userID=99854 Math Forum Profile] |- | [http://list.seqfan.eu/cgi-bin/mailman/listinfo/seqfan Fantasia Sequentia] | [http://list.seqfan.eu/pipermail/seqfan/ SeqFan Archive] |- | [http://www.research.att.com/~njas/sequences/Seis.html OEIS Land] | [http://www.research.att.com/~njas/sequences/?q=Awbrey Bolgia Mia] |- | [http://oeis.org/wiki/User:Jon_Awbrey OEIS Wiki Page] | [http://oeis.org/wiki/User_talk:Jon_Awbrey OEIS Wiki Talk] |- | [http://intersci.ss.uci.edu/wiki/index.php/User:Jon_Awbrey InterSciWiki Page] | [http://intersci.ss.uci.edu/wiki/index.php/User_talk:Jon_Awbrey InterSciWiki Talk] |- | [http://www.mathweb.org/wiki/User:Jon_Awbrey MathWeb Page] | [http://www.mathweb.org/wiki/User_talk:Jon_Awbrey MathWeb Talk] |- | [http://www.proofwiki.org/wiki/User:Jon_Awbrey ProofWiki Page] | [http://www.proofwiki.org/wiki/User_talk:Jon_Awbrey ProofWiki Talk] |- | [http://mathoverflow.net/ MathOverFlow] | [http://mathoverflow.net/users/1636/jon-awbrey MOFler Profile] |- | [http://www.p2pfoundation.net/User:JonAwbrey P2P Wiki Page] | [http://www.p2pfoundation.net/User_talk:JonAwbrey P2P Wiki Talk] |- | [http://vectors.usc.edu/ Vectors Project] | [http://vectors.usc.edu/thoughtmesh/ ThoughtMesh] |- | [http://altheim.4java.ca/ceryle/wiki/ Ceryle Project] | [http://altheim.4java.ca/ceryle/wiki/Wiki.jsp?page=JonAwbrey Ceryle Profile] |- | [http://forum.wolframscience.com/ NKS Forum] | [http://forum.wolframscience.com/member.php?s=&action=getinfo&userid=336 NKS Profile] |- | [http://www.wikinfo.org/index.php/User:Jon_Awbrey WikInfo Page] | [http://www.wikinfo.org/index.php/User_talk:Jon_Awbrey WikInfo Talk] |- | [http://www.getwiki.net/-User:Jon_Awbrey GetWiki Page] | [http://www.getwiki.net/-UserTalk:Jon_Awbrey GetWiki Talk] |- | [http://ontolog.cim3.net/ OntoLog Project] | [http://ontolog.cim3.net/cgi-bin/wiki.pl?JonAwbrey OntoLog Profile] |- | [http://semanticweb.org/wiki/User:Jon_Awbrey SemanticWeb Page] | [http://semanticweb.org/wiki/User_talk:Jon_Awbrey SemanticWeb Talk] |- | [http://zh.wikipedia.org/wiki/User:Jon_Awbrey 维基百科 : 用户页面] | [http://zh.wikipedia.org/wiki/User_talk:Jon_Awbrey 维基百科 : 讨论] |} ==Presentations and Publications== * Awbrey, S.M., and Awbrey, J.L. (May 2001), "Conceptual Barriers to Creating Integrative Universities", ''Organization : The Interdisciplinary Journal of Organization, Theory, and Society'' 8(2), Sage Publications, London, UK, pp. 269&ndash;284. [http://org.sagepub.com/cgi/content/abstract/8/2/269 Abstract]. * Awbrey, S.M., and Awbrey, J.L. (September 1999), "Organizations of Learning or Learning Organizations : The Challenge of Creating Integrative Universities for the Next Century", ''Second International Conference of the Journal ''Organization'' '', ''Re-Organizing Knowledge, Trans-Forming Institutions : Knowing, Knowledge, and the University in the 21st Century'', University of Massachusetts, Amherst, MA. [http://www.cspeirce.com/menu/library/aboutcsp/awbrey/integrat.htm Online]. * Awbrey, J.L., and Awbrey, S.M. (Autumn 1995), "Interpretation as Action : The Risk of Inquiry", ''Inquiry : Critical Thinking Across the Disciplines'' 15(1), pp. 40&ndash;52. [http://www.chss.montclair.edu/inquiry/fall95/awbrey.html Online]. * Awbrey, J.L., and Awbrey, S.M. (June 1992), "Interpretation as Action : The Risk of Inquiry", ''The Eleventh International Human Science Research Conference'', Oakland University, Rochester, Michigan. * Awbrey, S.M., and Awbrey, J.L. (May 1991), "An Architecture for Inquiry : Building Computer Platforms for Discovery", ''Proceedings of the Eighth International Conference on Technology and Education'', Toronto, Canada, pp. 874&ndash;875. [http://www.abccommunity.org/tmp-a.html Online]. * Awbrey, J.L., and Awbrey, S.M. (January 1991), "Exploring Research Data Interactively : Developing a Computer Architecture for Inquiry", Poster presented at the ''Annual Sigma Xi Research Forum'', University of Texas Medical Branch, Galveston, TX. * Awbrey, J.L., and Awbrey, S.M. (August 1990), "Exploring Research Data Interactively. Theme One : A Program of Inquiry", ''Proceedings of the Sixth Annual Conference on Applications of Artificial Intelligence and CD-ROM in Education and Training'', Society for Applied Learning Technology, Washington, DC, pp. 9&ndash;15. ==Education== * 1993&ndash;2003. [http://www2.oakland.edu/secs/dispprofile.asp?Fname=Fatma&Lname=Mili Graduate Study], [http://www2.oakland.edu/secs/dispprofile.asp?Fname=Mohamed&Lname=Zohdy Systems Engineering], [http://www.oakland.edu/?id=3099&sid=87 Oakland University]. * '''1989. [http://web.archive.org/web/20001206101800/http://www.msu.edu/dig/msumap/psychology.html M.A. Psychology]''', [http://web.archive.org/web/20000902103838/http://www.msu.edu/dig/msumap/beaumont.html Michigan State University]. * 1985&ndash;1986. [http://quod.lib.umich.edu/cgi/i/image/image-idx?id=S-BHL-X-BL001808%5DBL001808 Graduate Study, Mathematics], [http://www.umich.edu/ University of Michigan]. * 1985. [http://web.archive.org/web/20061230174652/http://www.uiuc.edu/navigation/buildings/altgeld.top.html Graduate Study, Mathematics], [http://www.uiuc.edu/ University of Illinois at Urbana&ndash;Champaign]. * 1984. [http://www.psych.uiuc.edu/graduate/ Graduate Study, Psychology], [http://www.uiuc.edu/ University of Illinois at Champaign&ndash;Urbana]. * '''1980. [http://www.mth.msu.edu/images/wells_medium.jpg M.A. Mathematics]''', [http://www.msu.edu/~hvac/survey/BeaumontTower.html Michigan State University]. * '''1976. [http://web.archive.org/web/20001206050600/http://www.msu.edu/dig/msumap/phillips.html B.A. Mathematical and Philosophical Method]''', <br> [http://www.enolagaia.com/JMC.html Justin Morrill College], [http://www.msu.edu/ Michigan State University]. cca553a09573afc4097cb23fb63b08efa5fd1c18 574 573 2010-09-30T16:20:05Z Jon Awbrey 3 update wikitext text/x-wiki <br> <p><font face="lucida calligraphy" size="7">Jon Awbrey</font></p> <br> __NOTOC__ ==Presently &hellip;== <center> [http://oeis.org/wiki/Riffs_and_Rotes R&R] [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems PERS] [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Information_%3D_Comprehension_%C3%97_Extension I = C &times; E] [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Peirce%27s_1870_Logic_Of_Relatives LOR 1870] [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositions_As_Types PAT Analogy] [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Analytic_Turing_Automata DATA Project] [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Cactus_Language Cactus Language] [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Inquiry Driven Systems] Logical Graphs : [http://knol.google.com/k/jon-awbrey/logical-graphs-1/3fkwvf69kridz/3 One] [http://knol.google.com/k/jon-awbrey/logical-graphs-2/3fkwvf69kridz/8 Two] [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Differential Logic : Introduction] [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Differential Propositional Calculus] [http://www.mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] [http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] </center> 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[http://mathoverflow.net/users/1636/jon-awbrey MOFler Profile] |- | [http://www.p2pfoundation.net/User:JonAwbrey P2P Wiki Page] | [http://www.p2pfoundation.net/User_talk:JonAwbrey P2P Wiki Talk] |- | [http://vectors.usc.edu/ Vectors Project] | [http://vectors.usc.edu/thoughtmesh/ ThoughtMesh] |- | [http://altheim.4java.ca/ceryle/wiki/ Ceryle Project] | [http://altheim.4java.ca/ceryle/wiki/Wiki.jsp?page=JonAwbrey Ceryle Profile] |- | [http://forum.wolframscience.com/ NKS Forum] | [http://forum.wolframscience.com/member.php?s=&action=getinfo&userid=336 NKS Profile] |- | [http://www.wikinfo.org/index.php/User:Jon_Awbrey WikInfo Page] | [http://www.wikinfo.org/index.php/User_talk:Jon_Awbrey WikInfo Talk] |- | [http://www.getwiki.net/-User:Jon_Awbrey GetWiki Page] | [http://www.getwiki.net/-UserTalk:Jon_Awbrey GetWiki Talk] |- | [http://ontolog.cim3.net/ OntoLog Project] | [http://ontolog.cim3.net/cgi-bin/wiki.pl?JonAwbrey OntoLog Profile] |- | [http://semanticweb.org/wiki/User:Jon_Awbrey SemanticWeb Page] | [http://semanticweb.org/wiki/User_talk:Jon_Awbrey SemanticWeb Talk] |- | [http://zh.wikipedia.org/wiki/User:Jon_Awbrey 维基百科 : 用户页面] | [http://zh.wikipedia.org/wiki/User_talk:Jon_Awbrey 维基百科 : 讨论] |} ==Presentations and Publications== * Awbrey, S.M., and Awbrey, J.L. (May 2001), "Conceptual Barriers to Creating Integrative Universities", ''Organization : The Interdisciplinary Journal of Organization, Theory, and Society'' 8(2), Sage Publications, London, UK, pp. 269&ndash;284. [http://org.sagepub.com/cgi/content/abstract/8/2/269 Abstract]. * Awbrey, S.M., and Awbrey, J.L. (September 1999), "Organizations of Learning or Learning Organizations : The Challenge of Creating Integrative Universities for the Next Century", ''Second International Conference of the Journal ''Organization'' '', ''Re-Organizing Knowledge, Trans-Forming Institutions : Knowing, Knowledge, and the University in the 21st Century'', University of Massachusetts, Amherst, MA. [http://www.cspeirce.com/menu/library/aboutcsp/awbrey/integrat.htm Online]. * Awbrey, J.L., and Awbrey, S.M. (Autumn 1995), "Interpretation as Action : The Risk of Inquiry", ''Inquiry : Critical Thinking Across the Disciplines'' 15(1), pp. 40&ndash;52. [http://www.chss.montclair.edu/inquiry/fall95/awbrey.html Online]. * Awbrey, J.L., and Awbrey, S.M. (June 1992), "Interpretation as Action : The Risk of Inquiry", ''The Eleventh International Human Science Research Conference'', Oakland University, Rochester, Michigan. * Awbrey, S.M., and Awbrey, J.L. (May 1991), "An Architecture for Inquiry : Building Computer Platforms for Discovery", ''Proceedings of the Eighth International Conference on Technology and Education'', Toronto, Canada, pp. 874&ndash;875. [http://www.abccommunity.org/tmp-a.html Online]. * Awbrey, J.L., and Awbrey, S.M. (January 1991), "Exploring Research Data Interactively : Developing a Computer Architecture for Inquiry", Poster presented at the ''Annual Sigma Xi Research Forum'', University of Texas Medical Branch, Galveston, TX. * Awbrey, J.L., and Awbrey, S.M. (August 1990), "Exploring Research Data Interactively. Theme One : A Program of Inquiry", ''Proceedings of the Sixth Annual Conference on Applications of Artificial Intelligence and CD-ROM in Education and Training'', Society for Applied Learning Technology, Washington, DC, pp. 9&ndash;15. ==Education== * 1993&ndash;2003. [http://www2.oakland.edu/secs/dispprofile.asp?Fname=Fatma&Lname=Mili Graduate Study], [http://www2.oakland.edu/secs/dispprofile.asp?Fname=Mohamed&Lname=Zohdy Systems Engineering], [http://www.oakland.edu/?id=3099&sid=87 Oakland University]. * '''1989. [http://web.archive.org/web/20001206101800/http://www.msu.edu/dig/msumap/psychology.html M.A. Psychology]''', [http://web.archive.org/web/20000902103838/http://www.msu.edu/dig/msumap/beaumont.html Michigan State University]. * 1985&ndash;1986. [http://quod.lib.umich.edu/cgi/i/image/image-idx?id=S-BHL-X-BL001808%5DBL001808 Graduate Study, Mathematics], [http://www.umich.edu/ University of Michigan]. * 1985. [http://web.archive.org/web/20061230174652/http://www.uiuc.edu/navigation/buildings/altgeld.top.html Graduate Study, Mathematics], [http://www.uiuc.edu/ University of Illinois at Urbana&ndash;Champaign]. * 1984. [http://www.psych.uiuc.edu/graduate/ Graduate Study, Psychology], [http://www.uiuc.edu/ University of Illinois at Champaign&ndash;Urbana]. * '''1980. [http://www.mth.msu.edu/images/wells_medium.jpg M.A. Mathematics]''', [http://www.msu.edu/~hvac/survey/BeaumontTower.html Michigan State University]. * '''1976. [http://web.archive.org/web/20001206050600/http://www.msu.edu/dig/msumap/phillips.html B.A. Mathematical and Philosophical Method]''', <br> [http://www.enolagaia.com/JMC.html Justin Morrill College], [http://www.msu.edu/ Michigan State University]. 258b563ef59873ae1538d244d0fe302df4604f77 0 187 575 453 2010-09-30T16:30:41Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''logical graph''' is a [[graph theory|graph-theoretic]] structure in one of the systems of graphical syntax that [[Charles Sanders Peirce]] developed for [[logic]]. In his papers on ''qualitative logic'', ''[[entitative graph]]s'', and ''existential graphs'', Peirce developed several versions of a graphical formalism, or a graph-theoretic formal language, designed to be interpreted for logic. In the century since Peirce initiated this line of development, a variety of formal systems have branched out from what is abstractly the same formal base of graph-theoretic structures. This article examines the common basis of these formal systems from a bird's eye view, focusing on those aspects of form that are shared by the entire family of algebras, calculi, or languages, however they happen to be viewed in a given application. ==Abstract point of view== {| width="100%" cellpadding="2" cellspacing="0" | width="60%" | &nbsp; | width="40%" | ''Wollust ward dem Wurm gegeben &hellip;'' |- | &nbsp; | align="right" | &mdash; Friedrich Schiller, ''An die Freude'' |} The bird's eye view in question is more formally known as the perspective of formal equivalence, from which remove one cannot see many distinctions that appear momentous from lower levels of abstraction. In particular, expressions of different formalisms whose syntactic structures are [[isomorphic]] from the standpoint of [[algebra]] or [[topology]] are not recognized as being different from each other in any significant sense. Though we may note in passing such historical details as the circumstance that Charles Sanders Peirce used a ''streamer-cross symbol'' where [[George Spencer Brown]] used a ''carpenter's square marker'', the theme of principal interest at the abstract level of form is neutral with regard to variations of that order. ==In lieu of a beginning== Consider the formal equations indicated in Figures&nbsp;1 and 2. {| align="center" border="0" cellpadding="10" cellspacing="0" | [[Image:Logical_Graph_Figure_1_Visible_Frame.jpg|500px]] || (1) |- | [[Image:Logical_Graph_Figure_2_Visible_Frame.jpg|500px]] || (2) |} For the time being these two forms of transformation may be referred to as ''[[axioms]]'' or ''initial equations''. ==Duality : logical and topological== There are two types of duality that have to be kept separately mind in the use of logical graphs &mdash; logical duality and topological duality. There is a standard way that graphs of the order that Peirce considered, those embedded in a continuous [[manifold]] like that commonly represented by a plane sheet of paper &mdash; with or without the paper bridges that Peirce used to augment its topological genus &mdash; can be represented in linear text as what are called ''parse strings'' or ''traversal strings'' and parsed into ''pointer structures'' in computer memory. A blank sheet of paper can be represented in linear text as a blank space, but that way of doing it tends to be confusing unless the logical expression under consideration is set off in a separate display. For example, consider the axiom or initial equation that is shown below: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_3_Visible_Frame.jpg|500px]] || (3) |} This can be written inline as <math>{}^{\backprime\backprime} \texttt{(~(~)~)} = \quad {}^{\prime\prime}</math> or set off in a text display as follows: {| align="center" cellpadding="10" | width="33%" | <math>\texttt{(~(~)~)}</math> | width="34%" | <math>=\!</math> | width="33%" | &nbsp; |} When we turn to representing the corresponding expressions in computer memory, where they can be manipulated with utmost facility, we begin by transforming the planar graphs into their topological duals. The planar regions of the original graph correspond to nodes (or points) of the [[dual graph]], and the boundaries between planar regions in the original graph correspond to edges (or lines) between the nodes of the dual graph. For example, overlaying the corresponding dual graphs on the plane-embedded graphs shown above, we get the following composite picture: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_4_Visible_Frame.jpg|500px]] || (4) |} Though it's not really there in the most abstract topology of the matter, for all sorts of pragmatic reasons we find ourselves compelled to single out the outermost region of the plane in a distinctive way and to mark it as the ''[[root node]]'' of the corresponding dual graph. In the present style of Figure the root nodes are marked by horizontal strike-throughs. Extracting the dual graphs from their composite matrices, we get this picture: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_5_Visible_Frame.jpg|500px]] || (5) |} It is easy to see the relationship between the parenthetical expressions of Peirce's logical graphs, that somewhat clippedly picture the ordered containments of their formal contents, and the associated dual graphs, that constitute the species of [[rooted tree]]s here to be described. In the case of our last example, a moment's contemplation of the following picture will lead us to see that we can get the corresponding parenthesis string by starting at the root of the tree, climbing up the left side of the tree until we reach the top, then climbing back down the right side of the tree until we return to the root, all the while reading off the symbols, in this case either <math>{}^{\backprime\backprime} \texttt{(} {}^{\prime\prime}</math> or <math>{}^{\backprime\backprime} \texttt{)} {}^{\prime\prime},</math> that we happen to encounter in our travels. {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_6_Visible_Frame.jpg|500px]] || (6) |} This ritual is called ''[[traversing]]'' the tree, and the string read off is called the ''[[traversal string]]'' of the tree. The reverse ritual, that passes from the string to the tree, is called ''[[parsing]]'' the string, and the tree constructed is called the ''[[parse graph]]'' of the string. The speakers thereof tend to be a bit loose in this language, often using ''[[parse string]]'' to mean the string that gets parsed into the associated graph. We have treated in some detail various forms of the initial equation or logical axiom that is formulated in string form as <math>{}^{\backprime\backprime} \texttt{(~(~)~)} = \quad {}^{\prime\prime}.</math> For the sake of comparison, let's record the plane-embedded and topological dual forms of the axiom that is formulated in string form as <math>{}^{\backprime\backprime} \texttt{(~)(~)} = \texttt{(~)} {}^{\prime\prime}.</math> First the plane-embedded maps: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_7_Visible_Frame.jpg|500px]] || (7) |} Next the plane-embedded maps and their dual trees superimposed: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_8_Visible_Frame.jpg|500px]] || (8) |} Finally the dual trees by themselves: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_9_Visible_Frame.jpg|500px]] || (9) |} And here are the parse trees with their traversal strings indicated: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_10_Visible_Frame.jpg|500px]] || (10) |} We have at this point enough material to begin thinking about the forms of [[analogy]], [[iconicity]], [[metaphor]], [[morphism]], whatever you want to call them, that are pertinent to the use of logical graphs in their various logical interpretations, for instance, those that Peirce described as ''[[entitative graph]]s'' and ''[[existential graph]]s''. ==Computational representation== The parse graphs that we've been looking at so far bring us one step closer to the pointer graphs that it takes to make these maps and trees live in computer memory, but they are still a couple of steps too abstract to detail the concrete species of dynamic data structures that we need. The time has come to flesh out the skeletons that we've drawn up to this point. Nodes in a graph represent ''records'' in computer memory. A record is a collection of data that can be conceived to reside at a specific ''address''. The address of a record is analogous to a demonstrative pronoun, on which account programmers commonly describe it as a ''pointer'' and semioticians recognize it as a type of sign called an ''index''. At the next level of concreteness, a pointer-record structure is represented as follows: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_11_Visible_Frame.jpg|500px]] || (11) |} This portrays the pointer <math>\mathit{index}_0\!</math> as the address of a record that contains the following data: {| align="center" cellpadding="10" | <math>\mathit{datum}_1, \mathit{datum}_2, \mathit{datum}_3, \ldots,\!</math> and so on. |} What makes it possible to represent graph-theoretical structures as data structures in computer memory is the fact that an address is just another datum, and so we may have a state of affairs like the following: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_12_Visible_Frame.jpg|500px]] || (12) |} Returning to the abstract level, it takes three nodes to represent the three data records illustrated above: one root node connected to a couple of adjacent nodes. The items of data that do not point any further up the tree are then treated as labels on the record-nodes where they reside, as shown below: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_13_Visible_Frame.jpg|500px]] || (13) |} Notice that drawing the arrows is optional with rooted trees like these, since singling out a unique node as the root induces a unique orientation on all the edges of the tree, with ''up'' being the same direction as ''away from the root''. ==Quick tour of the neighborhood== This much preparation allows us to take up the founding axioms or initial equations that determine the entire system of logical graphs. ===Primary arithmetic as semiotic system=== Though it may not seem too exciting, logically speaking, there are many reasons to make oneself at home with the system of forms that is represented indifferently, topologically speaking, by rooted trees, balanced strings of parentheses, or finite sets of non-intersecting simple closed curves in the plane. :* One reason is that it gives us a respectable example of a sign domain on which to cut our semiotic teeth, non-trivial in the sense that it contains a [[countable]] [[infinity]] of signs. :* Another reason is that it allows us to study a simple form of [[computation]] that is recognizable as a species of ''[[semiosis]]'', or sign-transforming process. This space of forms, along with the two axioms that induce its [[partition of a set|partition]] into exactly two [[equivalence class]]es, is what [[George Spencer Brown]] called the ''primary arithmetic''. The axioms of the primary arithmetic are shown below, as they appear in both graph and string forms, along with pairs of names that come in handy for referring to the two opposing directions of applying the axioms. {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_14_Banner.jpg|500px]] || (14) |- | [[Image:Logical_Graph_Figure_15_Banner.jpg|500px]] || (15) |} Let <math>S\!</math> be the set of rooted trees and let <math>S_0\!</math> be the 2-element subset of <math>S\!</math> that consists of a rooted node and a rooted edge. {| align="center" cellpadding="10" style="text-align:center" | <math>S\!</math> | <math>=\!</math> | <math>\{ \text{rooted trees} \}\!</math> |- | <math>S_0\!</math> | <math>=\!</math> | <math>\{ \ominus, \vert \} = \{</math>[[Image:Rooted Node.jpg|16px]], [[Image:Rooted Edge.jpg|12px]]<math>\}\!</math> |} Simple intuition, or a simple inductive proof, assures us that any rooted tree can be reduced by way of the arithmetic initials either to a root node [[Image:Rooted Node.jpg|16px]] or else to a rooted edge [[Image:Rooted Edge.jpg|12px]]&nbsp;. For example, consider the reduction that proceeds as follows: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_16.jpg|500px]] || (16) |} Regarded as a semiotic process, this amounts to a sequence of signs, every one after the first serving as the ''[[interpretant]]'' of its predecessor, ending in a final sign that may be taken as the canonical sign for their common object, in the upshot being the result of the computation process. Simple as it is, this exhibits the main features of any computation, namely, a semiotic process that proceeds from an obscure sign to a clear sign of the same object, being in its aim and effect an action on behalf of clarification. ===Primary algebra as pattern calculus=== Experience teaches that complex objects are best approached in a gradual, laminar, [[module|modular]] fashion, one step, one layer, one piece at a time, and it's just as much the case when the complexity of the object is irreducible, that is, when the articulations of the representation are necessarily at joints that are cloven disjointedly from nature, with some assembly required in the synthetic integrity of the intuition. That's one good reason for spending so much time on the first half of [[zeroth order logic]], represented here by the primary arithmetic, a level of formal structure that C.S. Peirce verged on intuiting at numerous points and times in his work on logical graphs, and that Spencer Brown named and brought more completely to life. There is one other reason for lingering a while longer in these primitive forests, and this is that an acquaintance with "bare trees", those as yet unadorned with literal or numerical labels, will provide a firm basis for understanding what's really at issue in such problems as the "ontological status of variables". It is probably best to illustrate this theme in the setting of a concrete case, which we can do by reviewing the previous example of reductive evaluation shown in Figure&nbsp;16. The observation of several ''semioses'', or sign-transformations, of roughly this shape will most likely lead an observer with any observational facility whatever to notice that it doesn't really matter what sorts of branches happen to sprout from the side of the root aside from the lone edge that also grows there &mdash; the end result will always be the same. Our observer might think to summarize the results of many such observations by introducing a label or variable to signify any shape of branch whatever, writing something like the following: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_17.jpg|500px]] || (17) |} Observations like that, made about an arithmetic of any variety, germinated by their summarizations, are the root of all algebra. Speaking of algebra, and having encountered already one example of an algebraic law, we might as well introduce the axioms of the ''primary algebra'', once again deriving their substance and their name from the works of [[Charles Sanders Peirce]] and [[George Spencer Brown]], respectively. {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_18.jpg|500px]] || (18) |- | [[Image:Logical_Graph_Figure_19.jpg|500px]] || (19) |} The choice of axioms for any formal system is to some degree a matter of aesthetics, as it is commonly the case that many different selections of formal rules will serve as axioms to derive all the rest as theorems. As it happens, the example of an algebraic law that we noticed first, <math>a(~) = (~),</math> as simple as it appears, proves to be provable as a theorem on the grounds of the foregoing axioms. We might also notice at this point a subtle difference between the primary arithmetic and the primary algebra with respect to the grounds of justification that we have naturally if tacitly adopted for their respective sets of axioms. The arithmetic axioms were introduced by fiat, in a quasi-[[apriori]] fashion, though of course it is only long prior experience with the practical uses of comparably developed generations of formal systems that would actually induce us to such a quasi-primal move. The algebraic axioms, in contrast, can be seen to derive their motive and their justice from the observation and summarization of patterns that are visible in the arithmetic spectrum. ==Formal development== What precedes this point is intended as an informal introduction to the axioms of the primary arithmetic and primary algebra, and hopefully provides the reader with an intuitive sense of their motivation and rationale. The next order of business is to give the exact forms of the axioms that are used in the following more formal development, devolving from Peirce's various systems of logical graphs via Spencer-Brown's ''Laws of Form'' (LOF). In formal proofs, a variation of the annotation scheme from LOF will be used to mark each step of the proof according to which axiom, or ''initial'', is being invoked to justify the corresponding step of syntactic transformation, whether it applies to graphs or to strings. ===Axioms=== The axioms are just four in number, divided into the ''arithmetic initials'', <math>I_1\!</math> and <math>I_2,\!</math> and the ''algebraic initials'', <math>J_1\!</math> and <math>J_2.\!</math> {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_20.jpg|500px]] || (20) |- | [[Image:Logical_Graph_Figure_21.jpg|500px]] || (21) |- | [[Image:Logical_Graph_Figure_22.jpg|500px]] || (22) |- | [[Image:Logical_Graph_Figure_23.jpg|500px]] || (23) |} One way of assigning logical meaning to the initial equations is known as the ''entitative interpretation'' (EN). Under EN, the axioms read as follows: {| align="center" cellpadding="10" | <math>\begin{array}{ccccc} I_1 & : & \operatorname{true}\ \operatorname{or}\ \operatorname{true} & = & \operatorname{true} \\ I_2 & : & \operatorname{not}\ \operatorname{true}\ & = & \operatorname{false} \\ J_1 & : & a\ \operatorname{or}\ \operatorname{not}\ a & = & \operatorname{true} \\ J_2 & : & (a\ \operatorname{or}\ b)\ \operatorname{and}\ (a\ \operatorname{or}\ c) & = & a\ \operatorname{or}\ (b\ \operatorname{and}\ c) \\ \end{array}</math> |} Another way of assigning logical meaning to the initial equations is known as the ''existential interpretation'' (EX). Under EX, the axioms read as follows: {| align="center" cellpadding="10" | <math>\begin{array}{ccccc} I_1 & : & \operatorname{false}\ \operatorname{and}\ \operatorname{false} & = & \operatorname{false} \\ I_2 & : & \operatorname{not}\ \operatorname{false} & = & \operatorname{true} \\ J_1 & : & a\ \operatorname{and}\ \operatorname{not}\ a & = & \operatorname{false} \\ J_2 & : & (a\ \operatorname{and}\ b)\ \operatorname{or}\ (a\ \operatorname{and}\ c) & = & a\ \operatorname{and}\ (b\ \operatorname{or}\ c) \\ \end{array}</math> |} All of the axioms in this set have the form of equations. This means that all of the inference steps that they allow are reversible. The proof annotation scheme employed below makes use of a double bar <math>\overline{\underline{~~~~~~}}</math> to mark this fact, although it will often be left to the reader to decide which of the two possible directions is the one required for applying the indicated axiom. ===Frequently used theorems=== The actual business of proof is a far more strategic affair than the simple cranking of inference rules might suggest. Part of the reason for this lies in the circumstance that the usual brands of inference rules combine the moving forward of a state of inquiry with the losing of information along the way that doesn't appear to be immediately relevant, at least, not as viewed in the local focus and the short run of the moment to moment proceedings of the proof in question. Over the long haul, this has the pernicious side-effect that one is forever strategically required to reconstruct much of the information that one had strategically thought to forget in earlier stages of the proof, if "before the proof started" can be counted as an earlier stage of the proof in view. For this reason, among others, it is very instructive to study equational inference rules of the sort that our axioms have just provided. Although equational forms of reasoning are paramount in mathematics, they are less familiar to the student of conventional logic textbooks, who may find a few surprises here. By way of gaining a minimal experience with how equational proofs look in the present forms of syntax, let us examine the proofs of a few essential theorems in the primary algebra. ====C₁. Double negation==== The first theorem goes under the names of ''Consequence&nbsp;1'' <math>(C_1)\!</math>, the ''double negation theorem'' (DNT), or ''Reflection''. {| align="center" cellpadding="10" | [[Image:Double Negation 1.0 Splash Page.png|500px]] || (24) |} The proof that follows is adapted from the one that was given by [[George Spencer Brown]] in his book ''Laws of Form'' (LOF) and credited to two of his students, John Dawes and D.A. Utting. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Double Negation 1.0 Marquee Title.png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 1.png|500px]] |- | [[Image:Equational Inference I2 Elicit (( )).png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 2.png|500px]] |- | [[Image:Equational Inference J1 Insert (a).png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 3.png|500px]] |- | [[Image:Equational Inference J2 Distribute ((a)).png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 4.png|500px]] |- | [[Image:Equational Inference J1 Delete (a).png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 5.png|500px]] |- | [[Image:Equational Inference J1 Insert a.png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 6.png|500px]] |- | [[Image:Equational Inference J2 Collect a.png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 7.png|500px]] |- | [[Image:Equational Inference J1 Delete ((a)).png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 8.png|500px]] |- | [[Image:Equational Inference I2 Cancel (( )).png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 9.png|500px]] |- | [[Image:Equational Inference Marquee QED.png|500px]] |} | (25) |} The steps of this proof are replayed in the following animation. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Double Negation 2.0 Animation.gif]] |} | (26) |} ====C₂. Generation theorem==== One theorem of frequent use goes under the nickname of the ''weed and seed theorem'' (WAST). The proof is just an exercise in mathematical induction, once a suitable basis is laid down, and it will be left as an exercise for the reader. What the WAST says is that a label can be freely distributed or freely erased anywhere in a subtree whose root is labeled with that label. The second in our list of frequently used theorems is in fact the base case of this weed and seed theorem. In LOF, it goes by the names of ''Consequence&nbsp;2'' <math>(C_2)\!</math> or ''Generation''. {| align="center" cellpadding="10" | [[Image:Generation Theorem 1.0 Splash Page.png|500px]] || (27) |} Here is a proof of the Generation Theorem. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Generation Theorem 1.0 Marquee Title.png|500px]] |- | [[Image:Generation Theorem 1.0 Storyboard 1.png|500px]] |- | [[Image:Equational Inference C1 Reflect a(b).png|500px]] |- | [[Image:Generation Theorem 1.0 Storyboard 2.png|500px]] |- | [[Image:Equational Inference I2 Elicit (( )).png|500px]] |- | [[Image:Generation Theorem 1.0 Storyboard 3.png|500px]] |- | [[Image:Equational Inference J1 Insert a.png|500px]] |- | [[Image:Generation Theorem 1.0 Storyboard 4.png|500px]] |- | [[Image:Equational Inference J2 Collect a.png|500px]] |- | [[Image:Generation Theorem 1.0 Storyboard 5.png|500px]] |- | [[Image:Equational Inference C1 Reflect a, b.png|500px]] |- | [[Image:Generation Theorem 1.0 Storyboard 6.png|500px]] |- | [[Image:Equational Inference Marquee QED.png|500px]] |} | (28) |} The steps of this proof are replayed in the following animation. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Generation Theorem 2.0 Animation.gif]] |} | (29) |} ====C₃. Dominant form theorem==== The third of the frequently used theorems of service to this survey is one that Spencer-Brown annotates as ''Consequence&nbsp;3'' <math>(C_3)\!</math> or ''Integration''. A better mnemonic might be ''dominance and recession theorem'' (DART), but perhaps the brevity of ''dominant form theorem'' (DFT) is sufficient reminder of its double-edged role in proofs. {| align="center" cellpadding="10" | [[Image:Dominant Form 1.0 Splash Page.png|500px]] || (30) |} Here is a proof of the Dominant Form Theorem. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Dominant Form 1.0 Marquee Title.png|500px]] |- | [[Image:Dominant Form 1.0 Storyboard 1.png|500px]] |- | [[Image:Equational Inference C2 Regenerate a.png|500px]] |- | [[Image:Dominant Form 1.0 Storyboard 2.png|500px]] |- | [[Image:Equational Inference J1 Delete a.png|500px]] |- | [[Image:Dominant Form 1.0 Storyboard 3.png|500px]] |- | [[Image:Equational Inference Marquee QED.png|500px]] |} | (31) |} The following animation provides an instant re*play. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Dominant Form 2.0 Animation.gif]] |} | (32) |} ===Exemplary proofs=== Based on the axioms given at the outest, and aided by the theorems recorded so far, it is possible to prove a multitude of much more complex theorems. A couple of all-time favorites are given next. ====Peirce's law==== : ''Main article'' : [[Peirce's law]] Peirce's law is commonly written in the following form: {| align="center" cellpadding="10" | <math>((p \Rightarrow q) \Rightarrow p) \Rightarrow p</math> |} The existential graph representation of Peirce's law is shown in Figure&nbsp;33. {| align="center" cellpadding="10" | [[Image:Peirce's Law 1.0 Splash Page.png|500px]] || (33) |} A graphical proof of Peirce's law is shown in Figure&nbsp;34. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Peirce's Law 1.0 Marquee Title.png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 1.png|500px]] |- | [[Image:Equational Inference Band Collect p.png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 2.png|500px]] |- | [[Image:Equational Inference Band Quit ((q)).png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 3.png|500px]] |- | [[Image:Equational Inference Band Cancel (( )).png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 4.png|500px]] |- | [[Image:Equational Inference Band Delete p.png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 5.png|500px]] |- | [[Image:Equational Inference Band Cancel (( )).png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 6.png|500px]] |- | [[Image:Equational Inference Marquee QED.png|500px]] |} | (34) |} The following animation replays the steps of the proof. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Peirce's Law 2.0 Animation.gif]] |} | (35) |} ====Praeclarum theorema==== An illustrious example of a propositional theorem is the ''praeclarum theorema'', the ''admirable'', ''shining'', or ''splendid'' theorem of [[Leibniz]]. {| align="center" cellspacing="6" width="90%" <!--QUOTE--> | <p>If ''a'' is ''b'' and ''d'' is ''c'', then ''ad'' will be ''bc''.</p> <p>This is a fine theorem, which is proved in this way:</p> <p>''a'' is ''b'', therefore ''ad'' is ''bd'' (by what precedes),</p> <p>''d'' is ''c'', therefore ''bd'' is ''bc'' (again by what precedes),</p> <p>''ad'' is ''bd'', and ''bd'' is ''bc'', therefore ''ad'' is ''bc''. Q.E.D.</p> <p>(Leibniz, ''Logical Papers'', p. 41).</p> |} Under the existential interpretation, the praeclarum theorema is represented by means of the following logical graph. {| align="center" cellpadding="10" | [[Image:Praeclarum Theorema 1.0 Splash Page.png|500px]] || (36) |} And here's a neat proof of that nice theorem. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Praeclarum Theorema 1.0 Marquee Title.png|500px]] |- | [[Image:Praeclarum Theorema 1.0 Storyboard 1.png|500px]] |- | [[Image:Equational Inference Rule Reflect ad(bc).png|500px]] |- | [[Image:Praeclarum Theorema 1.0 Storyboard 2.png|500px]] |- | [[Image:Equational Inference Rule Weed a, d.png|500px]] |- | [[Image:Praeclarum Theorema 1.0 Storyboard 3.png|500px]] |- | [[Image:Equational Inference Rule Reflect b, c.png|500px]] |- | [[Image:Praeclarum Theorema 1.0 Storyboard 4.png|500px]] |- | [[Image:Equational Inference Rule Weed bc.png|500px]] |- | [[Image:Praeclarum Theorema 1.0 Storyboard 5.png|500px]] |- | [[Image:Equational Inference Rule Quit abcd.png|500px]] |- | [[Image:Praeclarum Theorema 1.0 Storyboard 6.png|500px]] |- | [[Image:Equational Inference Rule Cancel (( )).png|500px]] |- | [[Image:Praeclarum Theorema 1.0 Storyboard 7.png|500px]] |- | [[Image:Equational Inference Marquee QED.png|500px]] |} | (37) |} The steps of the proof are replayed in the following animation. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Praeclarum Theorema 2.0 Animation.gif]] |} | (38) |} ====Two-thirds majority function==== Consider the following equation in boolean algebra, posted as a [http://mathoverflow.net/questions/9292/newbie-boolean-algebra-question problem for proof] at [http://mathoverflow.net/ MathOverFlow]. {| align="center" cellpadding="20" | <math>\begin{matrix} a b \bar{c} + a \bar{b} c + \bar{a} b c + a b c \\[6pt] \iff \\[6pt] a b + a c + b c \end{matrix}</math> | &nbsp;&nbsp;&nbsp;&nbsp; |} The required equation can be proven in the medium of logical graphs as shown in the following Figure. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Two-Thirds Majority Eq 1 Pf 1 Banner Title.png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 1.png|500px]] |- | [[Image:Equational Inference Bar Reflect ab, ac, bc.png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 2.png|500px]] |- | [[Image:Equational Inference Bar Distribute (abc).png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 3.png|500px]] |- | [[Image:Equational Inference Bar Collect ab, ac, bc.png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 4.png|500px]] |- | [[Image:Equational Inference Bar Quit (a), (b), (c).png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 5.png|500px]] |- | [[Image:Equational Inference Bar Cancel (( )).png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 6.png|500px]] |- | [[Image:Equational Inference Bar Weed ab, ac, bc.png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 7.png|500px]] |- | [[Image:Equational Inference Bar Delete a, b, c.png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 8.png|500px]] |- | [[Image:Equational Inference Bar Cancel (( )).png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 9.png|500px]] |- | [[Image:Equational Inference Banner QED.png|500px]] |} | (39) |} Here's an animated recap of the graphical transformations that occur in the above proof: {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Two-Thirds Majority Function 500 x 250 Animation.gif]] |} | (40) |} ==Bibliography== * [[Gottfried Leibniz|Leibniz, G.W.]] (1679&ndash;1686 ?), "Addenda to the Specimen of the Universal Calculus", pp. 40&ndash;46 in G.H.R. Parkinson (ed. and trans., 1966), ''Leibniz : Logical Papers'', Oxford University Press, London, UK. * [[Charles Peirce (Bibliography)|Peirce, C.S., Bibliography]]. * [[Charles Peirce|Peirce, C.S.]] (1931&ndash;1935, 1958), ''Collected Papers of Charles Sanders Peirce'', vols. 1&ndash;6, [[Charles Hartshorne]] and [[Paul Weiss]] (eds.), vols. 7&ndash;8, [[Arthur W. Burks]] (ed.), Harvard University Press, Cambridge, MA. Cited as (CP&nbsp;volume.paragraph). * Peirce, C.S. (1981&ndash;), ''Writings of Charles S. Peirce : A Chronological Edition'', [[Peirce Edition Project]] (eds.), Indiana University Press, Bloomington and Indianopolis, IN. Cited as (CE&nbsp;volume, page). * Peirce, C.S. (1885), "On the Algebra of Logic : A Contribution to the Philosophy of Notation", ''American Journal of Mathematics'' 7 (1885), 180&ndash;202. Reprinted as CP&nbsp;3.359&ndash;403 and CE&nbsp;5, 162&ndash;190. * Peirce, C.S. (''c.'' 1886), "Qualitative Logic", MS 736. Published as pp. 101&ndash;115 in Carolyn Eisele (ed., 1976), ''The New Elements of Mathematics by Charles S. Peirce, Volume 4, Mathematical Philosophy'', Mouton, The Hague. * Peirce, C.S. (1886 a), "Qualitative Logic", MS 582. Published as pp. 323&ndash;371 in ''Writings of Charles S. Peirce : A Chronological Edition, Volume 5, 1884&ndash;1886'', Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1993. * Peirce, C.S. (1886 b), "The Logic of Relatives : Qualitative and Quantitative", MS 584. Published as pp. 372&ndash;378 in ''Writings of Charles S. Peirce : A Chronological Edition, Volume 5, 1884&ndash;1886'', Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1993. * [[George Spencer Brown|Spencer Brown, George]] (1969), ''[[Laws of Form]]'', George Allen and Unwin, London, UK. ==Resources== * Bergman and Paavola (eds.), [http://www.helsinki.fi/science/commens/dictionary.html Commens Dictionary of Peirce's Terms] : [http://www.helsinki.fi/science/commens/terms/graphexis.html Existential Graph] : [http://www.helsinki.fi/science/commens/terms/graphlogi.html Logical Graph] * [http://dr-dau.net/ Dau, Frithjof] : [http://dr-dau.net/eg_readings.shtml Peirce's Existential Graphs &mdash; Readings and Links] : [http://web.archive.org/web/20070706192257/http://dr-dau.net/pc.shtml Leibniz's Splendid Theorem as Moving-Picture of Thought] * [http://www.math.uic.edu/~kauffman/ Kauffman, Louis H.] : [http://www.math.uic.edu/~kauffman/Arithmetic.htm Box Algebra, Boundary Mathematics, Logic, and Laws of Form] * [http://mathworld.wolfram.com/ MathWorld : A Wolfram Web Resource] : [http://mathworld.wolfram.com/about/author.html Weisstein, Eric W.], [http://mathworld.wolfram.com/Spencer-BrownForm.html Spencer-Brown Form] * [http://planetmath.org/ PlanetMath] : [http://planetmath.org/encyclopedia/LogicalGraph.html Logical Graph : Introduction] : [http://planetmath.org/encyclopedia/LogicalGraphFormalDevelopment.html Logical Graph : Formal Development] * Shoup, Richard (ed.), [http://www.lawsofform.org/ Laws of Form Web Site] : Spencer-Brown, George (1973), [http://www.lawsofform.org/aum/session1.html Transcript Session One], [http://www.lawsofform.org/aum/ AUM Conference], Esalen, CA * [http://www.jfsowa.com/ Sowa, John F.] : [http://www.jfsowa.com/cg/ Conceptual Graphs &mdash; Brief Summary] : [http://conceptualgraphs.org/ Conceptual Graphs &mdash; Resource Hub] : [http://conceptualstructures.org/ Conceptual Structures &mdash; Home Page] ==Translations== * [http://pt.wikipedia.org/wiki/Grafo_l%C3%B3gico Grafo lógico], [http://pt.wikipedia.org/ Portuguese Wikipedia]. ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://mywikibiz.com/Logical_graph Logical Graph @ MyWikiBiz] * [http://mathweb.org/wiki/Logical_graph Logical Graph @ MathWeb Wiki] * [http://netknowledge.org/wiki/Logical_graph Logical Graph @ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Logical_graph Logical Graph @ OER Commons] {{col-break}} * [http://p2pfoundation.net/Logical_Graph Logical Graph @ P2P Foundation] * [http://semanticweb.org/wiki/Logical_graph Logical Graph @ SemanticWeb] * [http://ref.subwiki.org/wiki/Logical_graph Logical Graph @ Subject Wikis] * [http://beta.wikiversity.org/wiki/Logical_graph Logical Graph @ Wikiversity Beta] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;] * [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;] ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Logical_graph Logical Graph], [http://mywikibiz.com/ MyWikiBiz] * [http://mathweb.org/wiki/Logical_graph Logical Graph], [http://mathweb.org/wiki/ MathWeb Wiki] * [http://netknowledge.org/wiki/Logical_graph Logical Graph], [http://netknowledge.org/ NetKnowledge] * [http://p2pfoundation.net/Logical_Graph Logical Graph], [http://p2pfoundation.net/ P2P Foundation] * [http://semanticweb.org/wiki/Logical_graph Logical Graph], [http://semanticweb.org/ Semantic Web] {{col-break}} * [http://proofwiki.org/wiki/Definition:Logical_Graph Logical Graph], [http://proofwiki.org/ ProofWiki] * [http://planetmath.org/encyclopedia/LogicalGraph.html Logical Graph : 1], [http://planetmath.org/ PlanetMath] * [http://planetmath.org/encyclopedia/LogicalGraphFormalDevelopment.html Logical Graph : 2], [http://planetmath.org/ PlanetMath] * [http://knol.google.com/k/logical-graphs-1 Logical Graph : 1], [http://knol.google.com/ Google Knol] * [http://knol.google.com/k/logical-graphs-2 Logical Graph : 2], [http://knol.google.com/ Google Knol] {{col-break}} * [http://beta.wikiversity.org/wiki/Logical_graph Logical Graph], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://getwiki.net/-Logical_Graph Logical Graph], [http://getwiki.net/ GetWiki] * [http://wikinfo.org/index.php/Logical_graph Logical Graph], [http://wikinfo.org/ Wikinfo] * [http://textop.org/wiki/index.php?title=Logical_graph Logical Graph], [http://textop.org/wiki/ Textop Wiki] * [http://en.wikipedia.org/wiki/Logical_graph Logical Graph], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Artificial Intelligence]] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Cybernetics]] [[Category:Equational Reasoning]] [[Category:Formal Languages]] [[Category:Formal Systems]] [[Category:Graph Theory]] [[Category:History of Logic]] [[Category:History of Mathematics]] [[Category:Knowledge Representation]] [[Category:Logic]] [[Category:Logical Graphs]] [[Category:Mathematics]] [[Category:Philosophy]] [[Category:Semiotics]] [[Category:Visualization]] 3d167447e2a56e9450e3a9ca9ce33bff69cb6563 0 351 576 2010-09-30T16:32:07Z Jon Awbrey 3 See [http://mywikibiz.com/Charles_Sanders_Peirce Charles Sanders Peirce] wikitext text/x-wiki See [http://mywikibiz.com/Charles_Sanders_Peirce Charles Sanders Peirce] df8ac09cabf7596d1b009a2797b15303ec6f1624 106 22 577 389 2010-11-09T01:11:07Z Vipul 2 wikitext text/x-wiki {| class="sortable" border="1" ! Wiki !! Main topic !! Number of pages (lower bound) !! State of development |- | [[Groupprops:Main Page|Groupprops]] || Group theory || 4500 || pre-alpha; easily browsable |- | [[Topospaces:Main Page|Topospaces]] || Topology (point set, algebraic) || 500 || pre-pre-alpha |- | [[Commalg:Main Page|Commalg]] || Commutative algebra || 400 || pre-pre-alpha |- | [[Diffgeom:Main Page|Diffgeom]] || Differential geometry || 400 || pre-pre-alpha |- | [[Number:Main Page|Number]] || Number theory || 200 || barely begun |- | [[Market:Main Page|Market]] || Economic theory of markets, choices, and prices || 100 || barely begun |- | [[Mech:Main Page|Mech]] || Classical mechanics || 50 || barely begun |} c6a03d075edbfb2ba2c7385151b04ad557dd0fe1 587 577 2011-01-05T00:35:04Z Vipul 2 wikitext text/x-wiki {| class="sortable" border="1" ! Wiki !! Main topic !! Number of pages (lower bound) !! State of development |- | [[Groupprops:Main Page|Groupprops]] || Group theory || 4500 || pre-alpha; easily browsable |- | [[Topospaces:Main Page|Topospaces]] || Topology (point set, algebraic) || 600 || pre-pre-alpha |- | [[Commalg:Main Page|Commalg]] || Commutative algebra || 400 || pre-pre-alpha |- | [[Diffgeom:Main Page|Diffgeom]] || Differential geometry || 400 || pre-pre-alpha |- | [[Number:Main Page|Number]] || Number theory || 200 || barely begun |- | [[Market:Main Page|Market]] || Economic theory of markets, choices, and prices || 100 || barely begun |- | [[Mech:Main Page|Mech]] || Classical mechanics || 50 || barely begun |} 2672b5ee04be952481c405a71432a5ab52194dd5 588 587 2011-01-05T00:45:24Z Vipul 2 wikitext text/x-wiki {| class="sortable" border="1" ! Wiki !! Main topic !! Number of pages (lower bound) !! Creation month !! State of development |- | [[Groupprops:Main Page|Groupprops]] || Group theory || 4500 || December 2006 || pre-alpha; easily browsable |- | [[Topospaces:Main Page|Topospaces]] || Topology (point set, algebraic) || 600 || May 2007 || pre-pre-alpha |- | [[Commalg:Main Page|Commalg]] || Commutative algebra || 400 || January 2007 || pre-pre-alpha |- | [[Diffgeom:Main Page|Diffgeom]] || Differential geometry || 400 || February 2007 || pre-pre-alpha |- | [[Number:Main Page|Number]] || Number theory || 200 || March 2009 || barely begun |- | [[Market:Main Page|Market]] || Economic theory of markets, choices, and prices || 100 || January 2009 || barely begun |- | [[Mech:Main Page|Mech]] || Classical mechanics || 50 || January 2009 || barely begun |} a39a5b82edf5aa21eee9ee1bd6bac9c6912712dd 589 588 2011-01-05T00:47:05Z Vipul 2 wikitext text/x-wiki {| class="sortable" border="1" ! Wiki !! Main topic !! Number of pages (lower bound) !! Creation month !! State of development |- | [[Groupprops:Main Page|Groupprops]] || Group theory || 4500 || December 2006 || pre-alpha; easily browsable |- | [[Topospaces:Main Page|Topospaces]] || Topology (point set, algebraic) || 600 || May 2007 || pre-pre-alpha |- | [[Commalg:Main Page|Commalg]] || Commutative algebra || 400 || January 2007 || pre-pre-alpha |- | [[Diffgeom:Main Page|Diffgeom]] || Differential geometry || 400 || February 2007 || pre-pre-alpha |- | [[Number:Main Page|Number]] || Number theory || 200 || March 2009 || barely begun |- | [[Market:Main Page|Market]] || Economic theory of markets, choices, and prices || 100 || January 2009 || barely begun |- | [[Mech:Main Page|Mech]] || Classical mechanics || 50 || January 2009 || barely begun |} '''NOTE''': The creation month need not coincide with the month of MediaWiki installation. In cases where the wiki was ported from elsewhere, the creation month may be earlier; in other cases, it may be later if the first edits happened long after installation. 048cfc4478081391250a93f556138295e1cd07e2 106 352 578 2010-11-09T01:18:44Z Vipul 2 Created page with '{| class="sortable" border="1" ! Entity !! Value |- | Link || http://groupprops.subwiki.org |- | Name || [[Groupprops:Main Page|Groupprops]], The Group Properties Wiki |- | Main ...' wikitext text/x-wiki {| class="sortable" border="1" ! Entity !! Value |- | Link || http://groupprops.subwiki.org |- | Name || [[Groupprops:Main Page|Groupprops]], The Group Properties Wiki |- | Main topic || Group theory |- | Number of pages || 4500+ |} 9480d8329daff8fbc0b2dac29d44626209531fb2 579 578 2010-11-09T01:20:28Z Vipul 2 wikitext text/x-wiki {| class="sortable" border="1" ! Entity !! Value |- | Link || http://groupprops.subwiki.org |- | Name || [[Groupprops:Main Page|Groupprops]], The Group Properties Wiki |- | Main topic || Group theory |- | Number of pages || 4500+ |- | Original creation || December 2006 |- | Migration to subwiki.org || May 2008 |} bfa9ac9b4463e15bfa91977bfdb5884539876e27 106 353 580 2010-11-09T01:24:12Z Vipul 2 Created page with '{| class="sortable" border="1" ! Entity !! Value |- | Link || http://topospaces.subwiki.org |- | Name || [[Topospaces:Main Page|Topospaces]], The Topology Wiki |- | Number of pag...' wikitext text/x-wiki {| class="sortable" border="1" ! Entity !! Value |- | Link || http://topospaces.subwiki.org |- | Name || [[Topospaces:Main Page|Topospaces]], The Topology Wiki |- | Number of pages || 500+ |- | Start date || May 2007 |- | Migration to subwiki.org || May 2008 |} 58c6d2d6cc9db77302e6658da7c2076f4551009f 581 580 2010-11-09T01:25:05Z Vipul 2 wikitext text/x-wiki {| class="sortable" border="1" ! Entity !! Value |- | Link || http://topospaces.subwiki.org |- | Name || [[Topospaces:Main Page|Topospaces]], The Topology Wiki |- | Main topic || Topology |- | Number of pages || 500+ |- | Start date || May 2007 |- | Migration to subwiki.org || May 2008 |} 8454e8ad49b5625e4514a76f6285547b8856743b 106 354 582 2010-11-09T01:56:04Z Vipul 2 Created page with 'The pinpoint principle is a basic design principle of subject wikis. The design principle is that a user should be able to, with minimal effort in learning the lay of the land of...' wikitext text/x-wiki The pinpoint principle is a basic design principle of subject wikis. The design principle is that a user should be able to, with minimal effort in learning the lay of the land of the subject wiki, get an answer to a specific question (of a variable degree of specificity) as quickly as possible. The gold standard is that the person should simply be able to type something in the go/search bar and land on the page that addresses the user's question, and has the right context and setup for further exploration of the topic. See also [[Subwiki:Road network versus specific routes]]. 2b74cf5de42affc58afefa0a022df1a9b92a302f 584 582 2010-11-09T02:03:51Z Vipul 2 wikitext text/x-wiki The pinpoint principle is a basic design principle of subject wikis. The design principle is that a user should be able to, with minimal effort in learning the lay of the land of the subject wiki, get an answer to a specific question (of a variable degree of specificity) as quickly as possible. The gold standard is that the person should simply be able to type something in the go/search bar and land on the page that addresses the user's question, and has the right context and setup for further exploration of the topic. See also [[Subwiki:Road network versus specific routes]]. ==Why the pinpoint principle?== The core reason is that people want quick answers to questions. ''The longer it takes for a specific question to be answered, the fewer questions a person is likely to ask.'' The fact that speed can have a decisive impact on the extent to which people are able to explore and learn is not new -- it was the subject of the [http://velocityconference.blip.tv Velocity Conference] where search engine people and other web folk gathered together to discuss the significance of Internet speed. 62fb30f0611c3e6cfd181b2b7483ac66a412cdd9 585 584 2010-11-09T02:06:35Z Vipul 2 wikitext text/x-wiki The pinpoint principle is a basic design principle of subject wikis. The design principle is that a user should be able to, with minimal effort in learning the lay of the land of the subject wiki, get an answer to a specific question (of a variable degree of specificity) as quickly as possible. The gold standard is that the person should simply be able to type something in the go/search bar and land on the page that addresses the user's question, and has the right context and setup for further exploration of the topic. See also [[Subwiki:Road network versus specific routes]]. ==Why the pinpoint principle?== ===Speed=== The core reason is that people want quick answers to questions. ''The longer it takes for a specific question to be answered, the fewer questions a person is likely to ask.'' The fact that speed can have a decisive impact on the extent to which people are able to explore and learn is not new -- it was the subject of the [http://velocityconference.blip.tv Velocity Conference] where search engine people and other web folk gathered together to discuss the significance of Internet speed. ===Accessibility=== Another reason is increased accessibility. A large number of textbooks and lecture notes on a wide variety of subjects are available to people learning a subject, yet these are not usually available in a ready-to-use form. Even though long textbooks have table of contents, index, and (in the case of online textbooks) search features, the organization is not fundamentally designed to contain information designed to answer a specific query from start to finish. c1f12d97a181dc2bb89cfa83cebe2986f583aaf4 106 355 583 2010-11-09T01:59:10Z Vipul 2 Created page with 'The [[subwiki:Subject wiki|subject wikis]] should, to the extent possible, be designed as road networks where the network is defined in terms of the pages and the links between t...' wikitext text/x-wiki The [[subwiki:Subject wiki|subject wikis]] should, to the extent possible, be designed as road networks where the network is defined in terms of the pages and the links between them. Road networks contrast with specific routes in the sense that a person traveling on a road network can start anywhere, choose to take any of a number of turns at each crossing, and stop anywhere. On the other hand, a specific route has a starting point and a destination and only fixed stopping points where a person can embark or disembark. The number of possible routes in a road network grows roughly exponentially in the number of nodes, thus allowing for richer and more varied exploration and opening up possibilities for use by more people. See also [[subwiki:pinpoint principle]]. d956e8589cbb7c15bccf7c60ac82f67f3848994d 106 179 586 388 2011-01-05T00:27:53Z Vipul 2 /* General-purpose versus specific-purpose */ wikitext text/x-wiki This page discusses some important differences between subject wikis and Wikipedia. ==About Wikipedia and similarities between subject wikis and Wikipedia== ===Quick introduction=== For many people using the Internet, "Wikipedia" is synonymous with "wiki" and also with the general appearance of a Wikipedia page. Thus, a lot of users who come to subject wiki pages simply think they have come to Wikipedia or some offshoot of Wikipedia. Before proceeding, it is important to clarify some distinctions. "Wikipedia" is a specific wiki-based [http://www.wikipedia.org multilingual encyclopedia] (since you're reading this, you have probably used the [[wp:Main Page|English-language version]] and possibly many other language versions). Each language Wikipedia can be viewed as a separate encyclopedia aiming to cover the entire range of encyclopedia-worthy knowledge. All the Wikipedias use a certain software called MediaWiki, which can be downloaded [[mediawikiwiki:Main Page|here]]. This software can also be used by other sister wikis to Wikipedia, such as [http://wikisource.org Wikisource], wikis independent of Wikipedia, such as those hosted at [http://www.wikia.com Wikia] or [http://www.wiki-site.com Wiki-site] or [http://editthis.info Editthis.info] or independent wikis such as [[wikitravel:Main Page|Wikitravel]], [http://wikileaks.org Wikileaks] or [http://www.sourcewatch.org SourceWatch]. Just because different wikis use the same underlying software and thus often have similar page layouts ''does not mean'' that they host similar content or are organized similarly. The differences between different wikis could be as small as the difference between different newspapers or as large as the difference between a newspaper and an academic journal. (To confuse matters further, MediaWiki is not the only software used to power wikis. You can see a fairly long list at the [[wikipedia:wiki software|wiki software page]] or the [[wikipedia:comparison of wiki software|comparison of wiki software page]] on Wikipedia. One example is [http://www.pmwiki.org/wiki/PmWiki/PmWiki PmWiki].) ===Distinctive features of Wikipedia=== Wikipedia is a wiki-based ''encyclopedia''. Both the choice of content and organization of content reflect the goal of being an encyclopedia. Specifically, pages on topics in Wikipedia have to meet criteria of [[wikipedia:wikipedia:notability|notability]] and facts in these pages must meet criteria of [[wikipedia:wikipedia:verifiability|verifiability]]. There are also guidelines against [[Wikipedia:Wikipedia:OR|putting original research on Wikipedia]], reflecting the fact that encyclopedias are secondary and tertiary sources of knowledge rather than primary sources. ==Two related reasons for the use of MediaWiki or similar software to maintain linked content== There are many reasons for choosing MediaWiki to organize, store, and maintain linked content on the web, but we focus here on two of them: * The ''organizational advantage'', which is the advantage of being able to maintain and organize a large amount of related content and the way the pieces of the content are interrelated. * The ''collaboration advantage'', which is the idea that the content can be edited collaboratively by large numbers of people. These are both related. The organizational advantage includes significant modularization, which allows different people to edit different pieces, making collaboration easier. Similarly, more collaboration helps with more organization. However, it is important to note that these two kinds of advantages, though related, are distinct. Further, some projects use MediaWiki or other wiki software primarily for the collaborative advantage -- the fact that large numbers of people can edit the same page, create new pages, etc. Here, wikis compete with mailing lists, discussion forums, and blog discussion threads. They have their advantages and disadvantages in these respects. Some of the earliest wikis, including Ward Cunningham's [http://c2.com/cgi/wiki WikiWikiWeb] and [http://www.meatballwiki.org/wiki/ MeatBallWiki] were focused on carrying out such discussions as a process to create new ideas. Other projects use MediaWiki primarily for the organizational advantage. In some of these cases, the greater ease of collaboration is crucial but still instrumental and not the final goal. In others, large-scale collaboration may be desirable but not crucial, and in yet others, large-scale collaboration may not be desirable. For instance, in the case of Wikipedia, the reason for using MediaWiki was to create a better-organized encyclopedia by combining the ease of organizing with the possibility of large-scale collaboration. Something similar is true for [[Wikitravel:Main Page|Wikitravel]]. The extent to which large-scale collaboration is crucial to the success of something like Wikipedia, and whether the collaborative process as it has currently developed on Wikipedia is intrinsic to its mission, are moot points. ==The key differences between subject wikis and Wikipedia== ===First impressions=== Wikipedia pages are typically designed well for ''accidental landing'' -- landing by people who aren't familiar with the topic and landed there by accident following links. In particular, Wikipedia pages start by setting a broad context, and ''easing the reader into the subject''. Subject wiki pages are designed for people who have some familiarity with the topic area, so less effort is spent at the beginning of the page clarifying the general background. For instance, Wikipedia's page on [[wikipedia:normal subgroup|normal subgroup]] begins as follows: ''In [[wikipedia:abstract algebra|abstract algebra]], a '''normal subgroup''' is a [[wikipedia:subgroup|subgroup]] ...'' The initial ''In [[wikipedia:abstract algebra|abstract algebra]]'' phrase is needed to set a ''context''. A similar phrase is not employed in the group properties wiki definition of [[groupprops:normal subgroup|normal subgroup]]. ===Specific organizational paradigms=== Each subject wiki uses a number of organizational paradigms specifically suited for the needs of the subject matter, though general ideas of the [[subwiki:type-based organization|type-based paradigm]], [[subwiki:property-theoretic organization|property-theoretic paradigm]], and [[subwiki:relational organization|relational paradigm]] are shared by a number of subject wikis. Even these general paradigms, however, require a cohesive and reasonably narrow subject matter domain. The paradigms on subject wikis guide the use of semantic information and category information, and allow for specific automatically generated information (for instance, automatically listing examples related to a fact, or properties between two given properties). These paradigms are also responsible for very similar article structures for articles of the same ''type'', but the article structures differ quite a bit for articles of different types. Once again, it is the focus on a narrow subject domain that permits the use of reasonably standard templates for articles of a given type. No similar organizational paradigms are used on Wikipedia. ==Differences in goals and policies== ===General-purpose versus specific-purpose=== Wikipedia calls itself ''the free encyclopaedia that anybody can edit''([[wp:Main Page|Wikipedia Main Page]]). It plans to be an ''encyclopaedia of everything'' with a ''let's all get together and do it'' attitude. Of course, much of the structure and organization of the wiki is determined by a small core group, but much of the activity is also carried out by the large mass of ordinary users. While the aim of Wikipedia is to be a general-purpose encyclopaedia, the subject wikis aim to be neither general-purpose nor encyclopedic. Rather, they aim to cover specific areas (of mathematics or other subjects) in a particular fashion. ===What deserves an article=== As mentioned above, each subject wiki has a clear notion of what deserves an article, which tends to mean that there are in general lots of articles. The policy on subject wikis (within the framework of material relevant to the subject) is more in line with, though hardly the same as, [[wp:WP:Separatism|the separatism policy]] that some Wikipedians believe in, as opposed to the [[wp:WP:Mergism|mergism policy]] that seems to be the more accepted one in Wikipedia, as it stands today. ===Neutral point of view=== Wikipedia claims to adhere to a [[wp:WP:NPOV|neutral point of view policy]], which basically means that all views are represented fairly, and are ''attributed to their adherents''. Individual subject wikis follow no such policy. In fact, they usually represent definite and distinctive points of view, in terms of the choices made on how to structure individual articles and how to shape organizational paradigms for the subject wiki. ===Original research policies=== Wikipedia has an [[wp:WP:OR|explicit ban on original research]], with a clear emphasis that whatever is put should be from verifiable sources. As a general rule, subject wikis does not ''require'' all articles to cite sources. Rather, we first try to get the definition and the facts there, and gradually fill in the best references for those facts. 60961f1c48d9180252c2e9b348d89a4db302eb89 106 356 590 2011-01-05T00:58:38Z Vipul 2 Created page with 'This page describes important historical milestones in the historical development of subject wikis. Due to imperfect records, precise dates are not available, but information is ...' wikitext text/x-wiki This page describes important historical milestones in the historical development of subject wikis. Due to imperfect records, precise dates are not available, but information is available up to the month: {| class="sortable" border="1" ! Month !! Event |- | December 2006 || The [[Groupprops:Main Page|Group Properties Wiki]] was created at this URL: http://editthis.info/groupprops |- | January/February 2007 || The Group Properties Wiki was moved to this URL: http://groupprops.wiki-site.com and other wikis such as Commalg and Diffgeom were created at wiki-site.com |- | May 2007 || The Topospaces (Topology) Wiki was created. |- | March 2008 || The subwiki.org domain name was purchased. |- | May 2008 || Groupprops and the other subject wikis were moved over to the subwiki.org domain, where they currently reside. |- | January 2009 || MediaWiki installations were done for many other subject wikis; the [http://blog.subwiki.org Subject Wikis Blog] was created. |- | September 2010 || For the first time, Groupprops logged 1000+ human pageviews in a single day. |- | December 2010 || Groupprops crossed 1 million pageviews as logged by MediaWiki (this also includes non-human pageviews) -- this does not include pageviews prior to May 2008. |} 6caf18fa2426da4c6bd91a8ec8f3bc300fac14a1 591 590 2011-01-05T01:01:59Z Vipul 2 wikitext text/x-wiki This page describes important historical milestones in the historical development of subject wikis. Due to imperfect records, precise dates are not available, but information is available up to the month: {| class="sortable" border="1" ! Month !! Event |- | December 2006 || The [[Groupprops:Main Page|Group Properties Wiki]] was created at this URL: http://editthis.info/groupprops |- | January/February 2007 || The Group Properties Wiki was moved to this URL: http://groupprops.wiki-site.com and other wikis such as Commalg and Diffgeom were created at wiki-site.com |- | May 2007 || The Topospaces (Topology) Wiki was created. |- | July 2007 || Groupprops logo design. |- | March 2008 || The subwiki.org domain name was purchased. |- | May 2008 || Groupprops and the other subject wikis were moved over to the subwiki.org domain, where they currently reside, with technical assistance from Harish of [http://4am.co.in 4AM]. |- | January 2009 || MediaWiki installations were done for many other subject wikis; the [http://blog.subwiki.org Subject Wikis Blog] was created. |- | May 2010 || Upgrade to MediaWiki 1.16, with introduction of Vector skin. |- | September 2010 || For the first time, Groupprops logged 1000+ human pageviews in a single day. This is now a regular feature for weekday traffic during weeks of the academic year. |- | December 2010 || Groupprops crossed 1 million pageviews as logged by MediaWiki (this also includes non-human pageviews) -- this does not include pageviews prior to May 2008. |} 2a1d11ee81a45e660ef11100e2c464c6e6fdeac1 592 591 2011-01-05T01:05:37Z Vipul 2 wikitext text/x-wiki This page describes important historical milestones in the historical development of subject wikis. Due to imperfect records, precise dates are not available, but information is available up to the month: {| class="sortable" border="1" ! Month !! Event !! Record |- | December 2006 || The [[Groupprops:Main Page|Group Properties Wiki]] was created at this URL: http://editthis.info/groupprops || |- | January/February 2007 || The Group Properties Wiki was moved to this URL: http://groupprops.wiki-site.com and other wikis such as Commalg and Diffgeom were created at wiki-site.com || |- | May 2007 || The Topospaces (Topology) Wiki was created. || |- | July 2007 || Groupprops logo design by Arpith. || |- | March 2008 || The subwiki.org domain name was purchased. || |- | May 2008 || Groupprops and the other subject wikis were moved over to the subwiki.org domain, where they currently reside, with technical assistance from Harish of [http://4am.co.in 4AM]. || |- | January 2009 || MediaWiki installations were done for many other subject wikis; the [http://blog.subwiki.org Subject Wikis Blog] was created. || [http://blog.subwiki.org/?p=3 Blog post on the beginning of the subject wikis blog] |- | May 2010 || Upgrade to MediaWiki 1.16, with introduction of Vector skin. || [http://blog.subwiki.org/?p=60 Blog post] |- | September 2010 || For the first time, Groupprops logged 1000+ human pageviews in a single day. This is now a regular feature for weekday traffic during weeks of the academic year. || [http://blog.subwiki.org/?p=65 Blog post] (two months after the milestone) |- | December 2010 || Groupprops crossed 1 million pageviews as logged by MediaWiki (this also includes non-human pageviews) -- this does not include pageviews prior to May 2008. |} 9e1a329567d89ce68e5797510dd8beb4e7812501 593 592 2011-01-05T01:05:59Z Vipul 2 wikitext text/x-wiki This page describes important historical milestones in the historical development of subject wikis. Due to imperfect records, precise dates are not available, but information is available up to the month: {| class="sortable" border="1" ! Month !! Event !! Record |- | December 2006 || The [[Groupprops:Main Page|Group Properties Wiki]] was created at this URL: http://editthis.info/groupprops || |- | January/February 2007 || The Group Properties Wiki was moved to this URL: http://groupprops.wiki-site.com and other wikis such as Commalg and Diffgeom were created at wiki-site.com || |- | May 2007 || The Topospaces (Topology) Wiki was created. || |- | July 2007 || Groupprops logo design by Arpith. || |- | March 2008 || The subwiki.org domain name was purchased. || |- | May 2008 || Groupprops and the other subject wikis were moved over to the subwiki.org domain, where they currently reside, with technical assistance from Harish of [http://4am.co.in 4AM]. || |- | January 2009 || MediaWiki installations were done for many other subject wikis; the [http://blog.subwiki.org Subject Wikis Blog] was created. || [http://blog.subwiki.org/?p=3 Blog post on the beginning of the subject wikis blog] |- | May 2010 || Upgrade to MediaWiki 1.16, with introduction of Vector skin. || [http://blog.subwiki.org/?p=60 Blog post] |- | September 2010 || For the first time, Groupprops logged 1000+ human pageviews in a single day. This is now a regular feature for weekday traffic during weeks of the academic year. || [http://blog.subwiki.org/?p=65 Blog post] (two months after the milestone) |- | December 2010 || Groupprops crossed 1 million pageviews as logged by MediaWiki (this also includes non-human pageviews) -- this does not include pageviews prior to May 2008. || |} dcd6af03cf9622d4033dc7e375a70be4d28c81d1 594 593 2011-01-05T01:12:50Z Vipul 2 wikitext text/x-wiki ==Events== This page describes important historical milestones in the historical development of subject wikis. Due to imperfect records, precise dates are not available, but information is available up to the month. {| class="sortable" border="1" ! Month !! Event !! Record |- | December 2006 || The [[Groupprops:Main Page|Group Properties Wiki]] was created at this URL: http://editthis.info/groupprops || |- | January/February 2007 || The Group Properties Wiki was moved to this URL: http://groupprops.wiki-site.com and other wikis such as Commalg and Diffgeom were created at wiki-site.com || |- | May 2007 || The Topospaces (Topology) Wiki was created. || |- | July 2007 || Groupprops logo design by Arpith. || |- | March 2008 || The subwiki.org domain name was purchased. || |- | May 2008 || Groupprops and the other subject wikis were moved over to the subwiki.org domain, where they currently reside, with technical assistance from Harish of [http://4am.co.in 4AM]. || |- | January 2009 || MediaWiki installations were done for many other subject wikis; the [http://blog.subwiki.org Subject Wikis Blog] was created. || [http://blog.subwiki.org/?p=3 Blog post on the beginning of the subject wikis blog] |- | May 2010 || Upgrade to MediaWiki 1.16, with introduction of Vector skin. || [http://blog.subwiki.org/?p=60 Blog post] |- | September 2010 || For the first time, Groupprops logged 1000+ human pageviews in a single day. This is now a regular feature for weekday traffic during weeks of the academic year. || [http://blog.subwiki.org/?p=65 Blog post] (two months after the milestone) |- | December 2010 || Groupprops crossed 1 million pageviews as logged by MediaWiki (this also includes non-human pageviews) -- this does not include pageviews prior to May 2008. || |} ==Usage statistics== These pageview counts are as measured by Google Analytics, and are intended to measure only human pageviews. Counts are rounded off to the nearest multiple of 1000. (Details for other wikis to be filled in). {| class="sortable" border="1" ! Subject wiki !! Pageviews in 2008 (starting May) !! Pageviews in 2009 !! Pageviews in 2010 |- | [[Groupprops:Main Page|Groupprops]] || 17000 || 99000 || 226000 |} 865f731aaf4b71856c870a7186877bb0a3072313 600 594 2011-02-03T21:05:03Z Vipul 2 /* Events */ wikitext text/x-wiki ==Events== This page describes important historical milestones in the historical development of subject wikis. Due to imperfect records, precise dates are not available, but information is available up to the month. {| class="sortable" border="1" ! Month !! Event !! Record |- | December 2006 || The [[Groupprops:Main Page|Group Properties Wiki]] was created at this URL: http://editthis.info/groupprops || |- | January/February 2007 || The Group Properties Wiki was moved to this URL: http://groupprops.wiki-site.com and other wikis such as Commalg and Diffgeom were created at wiki-site.com || |- | May 2007 || The Topospaces (Topology) Wiki was created. || |- | July 2007 || Groupprops logo design by Arpith. || |- | March 2008 || The subwiki.org domain name was purchased. || |- | May 2008 || Groupprops and the other subject wikis were moved over to the subwiki.org domain, where they currently reside, with technical assistance from Harish of [http://4am.co.in 4AM]. || |- | January 2009 || MediaWiki installations were done for many other subject wikis; the [http://blog.subwiki.org Subject Wikis Blog] was created. || [http://blog.subwiki.org/?p=3 Blog post on the beginning of the subject wikis blog] |- | May 2010 || Upgrade to MediaWiki 1.16, with introduction of Vector skin. || [http://blog.subwiki.org/?p=60 Blog post] |- | September 2010 || For the first time, Groupprops logged 1000+ human pageviews in a single day. This is now a regular feature for weekday traffic during weeks of the academic year. || [http://blog.subwiki.org/?p=65 Blog post] (two months after the milestone) |- | December 2010 || Groupprops crossed 1 million pageviews as logged by MediaWiki (this also includes non-human pageviews) -- this does not include pageviews prior to May 2008. || |- | February 2011 || Wikis upgraded to MediaWiki 1.16.2 || |} ==Usage statistics== These pageview counts are as measured by Google Analytics, and are intended to measure only human pageviews. Counts are rounded off to the nearest multiple of 1000. (Details for other wikis to be filled in). {| class="sortable" border="1" ! Subject wiki !! Pageviews in 2008 (starting May) !! Pageviews in 2009 !! Pageviews in 2010 |- | [[Groupprops:Main Page|Groupprops]] || 17000 || 99000 || 226000 |} 6cd424e42708b93c3a7ce6c9de113a86537d5c1c 601 600 2011-02-03T22:38:37Z Vipul 2 /* Usage statistics */ wikitext text/x-wiki ==Events== This page describes important historical milestones in the historical development of subject wikis. Due to imperfect records, precise dates are not available, but information is available up to the month. {| class="sortable" border="1" ! Month !! Event !! Record |- | December 2006 || The [[Groupprops:Main Page|Group Properties Wiki]] was created at this URL: http://editthis.info/groupprops || |- | January/February 2007 || The Group Properties Wiki was moved to this URL: http://groupprops.wiki-site.com and other wikis such as Commalg and Diffgeom were created at wiki-site.com || |- | May 2007 || The Topospaces (Topology) Wiki was created. || |- | July 2007 || Groupprops logo design by Arpith. || |- | March 2008 || The subwiki.org domain name was purchased. || |- | May 2008 || Groupprops and the other subject wikis were moved over to the subwiki.org domain, where they currently reside, with technical assistance from Harish of [http://4am.co.in 4AM]. || |- | January 2009 || MediaWiki installations were done for many other subject wikis; the [http://blog.subwiki.org Subject Wikis Blog] was created. || [http://blog.subwiki.org/?p=3 Blog post on the beginning of the subject wikis blog] |- | May 2010 || Upgrade to MediaWiki 1.16, with introduction of Vector skin. || [http://blog.subwiki.org/?p=60 Blog post] |- | September 2010 || For the first time, Groupprops logged 1000+ human pageviews in a single day. This is now a regular feature for weekday traffic during weeks of the academic year. || [http://blog.subwiki.org/?p=65 Blog post] (two months after the milestone) |- | December 2010 || Groupprops crossed 1 million pageviews as logged by MediaWiki (this also includes non-human pageviews) -- this does not include pageviews prior to May 2008. || |- | February 2011 || Wikis upgraded to MediaWiki 1.16.2 || |} ==Usage statistics== These pageview counts are as measured by Google Analytics, and are intended to measure only human pageviews. Counts are rounded off to the nearest multiple of 1000. (Details for other wikis to be filled in). {| class="sortable" border="1" ! Subject wiki !! Pageviews in 2008 (starting May) !! Pageviews in 2009 !! Pageviews in 2010 |- | [[Groupprops:Main Page|Groupprops]] || 17000 || 99000 || 226000 |- | [[Topospaces:Main Page|Topospaces]] || 420 || 9000 || 20000 |- | [[Commalg:Main Page|Commalg]] || 180 || 1600 || 3000 |} 8e006a70a61431e41e9951c0402c90f6a4750af1 106 357 595 2011-01-05T01:27:38Z Vipul 2 Created page with 'This page discusses some important similarities and differences between the subject wikis and other similar efforts. ==Comparison with other wiki-style knowledge base efforts== ...' wikitext text/x-wiki This page discusses some important similarities and differences between the subject wikis and other similar efforts. ==Comparison with other wiki-style knowledge base efforts== {| class="sortable" border="1" ! Title !! Subtitle !! Primary URL !! Funding model !! More information !! Comparison with subject wikis |- | Wikipedia || the free encyclopedia that anyone can edit || http://wikipedia.org (multilingual portal), http://en.wikipedia.org (English language version) || Server hosting, software improvement, and legal/organizational management funded through donations via the [http://wikimediafoundation.org Wikimedia Foundation] || [[Resource:Wikipedia]] || [[Subwiki:Subwiki versus Wikipedia]] |- | Tricki || a repository of mathematical know-how || http://www.tricki.org || Privately funded (?) || [[Resource:Tricki]] || [[Subwiki:Subwiki versus Tricki]] |- | ProofWiki || || http://www.proofwiki.org || Privately funded, also seek donations through [http://www.proofwiki.org/wiki/ProofWiki:Site_support site support page] || [[Resource:ProofWiki]] || [[ Subwiki:Subwiki versus ProofWiki]] |- | Planetmath || Math for the people, by the people || http://planetmath.org || Privately funded; seek donation/sponsorship through [http://aux.planetmath.org/doc/sponsor.html this webpage] || [[Resource:Planetmath]] || [[Subwiki:Subwiki versus Planetmath]] |- | Mathworld || the web's most extensive mathematics resource || http://mathworld.wolfram.com || Funded by Wolfram Research Inc., a private for-profit company || [[Resource:Mathworld]] || [[Subwiki:Subwiki versus Mathworld]] |- | Manifold Atlas Project || || http://www.map.him/uni-bonn.de || Sponsored by [http://www.hcm.uni-bonn.de Hausdorff Center for Mathematics] || || |- | Tangent Bundle: Mathematics Project || || http://www.mathematics.thetangentbundle.net || Unclear || || |} 17c6393f21216b5208e086049a4dc5785c11a2e7 596 595 2011-01-05T01:29:27Z Vipul 2 wikitext text/x-wiki This page discusses some important similarities and differences between the subject wikis and other similar efforts. ==Comparison with other wiki-style knowledge base efforts== {| class="sortable" border="1" ! Title !! Subtitle/slogan !! Primary URL !! Funding model !! More information !! Comparison with subject wikis |- | Wikipedia || the free encyclopedia that anyone can edit || http://wikipedia.org (multilingual portal), http://en.wikipedia.org (English language version) || Server hosting, software improvement, and legal/organizational management funded through donations via the [http://wikimediafoundation.org Wikimedia Foundation] || [[Resource:Wikipedia]] || [[Subwiki:Subwiki versus Wikipedia]] |- | Tricki || a repository of mathematical know-how || http://www.tricki.org || Privately funded (?) || [[Resource:Tricki]] || [[Subwiki:Subwiki versus Tricki]] |- | ProofWiki || || http://www.proofwiki.org || Privately funded, also seek donations through [http://www.proofwiki.org/wiki/ProofWiki:Site_support site support page] || [[Resource:ProofWiki]] || [[ Subwiki:Subwiki versus ProofWiki]] |- | Planetmath || Math for the people, by the people || http://planetmath.org || Privately funded; seek donation/sponsorship through [http://aux.planetmath.org/doc/sponsor.html this webpage] || [[Resource:Planetmath]] || [[Subwiki:Subwiki versus Planetmath]] |- | Mathworld || the web's most extensive mathematics resource || http://mathworld.wolfram.com || Funded by Wolfram Research Inc., a private for-profit company || [[Resource:Mathworld]] || [[Subwiki:Subwiki versus Mathworld]] |- | Manifold Atlas Project || || http://www.map.him/uni-bonn.de || Sponsored by [http://www.hcm.uni-bonn.de Hausdorff Center for Mathematics] || || |- | Tangent Bundle: Mathematics Project || || http://www.mathematics.thetangentbundle.net || Unclear || || |} 50cc348eca32ff753eb6cd9915a7dab35a3065f7 597 596 2011-01-05T01:41:46Z Vipul 2 wikitext text/x-wiki This page discusses some important similarities and differences between the subject wikis and other similar efforts. ==Comparison with other wiki-style knowledge base efforts== ===Tabular list of other wiki-style knowledge base efforts=== {| class="sortable" border="1" ! Title !! Subtitle/slogan !! Primary URL !! Funding model !! More information !! Comparison with subject wikis |- | Wikipedia || the free encyclopedia that anyone can edit || http://wikipedia.org (multilingual portal), http://en.wikipedia.org (English language version) || Server hosting, software improvement, and legal/organizational management funded through donations via the [http://wikimediafoundation.org Wikimedia Foundation] || [[Resource:Wikipedia]] || [[Subwiki:Subwiki versus Wikipedia]] |- | Tricki || a repository of mathematical know-how || http://www.tricki.org || Privately funded (?) || [[Resource:Tricki]] || [[Subwiki:Subwiki versus Tricki]] |- | ProofWiki || || http://www.proofwiki.org || Privately funded, also seek donations through [http://www.proofwiki.org/wiki/ProofWiki:Site_support site support page] || [[Resource:ProofWiki]] || [[ Subwiki:Subwiki versus ProofWiki]] |- | Planetmath || Math for the people, by the people || http://planetmath.org || Privately funded; seek donation/sponsorship through [http://aux.planetmath.org/doc/sponsor.html this webpage] || [[Resource:Planetmath]] || [[Subwiki:Subwiki versus Planetmath]] |- | Mathworld || the web's most extensive mathematics resource || http://mathworld.wolfram.com || Funded by Wolfram Research Inc., a private for-profit company || [[Resource:Mathworld]] || [[Subwiki:Subwiki versus Mathworld]] |- | Manifold Atlas Project || || http://www.map.him/uni-bonn.de || Sponsored by [http://www.hcm.uni-bonn.de Hausdorff Center for Mathematics] || || |- | Tangent Bundle: Mathematics Project || || http://www.mathematics.thetangentbundle.net || Unclear || || |} ===Key similarities=== The following are similarities between the subject wikis and many of these similar efforts. {| class="sortable" border="1" ! Trait for similarity !! How the traits are similar !! Exceptions in above list? |- | Accessibility of content || All content is freely available on the World Wide Web and accessible using a browser without the need for any subscription, login, or payment. || |- | Licensing of content || Content is licensed using an "open" license such as [http://creativecommons.org/licenses/by-sa/3.0/ CC-by-SA], CC-by-NC-SA or the GFDL. || Mathworld |- | Editability of content || Content can be edited by anybody, possibly subject to registration/login requirements || Mathworld (proposed articles or edits must be sent to a moderator who then may or may not put it up); Manifold Atlas Project allows editing but differentiates between approved and unapproved articles; Planetmath uses an ownership model where the person to create an article has authority over subsequent edits to that article. |- | One topic, one article || Articles are the fundamental unit of organization. Each "topic" gets its own article subject to various criteria || None, though there are differing interpretations of what constitutes a valid article topic. |} ===Key differences=== We focus on ways in which the subject wikis are different from all, or most, other alternatives. {{fillin}} 24c326d4e43a7c48dc43b3bf830b1cda56f84131 598 597 2011-01-05T01:42:31Z Vipul 2 /* Tabular list of other wiki-style knowledge base efforts */ wikitext text/x-wiki This page discusses some important similarities and differences between the subject wikis and other similar efforts. ==Comparison with other wiki-style knowledge base efforts== ===Tabular list of other wiki-style knowledge base efforts=== {| class="sortable" border="1" ! Title !! Subtitle/slogan !! Primary URL !! Funding model !! More information !! Comparison with subject wikis |- | Wikipedia || the free encyclopedia that anyone can edit || http://wikipedia.org (multilingual portal), http://en.wikipedia.org (English language version) || Server hosting, software improvement, and legal/organizational management funded through donations via the [http://wikimediafoundation.org Wikimedia Foundation] || [[Resource:Wikipedia]] || [[Subwiki:Subwiki versus Wikipedia]] |- | Tricki || a repository of mathematical know-how || http://www.tricki.org || Privately funded (?) || [[Resource:Tricki]] || [[Subwiki:Subwiki versus Tricki]] |- | ProofWiki || || http://www.proofwiki.org || Privately funded, also seek donations through [http://www.proofwiki.org/wiki/ProofWiki:Site_support site support page] || [[Resource:ProofWiki]] || [[ Subwiki:Subwiki versus ProofWiki]] |- | Planetmath || Math for the people, by the people || http://planetmath.org || Privately funded; seek donation/sponsorship through [http://aux.planetmath.org/doc/sponsor.html this webpage] || [[Resource:Planetmath]] || [[Subwiki:Subwiki versus Planetmath]] |- | Mathworld || the web's most extensive mathematics resource || http://mathworld.wolfram.com || Funded by [http://www.wolfram.com Wolfram Research Inc.], a private for-profit company || [[Resource:Mathworld]] || [[Subwiki:Subwiki versus Mathworld]] |- | Manifold Atlas Project || || http://www.map.him/uni-bonn.de || Sponsored by [http://www.hcm.uni-bonn.de Hausdorff Center for Mathematics] || || |- | Tangent Bundle: Mathematics Project || || http://www.mathematics.thetangentbundle.net || Unclear || || |} ===Key similarities=== The following are similarities between the subject wikis and many of these similar efforts. {| class="sortable" border="1" ! Trait for similarity !! How the traits are similar !! Exceptions in above list? |- | Accessibility of content || All content is freely available on the World Wide Web and accessible using a browser without the need for any subscription, login, or payment. || |- | Licensing of content || Content is licensed using an "open" license such as [http://creativecommons.org/licenses/by-sa/3.0/ CC-by-SA], CC-by-NC-SA or the GFDL. || Mathworld |- | Editability of content || Content can be edited by anybody, possibly subject to registration/login requirements || Mathworld (proposed articles or edits must be sent to a moderator who then may or may not put it up); Manifold Atlas Project allows editing but differentiates between approved and unapproved articles; Planetmath uses an ownership model where the person to create an article has authority over subsequent edits to that article. |- | One topic, one article || Articles are the fundamental unit of organization. Each "topic" gets its own article subject to various criteria || None, though there are differing interpretations of what constitutes a valid article topic. |} ===Key differences=== We focus on ways in which the subject wikis are different from all, or most, other alternatives. {{fillin}} 8281f2bb8e8778807f75fc5d8aff3547c1316572 599 598 2011-01-07T23:38:40Z Vipul 2 /* Comparison with other wiki-style knowledge base efforts */ wikitext text/x-wiki This page discusses some important similarities and differences between the subject wikis and other similar efforts. ==Comparison with other wiki-style knowledge base efforts== ===Tabular list of other wiki-style knowledge base efforts=== {| class="sortable" border="1" ! Title !! Subtitle/slogan !! Primary URL !! Funding model !! More information !! Comparison with subject wikis |- | Wikipedia || the free encyclopedia that anyone can edit || http://wikipedia.org (multilingual portal), http://en.wikipedia.org (English language version) || Server hosting, software improvement, and legal/organizational management funded through donations via the [http://wikimediafoundation.org Wikimedia Foundation] || [[Resource:Wikipedia]] || [[Subwiki:Subwiki versus Wikipedia]] |- | Tricki || a repository of mathematical know-how || http://www.tricki.org || Privately funded (?) || [[Resource:Tricki]] || [[Subwiki:Subwiki versus Tricki]] |- | nLab || || http://ncatlab.org/nlab || Privately funded || [[Resource:nLab]] || [[Subwiki:Subwiki versus nLab]] |- | Quantiki || || http://www.quantiki.org || ? || [[Resource:Quantiki]] || [[Subwiki:Subwiki versus Quantiki]] |- | ProofWiki || || http://www.proofwiki.org || Privately funded, also seek donations through [http://www.proofwiki.org/wiki/ProofWiki:Site_support site support page] || [[Resource:ProofWiki]] || [[ Subwiki:Subwiki versus ProofWiki]] |- | Planetmath || Math for the people, by the people || http://planetmath.org || Privately funded; seek donation/sponsorship through [http://aux.planetmath.org/doc/sponsor.html this webpage] || [[Resource:Planetmath]] || [[Subwiki:Subwiki versus Planetmath]] |- | Mathworld || the web's most extensive mathematics resource || http://mathworld.wolfram.com || Funded by [http://www.wolfram.com Wolfram Research Inc.], a private for-profit company || [[Resource:Mathworld]] || [[Subwiki:Subwiki versus Mathworld]] |- | Manifold Atlas Project || || http://www.map.him/uni-bonn.de || Sponsored by [http://www.hcm.uni-bonn.de Hausdorff Center for Mathematics] || || |- | Tangent Bundle: Mathematics Project || || http://www.mathematics.thetangentbundle.net || Unclear || || |} ===Key similarities=== The following are similarities between the subject wikis and many of these similar efforts. {| class="sortable" border="1" ! Trait for similarity !! How the traits are similar !! Exceptions in above list? |- | Accessibility of content || All content is freely available on the World Wide Web and accessible using a browser without the need for any subscription, login, or payment. || |- | Licensing of content || Content is licensed using an "open" license such as [http://creativecommons.org/licenses/by-sa/3.0/ CC-by-SA], CC-by-NC-SA or the GFDL. || Mathworld |- | Editability of content || Content can be edited by anybody, possibly subject to registration/login requirements || Mathworld (proposed articles or edits must be sent to a moderator who then may or may not put it up); Manifold Atlas Project allows editing but differentiates between approved and unapproved articles; Planetmath uses an ownership model where the person to create an article has authority over subsequent edits to that article. |- | One topic, one article || Articles are the fundamental unit of organization. Each "topic" gets its own article subject to various criteria || None, though there are differing interpretations of what constitutes a valid article topic. |} ===Key differences=== We focus on ways in which the subject wikis are different from all, or most, other alternatives. {{fillin}} f9cc7784a5f296e990676e646b68827498899ed2 106 356 602 601 2011-02-03T22:41:35Z Vipul 2 /* Usage statistics */ wikitext text/x-wiki ==Events== This page describes important historical milestones in the historical development of subject wikis. Due to imperfect records, precise dates are not available, but information is available up to the month. {| class="sortable" border="1" ! Month !! Event !! Record |- | December 2006 || The [[Groupprops:Main Page|Group Properties Wiki]] was created at this URL: http://editthis.info/groupprops || |- | January/February 2007 || The Group Properties Wiki was moved to this URL: http://groupprops.wiki-site.com and other wikis such as Commalg and Diffgeom were created at wiki-site.com || |- | May 2007 || The Topospaces (Topology) Wiki was created. || |- | July 2007 || Groupprops logo design by Arpith. || |- | March 2008 || The subwiki.org domain name was purchased. || |- | May 2008 || Groupprops and the other subject wikis were moved over to the subwiki.org domain, where they currently reside, with technical assistance from Harish of [http://4am.co.in 4AM]. || |- | January 2009 || MediaWiki installations were done for many other subject wikis; the [http://blog.subwiki.org Subject Wikis Blog] was created. || [http://blog.subwiki.org/?p=3 Blog post on the beginning of the subject wikis blog] |- | May 2010 || Upgrade to MediaWiki 1.16, with introduction of Vector skin. || [http://blog.subwiki.org/?p=60 Blog post] |- | September 2010 || For the first time, Groupprops logged 1000+ human pageviews in a single day. This is now a regular feature for weekday traffic during weeks of the academic year. || [http://blog.subwiki.org/?p=65 Blog post] (two months after the milestone) |- | December 2010 || Groupprops crossed 1 million pageviews as logged by MediaWiki (this also includes non-human pageviews) -- this does not include pageviews prior to May 2008. || |- | February 2011 || Wikis upgraded to MediaWiki 1.16.2 || |} ==Usage statistics== These pageview counts are as measured by Google Analytics, and are intended to measure only human pageviews. Counts are rounded off to the nearest multiple of 1000. (Details for other wikis to be filled in). {| class="sortable" border="1" ! Subject wiki !! Pageviews in 2008 (starting May) !! Pageviews in 2009 !! Pageviews in 2010 |- | [[Groupprops:Main Page|Groupprops]] || 17000 || 99000 || 226000 |- | [[Topospaces:Main Page|Topospaces]] || 420 || 9000 || 20000 |- | [[Commalg:Main Page|Commalg]] || 180 || 1600 || 3000 |- | [[Market:Main Page|Market]] || 0 || 900 || 5600 |} 8b48d6de80727b580073009f0547b55a97686f4b 603 602 2011-02-04T23:51:11Z Vipul 2 /* Events */ wikitext text/x-wiki ==Events== This page describes important historical milestones in the historical development of subject wikis. Due to imperfect records, precise dates are not available, but information is available up to the month. {| class="sortable" border="1" ! Month !! Event !! Record |- | December 2006 || The [[Groupprops:Main Page|Group Properties Wiki]] was created at this URL: http://editthis.info/groupprops || |- | January/February 2007 || The Group Properties Wiki was moved to this URL: http://groupprops.wiki-site.com and other wikis such as Commalg and Diffgeom were created at wiki-site.com || |- | May 2007 || The Topospaces (Topology) Wiki was created. || |- | July 2007 || Groupprops logo design by Arpith. || |- | March 2008 || The subwiki.org domain name was purchased. || |- | May 2008 || Groupprops and the other subject wikis were moved over to the subwiki.org domain, where they currently reside, with technical assistance from Harish of [http://4am.co.in 4AM]. || |- | January 2009 || MediaWiki installations were done for many other subject wikis; the [http://blog.subwiki.org Subject Wikis Blog] was created. || [http://blog.subwiki.org/?p=3 Blog post on the beginning of the subject wikis blog] |- | May 2010 || Upgrade to MediaWiki 1.16, with introduction of Vector skin. || [http://blog.subwiki.org/?p=60 Blog post] |- | September 2010 || For the first time, Groupprops logged 1000+ human pageviews in a single day. This is now a regular feature for weekday traffic during weeks of the academic year. || [http://blog.subwiki.org/?p=65 Blog post] (two months after the milestone) |- | December 2010 || Groupprops crossed 1 million pageviews as logged by MediaWiki (this also includes non-human pageviews) -- this does not include pageviews prior to May 2008. || |- | February 2011 || Wikis upgraded to MediaWiki 1.16.2 || [http://blog.subwiki.org/?p=82 Blog post] (a day after the upgrade) |} ==Usage statistics== These pageview counts are as measured by Google Analytics, and are intended to measure only human pageviews. Counts are rounded off to the nearest multiple of 1000. (Details for other wikis to be filled in). {| class="sortable" border="1" ! Subject wiki !! Pageviews in 2008 (starting May) !! Pageviews in 2009 !! Pageviews in 2010 |- | [[Groupprops:Main Page|Groupprops]] || 17000 || 99000 || 226000 |- | [[Topospaces:Main Page|Topospaces]] || 420 || 9000 || 20000 |- | [[Commalg:Main Page|Commalg]] || 180 || 1600 || 3000 |- | [[Market:Main Page|Market]] || 0 || 900 || 5600 |} e0c03818bbf2a1be8849237345c8d9f5b578d2a5 608 603 2011-04-17T22:39:53Z Vipul 2 wikitext text/x-wiki ==Events== This page describes important historical milestones in the historical development of subject wikis. Due to imperfect records, precise dates are not available, but information is available up to the month. {| class="sortable" border="1" ! Month !! Event !! Record |- | December 2006 || The [[Groupprops:Main Page|Group Properties Wiki]] was created at this URL: http://editthis.info/groupprops || |- | January/February 2007 || The Group Properties Wiki was moved to this URL: http://groupprops.wiki-site.com and other wikis such as Commalg and Diffgeom were created at wiki-site.com || |- | May 2007 || The Topospaces (Topology) Wiki was created. || |- | July 2007 || Groupprops logo design by Arpith. || |- | March 2008 || The subwiki.org domain name was purchased. || |- | May 2008 || Groupprops and the other subject wikis were moved over to the subwiki.org domain, where they currently reside, with technical assistance from Harish of [http://4am.co.in 4AM]. || |- | January 2009 || MediaWiki installations were done for many other subject wikis; the [http://blog.subwiki.org Subject Wikis Blog] was created. || [http://blog.subwiki.org/?p=3 Blog post on the beginning of the subject wikis blog] |- | May 2010 || Upgrade to MediaWiki 1.16, with introduction of Vector skin. || [http://blog.subwiki.org/?p=60 Blog post] |- | September 2010 || For the first time, Groupprops logged 1000+ human pageviews in a single day. This is now a regular feature for weekday traffic during weeks of the academic year. || [http://blog.subwiki.org/?p=65 Blog post] (two months after the milestone) |- | December 2010 || Groupprops crossed 1 million pageviews as logged by MediaWiki (this also includes non-human pageviews) -- this does not include pageviews prior to May 2008. || |- | February 2011 || Wikis upgraded to MediaWiki 1.16.2 || [http://blog.subwiki.org/?p=82 Blog post] (a day after the upgrade) |- | April 2011 || Wikis upgrade to MediaWiki 1.16.4 || |} ==Usage statistics== These pageview counts are as measured by Google Analytics, and are intended to measure only human pageviews. Counts are rounded off to the nearest multiple of 1000. (Details for other wikis to be filled in). {| class="sortable" border="1" ! Subject wiki !! Pageviews in 2008 (starting May) !! Pageviews in 2009 !! Pageviews in 2010 |- | [[Groupprops:Main Page|Groupprops]] || 17000 || 99000 || 226000 |- | [[Topospaces:Main Page|Topospaces]] || 420 || 9000 || 20000 |- | [[Commalg:Main Page|Commalg]] || 180 || 1600 || 3000 |- | [[Market:Main Page|Market]] || 0 || 900 || 5600 |} 210306044f50b42de38e67ef17d5e68914e0e641 609 608 2011-04-17T22:41:36Z Vipul 2 /* Events */ wikitext text/x-wiki ==Events== This page describes important historical milestones in the historical development of subject wikis. Due to imperfect records, precise dates are not available, but information is available up to the month. {| class="sortable" border="1" ! Month !! Event !! Record |- | December 2006 || The [[Groupprops:Main Page|Group Properties Wiki]] was created at this URL: http://editthis.info/groupprops || |- | January/February 2007 || The Group Properties Wiki was moved to this URL: http://groupprops.wiki-site.com and other wikis such as Commalg and Diffgeom were created at wiki-site.com || |- | May 2007 || The Topospaces (Topology) Wiki was created. || |- | July 2007 || Groupprops logo design by Arpith. || |- | March 2008 || The subwiki.org domain name was purchased. || |- | May 2008 || Groupprops and the other subject wikis were moved over to the subwiki.org domain, where they currently reside, with technical assistance from Harish of [http://4am.co.in 4AM]. || |- | January 2009 || MediaWiki installations were done for many other subject wikis; the [http://blog.subwiki.org Subject Wikis Blog] was created. || [http://blog.subwiki.org/?p=3 Blog post on the beginning of the subject wikis blog] |- | May 2010 || Upgrade to MediaWiki 1.16, with introduction of Vector skin. || [http://blog.subwiki.org/?p=60 Blog post] |- | September 2010 || For the first time, Groupprops logged 1000+ human pageviews in a single day. This is now a regular feature for weekday traffic during weeks of the academic year. || [http://blog.subwiki.org/?p=65 Blog post] (two months after the milestone) |- | December 2010 || Groupprops crossed 1 million pageviews as logged by MediaWiki (this also includes non-human pageviews) -- this does not include pageviews prior to May 2008. || |- | February 2011 || Upgrade to MediaWiki 1.16.2 || [http://blog.subwiki.org/?p=82 Blog post] (a day after the upgrade) |- | April 2011 || Upgrade to MediaWiki 1.16.4 || |} ==Usage statistics== These pageview counts are as measured by Google Analytics, and are intended to measure only human pageviews. Counts are rounded off to the nearest multiple of 1000. (Details for other wikis to be filled in). {| class="sortable" border="1" ! Subject wiki !! Pageviews in 2008 (starting May) !! Pageviews in 2009 !! Pageviews in 2010 |- | [[Groupprops:Main Page|Groupprops]] || 17000 || 99000 || 226000 |- | [[Topospaces:Main Page|Topospaces]] || 420 || 9000 || 20000 |- | [[Commalg:Main Page|Commalg]] || 180 || 1600 || 3000 |- | [[Market:Main Page|Market]] || 0 || 900 || 5600 |} 47fcc740569d7a33276b1f0d6beb02736ccdc8ed 610 609 2011-04-18T23:52:08Z Vipul 2 wikitext text/x-wiki ==Events== This page describes important historical milestones in the historical development of subject wikis. Due to imperfect records, precise dates are not available, but information is available up to the month. {| class="sortable" border="1" ! Month !! Event !! Record |- | December 2006 || The [[Groupprops:Main Page|Group Properties Wiki]] was created at this URL: http://editthis.info/groupprops || |- | January/February 2007 || The Group Properties Wiki was moved to this URL: http://groupprops.wiki-site.com and other wikis such as Commalg and Diffgeom were created at wiki-site.com || |- | May 2007 || The Topospaces (Topology) Wiki was created. || |- | July 2007 || Groupprops logo design by Arpith. || |- | March 2008 || The subwiki.org domain name was purchased. || |- | May 2008 || Groupprops and the other subject wikis were moved over to the subwiki.org domain, where they currently reside, with technical assistance from Harish of [http://4am.co.in 4AM]. || |- | January 2009 || MediaWiki installations were done for many other subject wikis; the [http://blog.subwiki.org Subject Wikis Blog] was created. || [http://blog.subwiki.org/?p=3 Blog post on the beginning of the subject wikis blog] |- | May 2010 || Upgrade to MediaWiki 1.16, with introduction of Vector skin. || [http://blog.subwiki.org/?p=60 Blog post] |- | September 2010 || For the first time, Groupprops logged 1000+ human pageviews in a single day. This is now a regular feature for weekday traffic during weeks of the academic year. || [http://blog.subwiki.org/?p=65 Blog post] (two months after the milestone) |- | December 2010 || Groupprops crossed 1 million pageviews as logged by MediaWiki (this also includes non-human pageviews) -- this does not include pageviews prior to May 2008. || |- | February 2011 || Upgrade to MediaWiki 1.16.2 || [http://blog.subwiki.org/?p=82 Blog post] (a day after the upgrade) |- | April 2011 || Upgrade to MediaWiki 1.16.4 || [http://blog.subwiki.org/2011/04/18/briefly-noted/ Blog post] (a day after the upgrade) |} ==Usage statistics== These pageview counts are as measured by Google Analytics, and are intended to measure only human pageviews. Counts are rounded off to the nearest multiple of 1000. (Details for other wikis to be filled in). {| class="sortable" border="1" ! Subject wiki !! Pageviews in 2008 (starting May) !! Pageviews in 2009 !! Pageviews in 2010 |- | [[Groupprops:Main Page|Groupprops]] || 17000 || 99000 || 226000 |- | [[Topospaces:Main Page|Topospaces]] || 420 || 9000 || 20000 |- | [[Commalg:Main Page|Commalg]] || 180 || 1600 || 3000 |- | [[Market:Main Page|Market]] || 0 || 900 || 5600 |} d48c68e0567bb1bd54ece36e253a698fb760e9ce 612 610 2011-06-21T23:51:57Z Vipul 2 /* Events */ wikitext text/x-wiki ==Events== This page describes important historical milestones in the historical development of subject wikis. Due to imperfect records, precise dates are not available, but information is available up to the month. {| class="sortable" border="1" ! Month !! Event !! Record |- | December 2006 || The [[Groupprops:Main Page|Group Properties Wiki]] was created at this URL: http://editthis.info/groupprops || |- | January/February 2007 || The Group Properties Wiki was moved to this URL: http://groupprops.wiki-site.com and other wikis such as Commalg and Diffgeom were created at wiki-site.com || |- | May 2007 || The Topospaces (Topology) Wiki was created. || |- | July 2007 || Groupprops logo design by Arpith. || |- | March 2008 || The subwiki.org domain name was purchased. || |- | May 2008 || Groupprops and the other subject wikis were moved over to the subwiki.org domain, where they currently reside, with technical assistance from Harish of [http://4am.co.in 4AM]. || |- | January 2009 || MediaWiki installations were done for many other subject wikis; the [http://blog.subwiki.org Subject Wikis Blog] was created. || [http://blog.subwiki.org/?p=3 Blog post on the beginning of the subject wikis blog] |- | May 2010 || Upgrade to MediaWiki 1.16, with introduction of Vector skin. || [http://blog.subwiki.org/?p=60 Blog post] |- | September 2010 || For the first time, Groupprops logged 1000+ human pageviews in a single day. This is now a regular feature for weekday traffic during weeks of the academic year. || [http://blog.subwiki.org/?p=65 Blog post] (two months after the milestone) |- | December 2010 || Groupprops crossed 1 million pageviews as logged by MediaWiki (this also includes non-human pageviews) -- this does not include pageviews prior to May 2008. || |- | February 2011 || Upgrade to MediaWiki 1.16.2 || [http://blog.subwiki.org/?p=82 Blog post] (a day after the upgrade) |- | April 2011 || Upgrade to MediaWiki 1.16.4 || [http://blog.subwiki.org/2011/04/18/briefly-noted/ Blog post] (a day after the upgrade) |- | June 2011 || Upgrade to MediaWiki 1.17.0 || |} ==Usage statistics== These pageview counts are as measured by Google Analytics, and are intended to measure only human pageviews. Counts are rounded off to the nearest multiple of 1000. (Details for other wikis to be filled in). {| class="sortable" border="1" ! Subject wiki !! Pageviews in 2008 (starting May) !! Pageviews in 2009 !! Pageviews in 2010 |- | [[Groupprops:Main Page|Groupprops]] || 17000 || 99000 || 226000 |- | [[Topospaces:Main Page|Topospaces]] || 420 || 9000 || 20000 |- | [[Commalg:Main Page|Commalg]] || 180 || 1600 || 3000 |- | [[Market:Main Page|Market]] || 0 || 900 || 5600 |} 23132de14043c019ad6ca05698c0b2bfef2f193c 106 352 604 579 2011-02-15T00:25:09Z Vipul 2 wikitext text/x-wiki {| class="sortable" border="1" ! Entity !! Value |- | Link || http://groupprops.subwiki.org |- | Name || [[Groupprops:Main Page|Groupprops]], The Group Properties Wiki |- | Main topic || Group theory |- | Number of pages || 4500+ |- | Original creation || December 2006 |- | Migration to subwiki.org || May 2008 |- | Current state || Pre-alpha; easily browsable |} 46b829530662f2238413bd7e9a60c810f3254f63 106 358 605 2011-02-16T00:09:12Z Vipul 2 Created page with "The ''tabular proof format'' is used in presenting a number of proofs in the subject wikis. The typical tabular proof format looks something like this: {| class="sortable" borde..." wikitext text/x-wiki The ''tabular proof format'' is used in presenting a number of proofs in the subject wikis. The typical tabular proof format looks something like this: {| class="sortable" border="1" ! Step no. !! Assertion/construction !! Facts used !! Given data used !! Previous steps used !! Explanation |- | 1 || (assertion/construction of first step goes here) || numbers/names of facts used in establishing the first assertion, if any, go here || given data needed for the assertion/construction go here || (none, since this is the first step) || more elaboration, if necessary |- | 2 || (assertion/construction of second step goes here) || numbers/names of facts used in establishing the second assertion, if any, go here || given data needed for the assertion/construction go here || Step (1) (if it relies on terminology/assertions made in step (1)), otherwise, nothing || more elaboration, if necessary |- | 3.. || ... || .. || .. || .. || .. |} This format is intended to allow people to peruse proofs at the desired level of depth, allowing for both eyeballing/scrolling and a detailed in-depth diagnostic understanding of proofs. Some aspects of the proof format are discussed below. ==Columns== ===Step number and previous steps used column=== The left column gives the step number. This is to allow easy cross-referencing to previous steps, which is done in the ''Previous steps used''. For instance, if the fourth row (with number 4) has, in the ''Previous steps used'' column, the numbers of Steps (1) and (3), this means that the justification for Step (4) relies on using the assertion/construction of Step (3). The step numbers are meant purely for an internal proof flow. ===Assertion/construction column=== The assertion/construction columns state the assertion/construction made in each step. The key idea is that a person can read ''only'' this column to get an idea of the proof, and ignore the ''explanation'' column. Further, whenever a step uses a previous step, it uses ''only'' the step as formulated in the assertion/construction column, and does not rely on the reasoning used in the ''explanation'' column. We can think of the assertion/construction column entry as the statement of an intermediate ''claim'' and the explanation column as the ''proof'' -- the key feature being that for later steps, it is the statement, not the proof, that matters. All new notation intended for reference later in the proof (e.g., a new letter for a particular object or function) ''must'' be introduced in the assertion/construction column. ===Facts used column=== The ''Facts used'' column indicates, for every step, the ''facts'' that need to be used to justify the assertion or carry out the construction. Usually, these facts are listed using fact numbers, though sometimes fact names are also given. The fact numbers ''do not refer to previous step numbers'', but rather to numbers used in a '''Facts used''' section that appears prior to the proof section on the page. The key distinction between '''Facts''' and ''Steps'' is that the facts have an independent existence outside of the statement being proved, and they usually have their own separate wiki page that is linked to in the '''Facts used''' section of the page. The advantage of the ''Facts used'' column is that it allows both a determination of what facts are used in a particular step, and a quick determination of what steps use a particular fact. The ''Facts used'' column does not usually contain additional information about ''how'' the fact is used to prove the assertion. That is done in the ''Explanation'' column. ===Given data used column=== This column provides, for each step, the ''given data'' used in establishing the assertion or construction made in that step. Given data here could include data about some object existing or satisfying some property. Given data differ from facts in that they do not have independent existence; they are just part of this particular step. They also differ from ''previous steps'' in that they are already given in the problem setup even before the proof begins. ===Explanation column=== This column explains how to arrive at the assertion or construction using the previous steps, given data, and facts. In some cases, no explanation is necessary -- these are cases where it is basically a modus ponens or other similar logical argument and there is no extra reasoning necessary. For instance, if the fact column states A implies B, and the given data column states that A is true in our case, then the assertion that B is true requires no additional justification. In some cases, more details may be helpful, and the explanation column provides these. It may also include variable re-assignments (for instance, the fact may be in terms of variable objects, and the explanation may inlude the values to assign those variables to use the fact). The explanation may also include some caveats or simple algebraic manipulation. Sometimes, explanations may be long enough that they are hidden using a show/hide feature, so that only those interested in seeing the explanation details will view the explanation. One key feature is that nothing ''within'' the explanation should be necessary for the later proof. If some intermediate conclusion reached within the explanation is to be used later, then the step should be split in two. e85b11e4146d974b1fcc61179c8930ea84de5931 606 605 2011-02-16T00:45:29Z Vipul 2 wikitext text/x-wiki The ''tabular proof format'' is used in presenting a number of proofs in the subject wikis. The typical tabular proof format looks something like this: {| class="sortable" border="1" ! Step no. !! Assertion/construction !! Facts used !! Given data used !! Previous steps used !! Explanation !! Commentary |- | 1 || (assertion/construction of first step goes here) || numbers/names of facts used in establishing the first assertion, if any, go here || given data needed for the assertion/construction go here || (none, since this is the first step) || more elaboration, if necessary || general commentary on the big picture, how the proof is flowing. |- | 2 || (assertion/construction of second step goes here) || numbers/names of facts used in establishing the second assertion, if any, go here || given data needed for the assertion/construction go here || Step (1) (if it relies on terminology/assertions made in step (1)), otherwise, nothing || more elaboration, if necessary || general commentary on the big picture, how the proof is flowing. |- | 3.. || ... || .. || .. || .. || .. || .. |} This format is intended to allow people to peruse proofs at the desired level of depth, allowing for both eyeballing/scrolling and a detailed in-depth diagnostic understanding of proofs. Some aspects of the proof format are discussed below. ==Columns== ===Step number and previous steps used column=== The left column gives the step number. This is to allow easy cross-referencing to previous steps, which is done in the ''Previous steps used''. For instance, if the fourth row (with number 4) has, in the ''Previous steps used'' column, the numbers of Steps (1) and (3), this means that the justification for Step (4) relies on using the assertion/construction of Step (3). The step numbers are meant purely for an internal proof flow. ===Assertion/construction column=== The assertion/construction columns state the assertion/construction made in each step. The key idea is that a person can read ''only'' this column to get an idea of the proof, and ignore the ''explanation'' column. Further, whenever a step uses a previous step, it uses ''only'' the step as formulated in the assertion/construction column, and does not rely on the reasoning used in the ''explanation'' column. We can think of the assertion/construction column entry as the statement of an intermediate ''claim'' and the explanation column as the ''proof'' -- the key feature being that for later steps, it is the statement, not the proof, that matters. All new notation intended for reference later in the proof (e.g., a new letter for a particular object or function) ''must'' be introduced in the assertion/construction column. ===Facts used column=== The ''Facts used'' column indicates, for every step, the ''facts'' that need to be used to justify the assertion or carry out the construction. Usually, these facts are listed using fact numbers, though sometimes fact names are also given. The fact numbers ''do not refer to previous step numbers'', but rather to numbers used in a '''Facts used''' section that appears prior to the proof section on the page. The key distinction between '''Facts''' and ''Steps'' is that the facts have an independent existence outside of the statement being proved, and they usually have their own separate wiki page that is linked to in the '''Facts used''' section of the page. The advantage of the ''Facts used'' column is that it allows both a determination of what facts are used in a particular step, and a quick determination of what steps use a particular fact. The ''Facts used'' column does not usually contain additional information about ''how'' the fact is used to prove the assertion. That is done in the ''Explanation'' column. ===Given data used column=== This column provides, for each step, the ''given data'' used in establishing the assertion or construction made in that step. Given data here could include data about some object existing or satisfying some property. Given data differ from facts in that they do not have independent existence; they are just part of this particular step. They also differ from ''previous steps'' in that they are already given in the problem setup even before the proof begins. ===Explanation column=== This column explains how to arrive at the assertion or construction using the previous steps, given data, and facts. In some cases, no explanation is necessary -- these are cases where it is basically a modus ponens or other similar logical argument and there is no extra reasoning necessary. For instance, if the fact column states A implies B, and the given data column states that A is true in our case, then the assertion that B is true requires no additional justification. In some cases, more details may be helpful, and the explanation column provides these. It may also include variable re-assignments (for instance, the fact may be in terms of variable objects, and the explanation may inlude the values to assign those variables to use the fact). The explanation may also include some caveats or simple algebraic manipulation. Sometimes, explanations may be long enough that they are hidden using a show/hide feature, so that only those interested in seeing the explanation details will view the explanation. One key feature is that nothing ''within'' the explanation should be necessary for the later proof. If some intermediate conclusion reached within the explanation is to be used later, then the step should be split in two. 7204febd536a27e974e6b653b59a55cd34289c26 607 606 2011-02-16T00:47:20Z Vipul 2 wikitext text/x-wiki The ''tabular proof format'' is used in presenting a number of proofs in the subject wikis. The typical tabular proof format looks something like this: {| class="sortable" border="1" ! Step no. !! Assertion/construction !! Facts used !! Given data used !! Previous steps used !! Explanation !! Commentary |- | 1 || (assertion/construction of first step goes here) || numbers/names of facts used in establishing the first assertion, if any, go here || given data needed for the assertion/construction go here || (none, since this is the first step) || more elaboration, if necessary || general commentary on the big picture, how the proof is flowing. |- | 2 || (assertion/construction of second step goes here) || numbers/names of facts used in establishing the second assertion, if any, go here || given data needed for the assertion/construction go here || Step (1) (if it relies on terminology/assertions made in step (1)), otherwise, nothing || more elaboration, if necessary || general commentary on the big picture, how the proof is flowing. |- | 3.. || ... || .. || .. || .. || .. || .. |} This format is intended to allow people to peruse proofs at the desired level of depth, allowing for both eyeballing/scrolling and a detailed in-depth diagnostic understanding of proofs. Some aspects of the proof format are discussed below. ==Columns== ===Step number and previous steps used column=== The left column gives the step number. This is to allow easy cross-referencing to previous steps, which is done in the ''Previous steps used''. For instance, if the fourth row (with number 4) has, in the ''Previous steps used'' column, the numbers of Steps (1) and (3), this means that the justification for Step (4) relies on using the assertion/construction of Step (3). The step numbers are meant purely for an internal proof flow. ===Assertion/construction column=== The assertion/construction columns state the assertion/construction made in each step. The key idea is that a person can read ''only'' this column to get an idea of the proof, and ignore the ''explanation'' column. Further, whenever a step uses a previous step, it uses ''only'' the step as formulated in the assertion/construction column, and does not rely on the reasoning used in the ''explanation'' column. We can think of the assertion/construction column entry as the statement of an intermediate ''claim'' and the explanation column as the ''proof'' -- the key feature being that for later steps, it is the statement, not the proof, that matters. All new notation intended for reference later in the proof (e.g., a new letter for a particular object or function) ''must'' be introduced in the assertion/construction column. ===Facts used column=== The ''Facts used'' column indicates, for every step, the ''facts'' that need to be used to justify the assertion or carry out the construction. Usually, these facts are listed using fact numbers, though sometimes fact names are also given. The fact numbers ''do not refer to previous step numbers'', but rather to numbers used in a '''Facts used''' section that appears prior to the proof section on the page. The key distinction between '''Facts''' and ''Steps'' is that the facts have an independent existence outside of the statement being proved, and they usually have their own separate wiki page that is linked to in the '''Facts used''' section of the page. The advantage of the ''Facts used'' column is that it allows both a determination of what facts are used in a particular step, and a quick determination of what steps use a particular fact. The ''Facts used'' column does not usually contain additional information about ''how'' the fact is used to prove the assertion. That is done in the ''Explanation'' column. ===Given data used column=== This column provides, for each step, the ''given data'' used in establishing the assertion or construction made in that step. Given data here could include data about some object existing or satisfying some property. Given data differ from facts in that they do not have independent existence; they are just part of this particular step. They also differ from ''previous steps'' in that they are already given in the problem setup even before the proof begins. ===Explanation column=== This column explains how to arrive at the assertion or construction using the previous steps, given data, and facts. In some cases, no explanation is necessary -- these are cases where it is basically a modus ponens or other similar logical argument and there is no extra reasoning necessary. For instance, if the fact column states A implies B, and the given data column states that A is true in our case, then the assertion that B is true requires no additional justification. In some cases, more details may be helpful, and the explanation column provides these. It may also include variable re-assignments (for instance, the fact may be in terms of variable objects, and the explanation may inlude the values to assign those variables to use the fact). The explanation may also include some caveats or simple algebraic manipulation. Sometimes, explanations may be long enough that they are hidden using a show/hide feature, so that only those interested in seeing the explanation details will view the explanation. One key feature is that nothing ''within'' the explanation should be necessary for the later proof. If some intermediate conclusion reached within the explanation is to be used later, then the step should be split in two. ===Commentary column=== This column is not to be found in all proofs. It is optional and can be thought of as a kind of running commentary that a knowledgeable person might try to provide orally while explaining a proof. Commentary can explain the big picture ideas, the nature and motivation of steps, and can look ahead to how a step will fit in with later stuff. cfcfdbde2e0d5880c082d105234e0dd1fa23a35b 106 22 611 589 2011-06-21T23:48:44Z Vipul 2 wikitext text/x-wiki {| class="sortable" border="1" ! Wiki !! Main topic !! Number of pages (lower bound) !! Creation month !! State of development |- | [[Groupprops:Main Page|Groupprops]] || Group theory || 4900 || December 2006 || pre-alpha; easily browsable |- | [[Topospaces:Main Page|Topospaces]] || Topology (point set, algebraic) || 600 || May 2007 || pre-pre-alpha |- | [[Commalg:Main Page|Commalg]] || Commutative algebra || 400 || January 2007 || pre-pre-alpha |- | [[Diffgeom:Main Page|Diffgeom]] || Differential geometry || 400 || February 2007 || pre-pre-alpha |- | [[Number:Main Page|Number]] || Number theory || 200 || March 2009 || barely begun |- | [[Market:Main Page|Market]] || Economic theory of markets, choices, and prices || 100 || January 2009 || barely begun |- | [[Mech:Main Page|Mech]] || Classical mechanics || 50 || January 2009 || barely begun |} '''NOTE''': The creation month need not coincide with the month of MediaWiki installation. In cases where the wiki was ported from elsewhere, the creation month may be earlier; in other cases, it may be later if the first edits happened long after installation. 4543be4b5e2e8b6ce3e48870272e7e54cbe5d76b 613 611 2011-09-05T13:47:47Z Vipul 2 wikitext text/x-wiki {| class="sortable" border="1" ! Wiki !! Main topic !! Number of pages (lower bound) !! Creation month !! State of development |- | [[Groupprops:Main Page|Groupprops]] || Group theory || 5200 || December 2006 || pre-alpha; easily browsable |- | [[Topospaces:Main Page|Topospaces]] || Topology (point set, algebraic) || 600 || May 2007 || pre-pre-alpha |- | [[Commalg:Main Page|Commalg]] || Commutative algebra || 400 || January 2007 || pre-pre-alpha |- | [[Diffgeom:Main Page|Diffgeom]] || Differential geometry || 400 || February 2007 || pre-pre-alpha |- | [[Number:Main Page|Number]] || Number theory || 200 || March 2009 || barely begun |- | [[Market:Main Page|Market]] || Economic theory of markets, choices, and prices || 100 || January 2009 || barely begun |- | [[Mech:Main Page|Mech]] || Classical mechanics || 50 || January 2009 || barely begun |- | [[Calculus:Main Page|Calculus]] || Calculus || 50 || August 2011 || barely begun |} '''NOTE''': The creation month need not coincide with the month of MediaWiki installation. In cases where the wiki was ported from elsewhere, the creation month may be earlier; in other cases, it may be later if the first edits happened long after installation. 55ae3ab172f1b7918e8f7026a1d4029c8fd9a793 626 613 2011-12-20T00:28:21Z Vipul 2 wikitext text/x-wiki {| class="sortable" border="1" ! Wiki !! Main topic !! Number of pages (lower bound) !! Creation month !! State of development |- | [[Groupprops:Main Page|Groupprops]] || Group theory || 5200 || December 2006 || alpha; easily browsable |- | [[Topospaces:Main Page|Topospaces]] || Topology (point set, algebraic) || 600 || May 2007 || pre-pre-alpha |- | [[Commalg:Main Page|Commalg]] || Commutative algebra || 400 || January 2007 || pre-pre-alpha |- | [[Diffgeom:Main Page|Diffgeom]] || Differential geometry || 400 || February 2007 || pre-pre-alpha |- | [[Number:Main Page|Number]] || Number theory || 200 || March 2009 || barely begun |- | [[Market:Main Page|Market]] || Economic theory of markets, choices, and prices || 100 || January 2009 || barely begun |- | [[Mech:Main Page|Mech]] || Classical mechanics || 50 || January 2009 || barely begun |- | [[Calculus:Main Page|Calculus]] || Calculus || 50 || August 2011 || barely begun |} '''NOTE''': The creation month need not coincide with the month of MediaWiki installation. In cases where the wiki was ported from elsewhere, the creation month may be earlier; in other cases, it may be later if the first edits happened long after installation. f34a6bd09314a0cd1686ce646252fac441428ca2 627 626 2011-12-20T00:30:21Z Vipul 2 wikitext text/x-wiki {| class="sortable" border="1" ! Wiki !! Main topic !! Number of pages (lower bound) !! Creation month !! State of development |- | [[Groupprops:Main Page|Groupprops]] || Group theory || 5200 || December 2006 || beta |- | [[Topospaces:Main Page|Topospaces]] || Topology (point set, algebraic) || 600 || May 2007 || alpha |- | [[Commalg:Main Page|Commalg]] || Commutative algebra || 400 || January 2007 || alpha |- | [[Diffgeom:Main Page|Diffgeom]] || Differential geometry || 400 || February 2007 || alpha |- | [[Number:Main Page|Number]] || Number theory || 200 || March 2009 || alpha |- | [[Market:Main Page|Market]] || Economic theory of markets, choices, and prices || 100 || January 2009 || alpha |- | [[Mech:Main Page|Mech]] || Classical mechanics || 50 || January 2009 || alpha |- | [[Calculus:Main Page|Calculus]] || Calculus || 50 || August 2011 || alpha |} '''NOTE''': The creation month need not coincide with the month of MediaWiki installation. In cases where the wiki was ported from elsewhere, the creation month may be earlier; in other cases, it may be later if the first edits happened long after installation. c4b8c1ff2acb0e974aa96a03397505a0344b6ce4 0 359 614 2011-11-23T22:45:09Z Vipul 2 Created page with "==Summary== ===Evaluation on key parameters=== '''Schistosomiasis Control Initiative''' ('''SCI''') is a recommended organization. {| class="sortable" border="1" ! Parameter !..." wikitext text/x-wiki ==Summary== ===Evaluation on key parameters=== '''Schistosomiasis Control Initiative''' ('''SCI''') is a recommended organization. {| class="sortable" border="1" ! Parameter !! One-word summary !! More details (section in this page) |- | Evidence of effectiveness || Strong || [[#Independent evidence of program effectiveness]] |- | Cost-effectiveness || Excellent || [[#Independent evidence of program effectiveness]] |- | Funding gap aka room for more funding || Significant || [[#Room for more funding]] |- | Transparency || ? || ? |- | Monitoring and evaluation || Strong || [[#Internal monitoring: large-scale programs]] |} ===Summary of who they are and what they do=== {| class="sortable" border="1" ! Item !! Value |- | Countries of operation || ? (African countries) |- | Headquarters || Imperial College, London, United Kingdom |- | Goals || Create or scale up [[mass drug administration program]]s for [[neglected tropical diseases]] (NTD), particularly [[schistosomiasis]] and [[soil-transmitted helminth]]s (STHs), in school-aged children and other groups determined to be at high risk. |- | Method summary || Advocate for the benefits of mass drug distributions to government officials.<br>Assist with planning and fund raising.<br>Deliver funding and drugs to governments.<br>Provide financial management and technical support.<br>Develop procedures for monitoring and evaluation. |- | Previous sources of funding || grant from [[Gates Foundation]] (private philanthropic foundation)<br>grants from [[USAID]] (US federal grantmaking agency)<br>grants from [[DFID]] (US federal grantmaking agency) |} ===Review process=== The review of SCI began in 2009. Below are details of the kind of inputs used in the review process. {| class="sortable" border="1" ! What we sought to learn !! How we learned it |- | Effectiveness and other effects of these kinds of programs || Study of the published literature |- | Details of the big picture of SCI's operations || Extensive communications with SCI Director Alan Fenwick to discuss SCI's methods and funding needs<br>meetings with other staff at SCI's headquarters in London<br>Requesting and reviewing SCI internal financial and organizational documents. |- | Feel for how SCI's programs are run on the ground || Visiting a national schistosomiasis control program meeting and demonstration mass drug administration in Malawi (site visit: October 2011) |} 2d4afa216700546370b8107af0cc810a8d5755c1 615 614 2011-11-23T23:06:06Z Vipul 2 wikitext text/x-wiki ==Summary== ===Evaluation on key parameters=== '''Schistosomiasis Control Initiative''' ('''SCI''') is a recommended organization. {| class="sortable" border="1" ! Parameter !! One-word summary !! More details (section in this page) |- | Evidence of effectiveness || Strong || [[#Independent evidence of program effectiveness]] |- | Cost-effectiveness || Excellent || [[#Independent evidence of program effectiveness]] |- | Funding gap aka room for more funding || Significant || [[#Room for more funding]] |- | Transparency || ? || ? |- | Monitoring and evaluation || Strong || [[#Internal monitoring: large-scale programs]] |} ===Summary of who they are and what they do=== {| class="sortable" border="1" ! Item !! Value |- | Countries of operation || ? (African countries) |- | Headquarters || Imperial College, London, United Kingdom |- | Goals || Create or scale up [[mass drug administration program]]s for [[neglected tropical diseases]] (NTD), particularly [[schistosomiasis]] and [[soil-transmitted helminth]]s (STHs), in school-aged children and other groups determined to be at high risk. |- | Method summary || Advocate for the benefits of mass drug distributions to government officials.<br>Assist with planning and fund raising.<br>Deliver funding and drugs to governments.<br>Provide financial management and technical support.<br>Develop procedures for monitoring and evaluation. |- | Previous sources of funding || grant from [[Gates Foundation]] (private philanthropic foundation)<br>grants from [[USAID]] (US federal grantmaking agency)<br>grants from [[DFID]] (UK central grantmaking agency) |} ===Review process=== The review of SCI began in 2009. Below are details of the kind of inputs used in the review process. {| class="sortable" border="1" ! What we sought to learn !! How we learned it |- | Effectiveness and other effects of these kinds of programs || Study of the published literature |- | Details of the big picture of SCI's operations || Extensive communications with SCI Director Alan Fenwick to discuss SCI's methods and funding needs<br>meetings with other staff at SCI's headquarters in London<br>Requesting and reviewing SCI internal financial and organizational documents. |- | Feel for how SCI's programs are run on the ground || Visiting a national schistosomiasis control program meeting and demonstration mass drug administration in Malawi (site visit: October 2011) |} ==Operations and history== Note to GiveWell staff: ignore the red links, they are meant to be stand-ins for your current internal links to pages. {{quotation|To understand this section better, follow the links to get more background information on these topics:<br>[[neglected tropical diseases]] (NTDs): diseases that currently afflict people largely in low-income tropical areas, particularly Africa, and are ''neglected'' by most research efforts in the developed world because they do not cause much trouble in developed world countries.<br>List of NTDs mentioned here: [[schistosomiasis]], [[soil-transmitted helminth]]s, [[lymphatic filariasis]], [[onchocerciasis]], and [[trachoma]].<br>[[mass drug administration program]]: {{fillin}} }} ===Broad description of goals and methods=== SCI's goal is to create or scale up [[mass drug administration program]]s for [[neglected tropical diseases]] (NTD), particularly [[schistosomiasis]] and [[soil-transmitted helminth]]s (STHs), in school-aged children and other groups determined to be at high risk. They list the following five methods: * Advocate for the benefits of mass drug distributions to government officials in the African countries * Assist with planning and fund raising * Deliver funding and drugs to governments * Provide financial management and technical support * Develop procedures for monitoring and evaluation ===History of operations and funding=== Below are some of the important funders and details of the programs funded (the ?s reflect things that weren't clear from GiveWell's page but can be dug up from the original sources, I'm sure): {| class="sortable" border="1" ! Funder !! Year of funding !! Amount of funding !! NTDs tackled !! Countries tackled !! Expiry of grant !! Additional comments |- | [[Gates Foundation]] (private philanthropic foundation that makes grants for developing world health charities) || 2002 || $32 million || Schistosomiasis and soil-transmitted helminths || six African countries: {{fillin}} || unspecified || This was a founding grant. |- | [[USAID]] (US federal grantmaking agency) and [[Gates Foundation]] || 2006 || unspecified || lymphatic filariasis, onchocerciasis, and trachoma, in addition to schistosomiasis and STHs || eight African countries: {{fillin}} || Till 2011 (five year grant) |- | ? || 2007 || unspecified || lymphatic filariasis, onchocerciasis, and trachoma, in addition to schistosomiasis and STHs || Rwanda and Burundi || Till 2011 || This was intended to expand the program from the eight current countries of operation to these two new countries. |- | [[DFID]] (UK central grantmaking agency) || 2010 || unspecified || schistosomiasis and soil-transmitted helminths || in eight countries (same eight?) || Till 2015 (?), five year grant || Other NTDs are not covered by the grant, <toggledisplay>though DFID also provided funding to the Liverpool School of Tropical Medicine to integrate treatment for lymphatic filariasis with SCI-funded schistosomiasis and STH programs in six countries</toggledisplay> |} b1be6d57ae1f2d3c323174fd55785995cf6d032c 618 615 2011-11-23T23:10:23Z Vipul 2 /* History of operations and funding */ wikitext text/x-wiki ==Summary== ===Evaluation on key parameters=== '''Schistosomiasis Control Initiative''' ('''SCI''') is a recommended organization. {| class="sortable" border="1" ! Parameter !! One-word summary !! More details (section in this page) |- | Evidence of effectiveness || Strong || [[#Independent evidence of program effectiveness]] |- | Cost-effectiveness || Excellent || [[#Independent evidence of program effectiveness]] |- | Funding gap aka room for more funding || Significant || [[#Room for more funding]] |- | Transparency || ? || ? |- | Monitoring and evaluation || Strong || [[#Internal monitoring: large-scale programs]] |} ===Summary of who they are and what they do=== {| class="sortable" border="1" ! Item !! Value |- | Countries of operation || ? (African countries) |- | Headquarters || Imperial College, London, United Kingdom |- | Goals || Create or scale up [[mass drug administration program]]s for [[neglected tropical diseases]] (NTD), particularly [[schistosomiasis]] and [[soil-transmitted helminth]]s (STHs), in school-aged children and other groups determined to be at high risk. |- | Method summary || Advocate for the benefits of mass drug distributions to government officials.<br>Assist with planning and fund raising.<br>Deliver funding and drugs to governments.<br>Provide financial management and technical support.<br>Develop procedures for monitoring and evaluation. |- | Previous sources of funding || grant from [[Gates Foundation]] (private philanthropic foundation)<br>grants from [[USAID]] (US federal grantmaking agency)<br>grants from [[DFID]] (UK central grantmaking agency) |} ===Review process=== The review of SCI began in 2009. Below are details of the kind of inputs used in the review process. {| class="sortable" border="1" ! What we sought to learn !! How we learned it |- | Effectiveness and other effects of these kinds of programs || Study of the published literature |- | Details of the big picture of SCI's operations || Extensive communications with SCI Director Alan Fenwick to discuss SCI's methods and funding needs<br>meetings with other staff at SCI's headquarters in London<br>Requesting and reviewing SCI internal financial and organizational documents. |- | Feel for how SCI's programs are run on the ground || Visiting a national schistosomiasis control program meeting and demonstration mass drug administration in Malawi (site visit: October 2011) |} ==Operations and history== Note to GiveWell staff: ignore the red links, they are meant to be stand-ins for your current internal links to pages. {{quotation|To understand this section better, follow the links to get more background information on these topics:<br>[[neglected tropical diseases]] (NTDs): diseases that currently afflict people largely in low-income tropical areas, particularly Africa, and are ''neglected'' by most research efforts in the developed world because they do not cause much trouble in developed world countries.<br>List of NTDs mentioned here: [[schistosomiasis]], [[soil-transmitted helminth]]s, [[lymphatic filariasis]], [[onchocerciasis]], and [[trachoma]].<br>[[mass drug administration program]]: {{fillin}} }} ===Broad description of goals and methods=== SCI's goal is to create or scale up [[mass drug administration program]]s for [[neglected tropical diseases]] (NTD), particularly [[schistosomiasis]] and [[soil-transmitted helminth]]s (STHs), in school-aged children and other groups determined to be at high risk. They list the following five methods: * Advocate for the benefits of mass drug distributions to government officials in the African countries * Assist with planning and fund raising * Deliver funding and drugs to governments * Provide financial management and technical support * Develop procedures for monitoring and evaluation ===History of operations and funding=== Below are some of the important funders and details of the programs funded (the ?s reflect things that weren't clear from GiveWell's page but can be dug up from the original sources, I'm sure): {| class="sortable" border="1" ! Funder !! Year of funding !! Amount of funding !! NTDs tackled !! Countries tackled !! Expiry of grant !! Additional comments |- | [[Gates Foundation]] (private philanthropic foundation that makes grants for developing world health charities) || 2002 || $32 million || Schistosomiasis and soil-transmitted helminths || six African countries: {{fillin}} || unspecified || This was a founding grant. |- | [[USAID]] (US federal grantmaking agency) and [[Gates Foundation]] || 2006 || unspecified || lymphatic filariasis, onchocerciasis, and trachoma, in addition to schistosomiasis and STHs || eight African countries: {{fillin}} || Till 2011 (five year grant) || |- | ? || 2007 || unspecified || lymphatic filariasis, onchocerciasis, and trachoma, in addition to schistosomiasis and STHs || Rwanda and Burundi || Till 2011 || This was intended to expand the program from the eight current countries of operation to these two new countries. |- | [[DFID]] (UK central grantmaking agency) || 2010 || unspecified || schistosomiasis and soil-transmitted helminths || in eight countries (same eight?) || Till 2015 (?), five year grant || Other NTDs are not covered by the grant, <toggledisplay>though DFID also provided funding to the Liverpool School of Tropical Medicine to integrate treatment for lymphatic filariasis with SCI-funded schistosomiasis and STH programs in six countries</toggledisplay> |} 039b82a0c06b4bf53242821d830d01adba4a7bf2 619 618 2011-11-23T23:12:15Z Vipul 2 /* History of operations and funding */ wikitext text/x-wiki ==Summary== ===Evaluation on key parameters=== '''Schistosomiasis Control Initiative''' ('''SCI''') is a recommended organization. {| class="sortable" border="1" ! Parameter !! One-word summary !! More details (section in this page) |- | Evidence of effectiveness || Strong || [[#Independent evidence of program effectiveness]] |- | Cost-effectiveness || Excellent || [[#Independent evidence of program effectiveness]] |- | Funding gap aka room for more funding || Significant || [[#Room for more funding]] |- | Transparency || ? || ? |- | Monitoring and evaluation || Strong || [[#Internal monitoring: large-scale programs]] |} ===Summary of who they are and what they do=== {| class="sortable" border="1" ! Item !! Value |- | Countries of operation || ? (African countries) |- | Headquarters || Imperial College, London, United Kingdom |- | Goals || Create or scale up [[mass drug administration program]]s for [[neglected tropical diseases]] (NTD), particularly [[schistosomiasis]] and [[soil-transmitted helminth]]s (STHs), in school-aged children and other groups determined to be at high risk. |- | Method summary || Advocate for the benefits of mass drug distributions to government officials.<br>Assist with planning and fund raising.<br>Deliver funding and drugs to governments.<br>Provide financial management and technical support.<br>Develop procedures for monitoring and evaluation. |- | Previous sources of funding || grant from [[Gates Foundation]] (private philanthropic foundation)<br>grants from [[USAID]] (US federal grantmaking agency)<br>grants from [[DFID]] (UK central grantmaking agency) |} ===Review process=== The review of SCI began in 2009. Below are details of the kind of inputs used in the review process. {| class="sortable" border="1" ! What we sought to learn !! How we learned it |- | Effectiveness and other effects of these kinds of programs || Study of the published literature |- | Details of the big picture of SCI's operations || Extensive communications with SCI Director Alan Fenwick to discuss SCI's methods and funding needs<br>meetings with other staff at SCI's headquarters in London<br>Requesting and reviewing SCI internal financial and organizational documents. |- | Feel for how SCI's programs are run on the ground || Visiting a national schistosomiasis control program meeting and demonstration mass drug administration in Malawi (site visit: October 2011) |} ==Operations and history== Note to GiveWell staff: ignore the red links, they are meant to be stand-ins for your current internal links to pages. {{quotation|To understand this section better, follow the links to get more background information on these topics:<br>[[neglected tropical diseases]] (NTDs): diseases that currently afflict people largely in low-income tropical areas, particularly Africa, and are ''neglected'' by most research efforts in the developed world because they do not cause much trouble in developed world countries.<br>List of NTDs mentioned here: [[schistosomiasis]], [[soil-transmitted helminth]]s, [[lymphatic filariasis]], [[onchocerciasis]], and [[trachoma]].<br>[[mass drug administration program]]: {{fillin}} }} ===Broad description of goals and methods=== SCI's goal is to create or scale up [[mass drug administration program]]s for [[neglected tropical diseases]] (NTD), particularly [[schistosomiasis]] and [[soil-transmitted helminth]]s (STHs), in school-aged children and other groups determined to be at high risk. They list the following five methods: * Advocate for the benefits of mass drug distributions to government officials in the African countries * Assist with planning and fund raising * Deliver funding and drugs to governments * Provide financial management and technical support * Develop procedures for monitoring and evaluation ===History of operations and funding=== Below are some of the important funders and details of the programs funded (the ?s reflect things that weren't clear from GiveWell's page but can be dug up from the original sources, I'm sure): {| class="sortable" border="1" ! Funder !! Year of funding !! Amount of funding !! NTDs tackled !! Countries tackled !! Expiry of grant !! Additional comments |- | [[Gates Foundation]] (private philanthropic foundation that makes grants for developing world health charities) || 2002 || $32 million || schistosomiasis, STHs || six African countries: {{fillin}} || unspecified || This was a founding grant. |- | [[USAID]] (US federal grantmaking agency) and [[Gates Foundation]] || 2006 || unspecified || schistosomiasis, STHs, lymphatic filariasis, onchocerciasis, and trachoma, || eight African countries: {{fillin}} || Till 2011 (five year grant) || |- | ? || 2007 || unspecified || schistosomiasis, STHs, lymphatic filariasis, onchocerciasis, and trachoma, || Rwanda and Burundi || Till 2011 || This was intended to expand the program from the eight current countries of operation to these two new countries. |- | [[DFID]] (UK central grantmaking agency) || 2010 || unspecified || schistosomiasis, STHs || in eight countries (same eight?) || Till 2015 (?), five year grant || Other NTDs are not covered by the grant, <toggledisplay>though DFID also provided funding to the Liverpool School of Tropical Medicine to integrate treatment for lymphatic filariasis with SCI-funded schistosomiasis and STH programs in six countries</toggledisplay> |} 91edec07158002ae146c7415b7f06e2f5ee507ff 620 619 2011-11-23T23:13:07Z Vipul 2 /* Operations and history */ wikitext text/x-wiki ==Summary== ===Evaluation on key parameters=== '''Schistosomiasis Control Initiative''' ('''SCI''') is a recommended organization. {| class="sortable" border="1" ! Parameter !! One-word summary !! More details (section in this page) |- | Evidence of effectiveness || Strong || [[#Independent evidence of program effectiveness]] |- | Cost-effectiveness || Excellent || [[#Independent evidence of program effectiveness]] |- | Funding gap aka room for more funding || Significant || [[#Room for more funding]] |- | Transparency || ? || ? |- | Monitoring and evaluation || Strong || [[#Internal monitoring: large-scale programs]] |} ===Summary of who they are and what they do=== {| class="sortable" border="1" ! Item !! Value |- | Countries of operation || ? (African countries) |- | Headquarters || Imperial College, London, United Kingdom |- | Goals || Create or scale up [[mass drug administration program]]s for [[neglected tropical diseases]] (NTD), particularly [[schistosomiasis]] and [[soil-transmitted helminth]]s (STHs), in school-aged children and other groups determined to be at high risk. |- | Method summary || Advocate for the benefits of mass drug distributions to government officials.<br>Assist with planning and fund raising.<br>Deliver funding and drugs to governments.<br>Provide financial management and technical support.<br>Develop procedures for monitoring and evaluation. |- | Previous sources of funding || grant from [[Gates Foundation]] (private philanthropic foundation)<br>grants from [[USAID]] (US federal grantmaking agency)<br>grants from [[DFID]] (UK central grantmaking agency) |} ===Review process=== The review of SCI began in 2009. Below are details of the kind of inputs used in the review process. {| class="sortable" border="1" ! What we sought to learn !! How we learned it |- | Effectiveness and other effects of these kinds of programs || Study of the published literature |- | Details of the big picture of SCI's operations || Extensive communications with SCI Director Alan Fenwick to discuss SCI's methods and funding needs<br>meetings with other staff at SCI's headquarters in London<br>Requesting and reviewing SCI internal financial and organizational documents. |- | Feel for how SCI's programs are run on the ground || Visiting a national schistosomiasis control program meeting and demonstration mass drug administration in Malawi (site visit: October 2011) |} ==Operations and history== Note to GiveWell staff: ignore the red links, they are meant to be stand-ins for your current internal links to pages. {{quotation|To understand this section better, follow the links to get more background information on these topics:<br>[[neglected tropical diseases]] (NTDs): diseases that currently afflict people largely in low-income tropical areas, particularly Africa, and are ''neglected'' by most research efforts in the developed world because they do not cause much trouble in developed world countries.<br>List of NTDs mentioned here: [[schistosomiasis]], [[soil-transmitted helminth]]s (acronym '''STH'''s), [[lymphatic filariasis]], [[onchocerciasis]], and [[trachoma]].<br>[[mass drug administration program]]: {{fillin}} }} ===Broad description of goals and methods=== SCI's goal is to create or scale up [[mass drug administration program]]s for [[neglected tropical diseases]] (NTD), particularly [[schistosomiasis]] and [[soil-transmitted helminth]]s (STHs), in school-aged children and other groups determined to be at high risk. They list the following five methods: * Advocate for the benefits of mass drug distributions to government officials in the African countries * Assist with planning and fund raising * Deliver funding and drugs to governments * Provide financial management and technical support * Develop procedures for monitoring and evaluation ===History of operations and funding=== Below are some of the important funders and details of the programs funded (the ?s reflect things that weren't clear from GiveWell's page but can be dug up from the original sources, I'm sure): {| class="sortable" border="1" ! Funder !! Year of funding !! Amount of funding !! Neglected Tropical Diseases (NTDs) tackled !! Countries tackled !! Expiry of grant !! Additional comments |- | [[Gates Foundation]] (private philanthropic foundation that makes grants for developing world health charities) || 2002 || $32 million || schistosomiasis, STHs || six African countries: {{fillin}} || unspecified || This was a founding grant. |- | [[USAID]] (US federal grantmaking agency) and [[Gates Foundation]] || 2006 || unspecified || schistosomiasis, STHs, lymphatic filariasis, onchocerciasis, and trachoma || eight African countries: {{fillin}} || Till 2011 (five year grant) || |- | ? || 2007 || unspecified || schistosomiasis, STHs, lymphatic filariasis, onchocerciasis, and trachoma || Rwanda and Burundi || Till 2011 || This was intended to expand the program from the eight current countries of operation to these two new countries. |- | [[DFID]] (UK central grantmaking agency) || 2010 || unspecified || schistosomiasis, STHs || in eight countries (same eight?) || Till 2015 (?), five year grant || Other NTDs are not covered by the grant, <toggledisplay>though DFID also provided funding to the Liverpool School of Tropical Medicine to integrate treatment for lymphatic filariasis with SCI-funded schistosomiasis and STH programs in six countries</toggledisplay> |} 5519f4ccc87cb8c5fb4d77f1f5211a549c9e297d 621 620 2011-11-23T23:13:47Z Vipul 2 /* History of operations and funding */ wikitext text/x-wiki ==Summary== ===Evaluation on key parameters=== '''Schistosomiasis Control Initiative''' ('''SCI''') is a recommended organization. {| class="sortable" border="1" ! Parameter !! One-word summary !! More details (section in this page) |- | Evidence of effectiveness || Strong || [[#Independent evidence of program effectiveness]] |- | Cost-effectiveness || Excellent || [[#Independent evidence of program effectiveness]] |- | Funding gap aka room for more funding || Significant || [[#Room for more funding]] |- | Transparency || ? || ? |- | Monitoring and evaluation || Strong || [[#Internal monitoring: large-scale programs]] |} ===Summary of who they are and what they do=== {| class="sortable" border="1" ! Item !! Value |- | Countries of operation || ? (African countries) |- | Headquarters || Imperial College, London, United Kingdom |- | Goals || Create or scale up [[mass drug administration program]]s for [[neglected tropical diseases]] (NTD), particularly [[schistosomiasis]] and [[soil-transmitted helminth]]s (STHs), in school-aged children and other groups determined to be at high risk. |- | Method summary || Advocate for the benefits of mass drug distributions to government officials.<br>Assist with planning and fund raising.<br>Deliver funding and drugs to governments.<br>Provide financial management and technical support.<br>Develop procedures for monitoring and evaluation. |- | Previous sources of funding || grant from [[Gates Foundation]] (private philanthropic foundation)<br>grants from [[USAID]] (US federal grantmaking agency)<br>grants from [[DFID]] (UK central grantmaking agency) |} ===Review process=== The review of SCI began in 2009. Below are details of the kind of inputs used in the review process. {| class="sortable" border="1" ! What we sought to learn !! How we learned it |- | Effectiveness and other effects of these kinds of programs || Study of the published literature |- | Details of the big picture of SCI's operations || Extensive communications with SCI Director Alan Fenwick to discuss SCI's methods and funding needs<br>meetings with other staff at SCI's headquarters in London<br>Requesting and reviewing SCI internal financial and organizational documents. |- | Feel for how SCI's programs are run on the ground || Visiting a national schistosomiasis control program meeting and demonstration mass drug administration in Malawi (site visit: October 2011) |} ==Operations and history== Note to GiveWell staff: ignore the red links, they are meant to be stand-ins for your current internal links to pages. {{quotation|To understand this section better, follow the links to get more background information on these topics:<br>[[neglected tropical diseases]] (NTDs): diseases that currently afflict people largely in low-income tropical areas, particularly Africa, and are ''neglected'' by most research efforts in the developed world because they do not cause much trouble in developed world countries.<br>List of NTDs mentioned here: [[schistosomiasis]], [[soil-transmitted helminth]]s (acronym '''STH'''s), [[lymphatic filariasis]], [[onchocerciasis]], and [[trachoma]].<br>[[mass drug administration program]]: {{fillin}} }} ===Broad description of goals and methods=== SCI's goal is to create or scale up [[mass drug administration program]]s for [[neglected tropical diseases]] (NTD), particularly [[schistosomiasis]] and [[soil-transmitted helminth]]s (STHs), in school-aged children and other groups determined to be at high risk. They list the following five methods: * Advocate for the benefits of mass drug distributions to government officials in the African countries * Assist with planning and fund raising * Deliver funding and drugs to governments * Provide financial management and technical support * Develop procedures for monitoring and evaluation ===History of operations and funding=== Below are some of the important funders and details of the programs funded (the ?s reflect things that weren't clear from GiveWell's page but can be dug up from the original sources, I'm sure): {| class="sortable" border="1" ! Funder !! Year of funding !! Amount of funding !! Neglected Tropical Diseases (NTDs) tackled !! Countries tackled !! Expiry of grant !! Additional comments |- | [[Gates Foundation]] (private philanthropic foundation that makes grants for developing world health charities) || 2002 || $32 million || schistosomiasis, STHs || six African countries: {{fillin}} || unspecified || This was a founding grant. |- | [[USAID]] (US federal grantmaking agency) and [[Gates Foundation]] || 2006 || ? || schistosomiasis, STHs, lymphatic filariasis, onchocerciasis, and trachoma || eight African countries: {{fillin}} || Till 2011 (five year grant) || |- | ? || 2007 || ? || schistosomiasis, STHs, lymphatic filariasis, onchocerciasis, and trachoma || Rwanda and Burundi || Till 2011 || This was intended to expand the program from the eight current countries of operation to these two new countries. |- | [[DFID]] (UK central grantmaking agency) || 2010 || ? || schistosomiasis, STHs || in eight countries (same eight?) || Till 2015 (?), five year grant || Other NTDs are not covered by the grant, <toggledisplay>though DFID also provided funding to the Liverpool School of Tropical Medicine to integrate treatment for lymphatic filariasis with SCI-funded schistosomiasis and STH programs in six countries</toggledisplay> |} 82b1292d4f59a0fbc16a3c7bc68d85cd8220bc31 622 621 2011-11-23T23:14:18Z Vipul 2 /* History of operations and funding */ wikitext text/x-wiki ==Summary== ===Evaluation on key parameters=== '''Schistosomiasis Control Initiative''' ('''SCI''') is a recommended organization. {| class="sortable" border="1" ! Parameter !! One-word summary !! More details (section in this page) |- | Evidence of effectiveness || Strong || [[#Independent evidence of program effectiveness]] |- | Cost-effectiveness || Excellent || [[#Independent evidence of program effectiveness]] |- | Funding gap aka room for more funding || Significant || [[#Room for more funding]] |- | Transparency || ? || ? |- | Monitoring and evaluation || Strong || [[#Internal monitoring: large-scale programs]] |} ===Summary of who they are and what they do=== {| class="sortable" border="1" ! Item !! Value |- | Countries of operation || ? (African countries) |- | Headquarters || Imperial College, London, United Kingdom |- | Goals || Create or scale up [[mass drug administration program]]s for [[neglected tropical diseases]] (NTD), particularly [[schistosomiasis]] and [[soil-transmitted helminth]]s (STHs), in school-aged children and other groups determined to be at high risk. |- | Method summary || Advocate for the benefits of mass drug distributions to government officials.<br>Assist with planning and fund raising.<br>Deliver funding and drugs to governments.<br>Provide financial management and technical support.<br>Develop procedures for monitoring and evaluation. |- | Previous sources of funding || grant from [[Gates Foundation]] (private philanthropic foundation)<br>grants from [[USAID]] (US federal grantmaking agency)<br>grants from [[DFID]] (UK central grantmaking agency) |} ===Review process=== The review of SCI began in 2009. Below are details of the kind of inputs used in the review process. {| class="sortable" border="1" ! What we sought to learn !! How we learned it |- | Effectiveness and other effects of these kinds of programs || Study of the published literature |- | Details of the big picture of SCI's operations || Extensive communications with SCI Director Alan Fenwick to discuss SCI's methods and funding needs<br>meetings with other staff at SCI's headquarters in London<br>Requesting and reviewing SCI internal financial and organizational documents. |- | Feel for how SCI's programs are run on the ground || Visiting a national schistosomiasis control program meeting and demonstration mass drug administration in Malawi (site visit: October 2011) |} ==Operations and history== Note to GiveWell staff: ignore the red links, they are meant to be stand-ins for your current internal links to pages. {{quotation|To understand this section better, follow the links to get more background information on these topics:<br>[[neglected tropical diseases]] (NTDs): diseases that currently afflict people largely in low-income tropical areas, particularly Africa, and are ''neglected'' by most research efforts in the developed world because they do not cause much trouble in developed world countries.<br>List of NTDs mentioned here: [[schistosomiasis]], [[soil-transmitted helminth]]s (acronym '''STH'''s), [[lymphatic filariasis]], [[onchocerciasis]], and [[trachoma]].<br>[[mass drug administration program]]: {{fillin}} }} ===Broad description of goals and methods=== SCI's goal is to create or scale up [[mass drug administration program]]s for [[neglected tropical diseases]] (NTD), particularly [[schistosomiasis]] and [[soil-transmitted helminth]]s (STHs), in school-aged children and other groups determined to be at high risk. They list the following five methods: * Advocate for the benefits of mass drug distributions to government officials in the African countries * Assist with planning and fund raising * Deliver funding and drugs to governments * Provide financial management and technical support * Develop procedures for monitoring and evaluation ===History of operations and funding=== Below are some of the important funders and details of the programs funded (the ?s reflect things that weren't clear from GiveWell's page but can be dug up from the original sources, I'm sure): {| class="sortable" border="1" ! Funder !! Year of funding !! Amount of funding !! Neglected Tropical Diseases (NTDs) tackled !! Countries tackled !! Expiry of grant !! Additional comments |- | [[Gates Foundation]] (private philanthropic foundation that makes grants for developing world health charities) || 2002 || $32 million || schistosomiasis, STHs || six African countries: {{fillin}} || unspecified || This was a founding grant. |- | [[USAID]] (US federal grantmaking agency) and [[Gates Foundation]] || 2006 || ? || schistosomiasis, STHs, lymphatic filariasis, onchocerciasis, and trachoma || eight African countries: {{fillin}} || Till 2011 (five year grant) || |- | ? || 2007 || ? || schistosomiasis, STHs, lymphatic filariasis, onchocerciasis, and trachoma || Rwanda and Burundi || Till 2011 || This was intended to expand the program from the eight current countries of operation to these two new countries. |- | [[DFID]] (UK central grantmaking agency) || 2010 || ? || schistosomiasis, STHs || eight African countries (same eight?) || Till 2015 (?), five year grant || Other NTDs are not covered by the grant, <toggledisplay>though DFID also provided funding to the Liverpool School of Tropical Medicine to integrate treatment for lymphatic filariasis with SCI-funded schistosomiasis and STH programs in six countries</toggledisplay> |} 75b68778d3c969ba486605010f88305742ebb402 623 622 2011-11-23T23:21:53Z Vipul 2 /* Summary of who they are and what they do */ wikitext text/x-wiki ==Summary== ===Evaluation on key parameters=== '''Schistosomiasis Control Initiative''' ('''SCI''') is a recommended organization. {| class="sortable" border="1" ! Parameter !! One-word summary !! More details (section in this page) |- | Evidence of effectiveness || Strong || [[#Independent evidence of program effectiveness]] |- | Cost-effectiveness || Excellent || [[#Independent evidence of program effectiveness]] |- | Funding gap aka room for more funding || Significant || [[#Room for more funding]] |- | Transparency || ? || ? |- | Monitoring and evaluation || Strong || [[#Internal monitoring: large-scale programs]] |} ===Summary of who they are and what they do=== {| class="sortable" border="1" ! Item !! Value |- | Countries of operation || 8-10 African countries |- | Headquarters || Imperial College, London, United Kingdom |- | Goals || Create or scale up [[mass drug administration program]]s for [[neglected tropical diseases]] (NTD), particularly [[schistosomiasis]] and [[soil-transmitted helminth]]s (STHs), in school-aged children and other groups determined to be at high risk. |- | Method summary || Advocate for the benefits of mass drug distributions to government officials.<br>Assist with planning and fund raising.<br>Deliver funding and drugs to governments.<br>Provide financial management and technical support.<br>Develop procedures for monitoring and evaluation. |- | Previous sources of funding || grant from [[Gates Foundation]] (private philanthropic foundation)<br>grants from [[USAID]] (US federal grantmaking agency)<br>grants from [[DFID]] (UK central grantmaking agency) |} ===Review process=== The review of SCI began in 2009. Below are details of the kind of inputs used in the review process. {| class="sortable" border="1" ! What we sought to learn !! How we learned it |- | Effectiveness and other effects of these kinds of programs || Study of the published literature |- | Details of the big picture of SCI's operations || Extensive communications with SCI Director Alan Fenwick to discuss SCI's methods and funding needs<br>meetings with other staff at SCI's headquarters in London<br>Requesting and reviewing SCI internal financial and organizational documents. |- | Feel for how SCI's programs are run on the ground || Visiting a national schistosomiasis control program meeting and demonstration mass drug administration in Malawi (site visit: October 2011) |} ==Operations and history== Note to GiveWell staff: ignore the red links, they are meant to be stand-ins for your current internal links to pages. {{quotation|To understand this section better, follow the links to get more background information on these topics:<br>[[neglected tropical diseases]] (NTDs): diseases that currently afflict people largely in low-income tropical areas, particularly Africa, and are ''neglected'' by most research efforts in the developed world because they do not cause much trouble in developed world countries.<br>List of NTDs mentioned here: [[schistosomiasis]], [[soil-transmitted helminth]]s (acronym '''STH'''s), [[lymphatic filariasis]], [[onchocerciasis]], and [[trachoma]].<br>[[mass drug administration program]]: {{fillin}} }} ===Broad description of goals and methods=== SCI's goal is to create or scale up [[mass drug administration program]]s for [[neglected tropical diseases]] (NTD), particularly [[schistosomiasis]] and [[soil-transmitted helminth]]s (STHs), in school-aged children and other groups determined to be at high risk. They list the following five methods: * Advocate for the benefits of mass drug distributions to government officials in the African countries * Assist with planning and fund raising * Deliver funding and drugs to governments * Provide financial management and technical support * Develop procedures for monitoring and evaluation ===History of operations and funding=== Below are some of the important funders and details of the programs funded (the ?s reflect things that weren't clear from GiveWell's page but can be dug up from the original sources, I'm sure): {| class="sortable" border="1" ! Funder !! Year of funding !! Amount of funding !! Neglected Tropical Diseases (NTDs) tackled !! Countries tackled !! Expiry of grant !! Additional comments |- | [[Gates Foundation]] (private philanthropic foundation that makes grants for developing world health charities) || 2002 || $32 million || schistosomiasis, STHs || six African countries: {{fillin}} || unspecified || This was a founding grant. |- | [[USAID]] (US federal grantmaking agency) and [[Gates Foundation]] || 2006 || ? || schistosomiasis, STHs, lymphatic filariasis, onchocerciasis, and trachoma || eight African countries: {{fillin}} || Till 2011 (five year grant) || |- | ? || 2007 || ? || schistosomiasis, STHs, lymphatic filariasis, onchocerciasis, and trachoma || Rwanda and Burundi || Till 2011 || This was intended to expand the program from the eight current countries of operation to these two new countries. |- | [[DFID]] (UK central grantmaking agency) || 2010 || ? || schistosomiasis, STHs || eight African countries (same eight?) || Till 2015 (?), five year grant || Other NTDs are not covered by the grant, <toggledisplay>though DFID also provided funding to the Liverpool School of Tropical Medicine to integrate treatment for lymphatic filariasis with SCI-funded schistosomiasis and STH programs in six countries</toggledisplay> |} 74e6a8c68406b622f0dc0ce73bcb1b67b348c5b3 624 623 2011-11-23T23:22:58Z Vipul 2 /* Review process */ wikitext text/x-wiki ==Summary== ===Evaluation on key parameters=== '''Schistosomiasis Control Initiative''' ('''SCI''') is a recommended organization. {| class="sortable" border="1" ! Parameter !! One-word summary !! More details (section in this page) |- | Evidence of effectiveness || Strong || [[#Independent evidence of program effectiveness]] |- | Cost-effectiveness || Excellent || [[#Independent evidence of program effectiveness]] |- | Funding gap aka room for more funding || Significant || [[#Room for more funding]] |- | Transparency || ? || ? |- | Monitoring and evaluation || Strong || [[#Internal monitoring: large-scale programs]] |} ===Summary of who they are and what they do=== {| class="sortable" border="1" ! Item !! Value |- | Countries of operation || 8-10 African countries |- | Headquarters || Imperial College, London, United Kingdom |- | Goals || Create or scale up [[mass drug administration program]]s for [[neglected tropical diseases]] (NTD), particularly [[schistosomiasis]] and [[soil-transmitted helminth]]s (STHs), in school-aged children and other groups determined to be at high risk. |- | Method summary || Advocate for the benefits of mass drug distributions to government officials.<br>Assist with planning and fund raising.<br>Deliver funding and drugs to governments.<br>Provide financial management and technical support.<br>Develop procedures for monitoring and evaluation. |- | Previous sources of funding || grant from [[Gates Foundation]] (private philanthropic foundation)<br>grants from [[USAID]] (US federal grantmaking agency)<br>grants from [[DFID]] (UK central grantmaking agency) |} ===Review process=== The review of SCI began in 2009. Below are details of the kind of inputs used in the review process. {| class="sortable" border="1" ! What we sought to learn !! How we learned it |- | Effectiveness and other effects of these kinds of programs || Study of the published literature |- | Details and big picture of SCI's operations || Extensive communications with SCI Director Alan Fenwick to discuss SCI's methods and funding needs<br>meetings with other staff at SCI's headquarters in London<br>Requesting and reviewing SCI internal financial and organizational documents. |- | Feel for how SCI's programs are run on the ground || Visiting a national schistosomiasis control program meeting and demonstration mass drug administration in Malawi (site visit: October 2011) |} ==Operations and history== Note to GiveWell staff: ignore the red links, they are meant to be stand-ins for your current internal links to pages. {{quotation|To understand this section better, follow the links to get more background information on these topics:<br>[[neglected tropical diseases]] (NTDs): diseases that currently afflict people largely in low-income tropical areas, particularly Africa, and are ''neglected'' by most research efforts in the developed world because they do not cause much trouble in developed world countries.<br>List of NTDs mentioned here: [[schistosomiasis]], [[soil-transmitted helminth]]s (acronym '''STH'''s), [[lymphatic filariasis]], [[onchocerciasis]], and [[trachoma]].<br>[[mass drug administration program]]: {{fillin}} }} ===Broad description of goals and methods=== SCI's goal is to create or scale up [[mass drug administration program]]s for [[neglected tropical diseases]] (NTD), particularly [[schistosomiasis]] and [[soil-transmitted helminth]]s (STHs), in school-aged children and other groups determined to be at high risk. They list the following five methods: * Advocate for the benefits of mass drug distributions to government officials in the African countries * Assist with planning and fund raising * Deliver funding and drugs to governments * Provide financial management and technical support * Develop procedures for monitoring and evaluation ===History of operations and funding=== Below are some of the important funders and details of the programs funded (the ?s reflect things that weren't clear from GiveWell's page but can be dug up from the original sources, I'm sure): {| class="sortable" border="1" ! Funder !! Year of funding !! Amount of funding !! Neglected Tropical Diseases (NTDs) tackled !! Countries tackled !! Expiry of grant !! Additional comments |- | [[Gates Foundation]] (private philanthropic foundation that makes grants for developing world health charities) || 2002 || $32 million || schistosomiasis, STHs || six African countries: {{fillin}} || unspecified || This was a founding grant. |- | [[USAID]] (US federal grantmaking agency) and [[Gates Foundation]] || 2006 || ? || schistosomiasis, STHs, lymphatic filariasis, onchocerciasis, and trachoma || eight African countries: {{fillin}} || Till 2011 (five year grant) || |- | ? || 2007 || ? || schistosomiasis, STHs, lymphatic filariasis, onchocerciasis, and trachoma || Rwanda and Burundi || Till 2011 || This was intended to expand the program from the eight current countries of operation to these two new countries. |- | [[DFID]] (UK central grantmaking agency) || 2010 || ? || schistosomiasis, STHs || eight African countries (same eight?) || Till 2015 (?), five year grant || Other NTDs are not covered by the grant, <toggledisplay>though DFID also provided funding to the Liverpool School of Tropical Medicine to integrate treatment for lymphatic filariasis with SCI-funded schistosomiasis and STH programs in six countries</toggledisplay> |} 311885c47b0b0bae8909a11fcbd10a38e6ab9182 625 624 2011-11-23T23:43:07Z Vipul 2 wikitext text/x-wiki ==Summary== ===Evaluation on key parameters=== '''Schistosomiasis Control Initiative''' ('''SCI''') is a recommended organization. {| class="sortable" border="1" ! Parameter !! One-word summary !! More details (section in this page) |- | Evidence of effectiveness || Strong || [[#Independent evidence of program effectiveness]] |- | Cost-effectiveness || Excellent || [[#Independent evidence of program effectiveness]] |- | Funding gap aka room for more funding || Significant || [[#Room for more funding]] |- | Transparency || ? || ? |- | Monitoring and evaluation || Strong || [[#Internal monitoring: large-scale programs]] |} ===Summary of who they are and what they do=== {| class="sortable" border="1" ! Item !! Value |- | Countries of operation || 8-10 African countries |- | Headquarters || Imperial College, London, United Kingdom |- | Goals || Create or scale up [[mass drug administration program]]s for [[neglected tropical diseases]] (NTD), particularly [[schistosomiasis]] and [[soil-transmitted helminth]]s (STHs), in school-aged children and other groups determined to be at high risk. |- | Method summary || Advocate for the benefits of mass drug distributions to government officials.<br>Assist with planning and fund raising.<br>Deliver funding and drugs to governments.<br>Provide financial management and technical support.<br>Develop procedures for monitoring and evaluation. |- | Previous sources of funding || grant from [[Gates Foundation]] (private philanthropic foundation)<br>grants from [[USAID]] (US federal grantmaking agency)<br>grants from [[DFID]] (UK central grantmaking agency) |} ===Review process=== The review of SCI began in 2009. Below are details of the kind of inputs used in the review process. {| class="sortable" border="1" ! What we sought to learn !! How we learned it |- | Effectiveness and other effects of these kinds of programs || Study of the published literature |- | Details and big picture of SCI's operations || Extensive communications with SCI Director Alan Fenwick to discuss SCI's methods and funding needs<br>meetings with other staff at SCI's headquarters in London<br>Requesting and reviewing SCI internal financial and organizational documents. |- | Feel for how SCI's programs are run on the ground || Visiting a national schistosomiasis control program meeting and demonstration mass drug administration in Malawi (site visit: October 2011) |} ==Operations and history== Note to GiveWell staff: ignore the red links, they are meant to be stand-ins for your current internal links to pages. {{quotation|To understand this section better, follow the links to get more background information on these topics:<br>[[neglected tropical diseases]] (NTDs): diseases that currently afflict people largely in low-income tropical areas, particularly Africa, and are ''neglected'' by most research efforts in the developed world because they do not cause much trouble in developed world countries.<br>List of NTDs mentioned here: [[schistosomiasis]], [[soil-transmitted helminth]]s (acronym '''STH'''s), [[lymphatic filariasis]], [[onchocerciasis]], and [[trachoma]].<br>[[mass drug administration program]]: {{fillin}} }} ===Broad description of goals and methods=== SCI's goal is to create or scale up [[mass drug administration program]]s for [[neglected tropical diseases]] (NTD), particularly [[schistosomiasis]] and [[soil-transmitted helminth]]s (STHs), in school-aged children and other groups determined to be at high risk. They list the following five methods: * Advocate for the benefits of mass drug distributions to government officials in the African countries * Assist with planning and fund raising * Deliver funding and drugs to governments * Provide financial management and technical support * Develop procedures for monitoring and evaluation ===History of operations and funding=== Below are some of the important funders and details of the programs funded (the ?s reflect things that weren't clear from GiveWell's page but can be dug up from the original sources, I'm sure): {| class="sortable" border="1" ! Funder !! Year of funding !! Amount of funding !! Neglected Tropical Diseases (NTDs) tackled !! Countries tackled !! Expiry of grant !! Additional comments |- | [[Gates Foundation]] (private philanthropic foundation that makes grants for developing world health charities) || 2002 || $32 million || schistosomiasis, STHs || six African countries: {{fillin}} || unspecified || This was a founding grant. |- | [[USAID]] (US federal grantmaking agency) and [[Gates Foundation]] || 2006 || ? || schistosomiasis, STHs, lymphatic filariasis, onchocerciasis, and trachoma || eight African countries: {{fillin}} || Till 2011 (five year grant) || |- | ? || 2007 || ? || schistosomiasis, STHs, lymphatic filariasis, onchocerciasis, and trachoma || Rwanda and Burundi || Till 2011 || This was intended to expand the program from the eight current countries of operation to these two new countries. |- | [[DFID]] (UK central grantmaking agency) || 2010 || ? || schistosomiasis, STHs || eight African countries (same eight?) || Till 2015 (?), five year grant || Other NTDs are not covered by the grant, <toggledisplay>though DFID also provided funding to the Liverpool School of Tropical Medicine to integrate treatment for lymphatic filariasis with SCI-funded schistosomiasis and STH programs in six countries</toggledisplay> |} ===Other projects=== {{fillin}} -- long and tedious ===Overall spending beakdown=== {{fillin}} -- long and tedious ==Does it work?== ===Independent evidence of program effectiveness=== {{fillin}} 361bab5ec20e829d1ce87454c70a41b7887eb69c 639 625 2013-02-19T07:30:11Z Vipul 2 wikitext text/x-wiki ==Summary== ===Evaluation on key parameters=== '''Schistosomiasis Control Initiative''' ('''SCI''') is a recommended organization. {| class="sortable" border="1" ! Parameter !! One-word summary !! More details (section in this page) |- | Evidence of effectiveness || Strong || [[#Independent evidence of program effectiveness]] |- | Cost-effectiveness || Excellent || [[#Independent evidence of program effectiveness]] |- | Funding gap aka room for more funding || Significant || [[#Room for more funding]] |- | Transparency || ? || ? |- | Monitoring and evaluation || Strong || [[#Internal monitoring: large-scale programs]] |} ===Summary of who they are and what they do=== {| class="sortable" border="1" ! Item !! Value |- | Countries of operation || 8-10 African countries (which countries? Difficult to glean) |- | Headquarters || Imperial College, London, United Kingdom |- | Goals || Create or scale up [[mass drug administration program]]s for [[neglected tropical diseases]] (NTD), particularly [[schistosomiasis]] and [[soil-transmitted helminth]]s (STHs), in school-aged children and other groups determined to be at high risk. |- | Method summary || Advocate for the benefits of mass drug distributions to government officials.<br>Assist with planning and fund raising.<br>Deliver funding and drugs to governments.<br>Provide financial management and technical support.<br>Develop procedures for monitoring and evaluation. |- | Previous sources of funding || grant from [[Gates Foundation]] (private philanthropic foundation)<br>grants from [[USAID]] (US federal grantmaking agency)<br>grants from [[DFID]] (UK central grantmaking agency). More in the [[#History of operations and funding]] section. |} ===Review process=== The review of SCI began in 2009. Below are details of the kind of inputs used in the review process. {| class="sortable" border="1" ! What we sought to learn !! How we learned it |- | Effectiveness and other effects of these kinds of programs || Study of the published literature |- | Details and big picture of SCI's operations || Extensive communications with SCI Director Alan Fenwick to discuss SCI's methods and funding needs<br>meetings with other staff at SCI's headquarters in London<br>Requesting and reviewing SCI internal financial and organizational documents. |- | Feel for how SCI's programs are run on the ground || Visiting a national schistosomiasis control program meeting and demonstration mass drug administration in Malawi (site visit: October 2011) |} ==Operations and history== {{quotation|To understand this section better, follow the links to get more background information on these topics:<br>[[neglected tropical diseases]] (NTDs): diseases that currently afflict people largely in low-income tropical areas, particularly Africa, and are ''neglected'' by most research efforts in the developed world because they do not cause much trouble in developed world countries.<br>List of NTDs mentioned here: [[schistosomiasis]], [[soil-transmitted helminth]]s (acronym '''STH'''s), [[lymphatic filariasis]], [[onchocerciasis]], and [[trachoma]].<br>[[mass drug administration program]]: {{fillin}} }} ===Broad description of goals and methods=== SCI's goal is to create or scale up [[mass drug administration program]]s for [[neglected tropical diseases]] (NTD), particularly [[schistosomiasis]] and [[soil-transmitted helminth]]s (STHs), in school-aged children and other groups determined to be at high risk. They list the following five methods: * Advocate for the benefits of mass drug distributions to government officials in the African countries * Assist with planning and fund raising * Deliver funding and drugs to governments * Provide financial management and technical support * Develop procedures for monitoring and evaluation ===History of operations and funding=== Below are some of the important funders and details of the programs funded (the ?s reflect things that weren't clear from GiveWell's page but can be dug up from the original sources, I'm sure): {| class="sortable" border="1" ! Funder !! Year of funding !! Amount of funding !! Neglected Tropical Diseases (NTDs) tackled !! Countries tackled !! Expiry of grant !! Additional comments |- | [[Gates Foundation]] (private philanthropic foundation that makes grants for developing world health charities) || 2002 || $32 million || schistosomiasis, STHs || six African countries: {{fillin}} || unspecified || This was a founding grant. |- | [[USAID]] (US federal grantmaking agency) and [[Gates Foundation]] || 2006 || ? || schistosomiasis, STHs, lymphatic filariasis, onchocerciasis, and trachoma || eight African countries: {{fillin}} || Till 2011 (five year grant) || |- | ? || 2007 || ? || schistosomiasis, STHs, lymphatic filariasis, onchocerciasis, and trachoma || Rwanda and Burundi || Till 2011 || This was intended to expand the program from the eight current countries of operation to these two new countries. |- | [[DFID]] (UK central grantmaking agency) || 2010 || ? || schistosomiasis, STHs || eight African countries (same eight?) || Till 2015 (?), five year grant || Other NTDs are not covered by the grant, <toggledisplay>though DFID also provided funding to the Liverpool School of Tropical Medicine to integrate treatment for lymphatic filariasis with SCI-funded schistosomiasis and STH programs in six countries</toggledisplay> |- ! Total !! -- !! add up !! -- !! -- |} ===Other projects=== {{fillin}} -- long and tedious ===Overall spending breakdown=== {{fillin}} -- long and tedious ==Does it work?== ===Independent evidence of program effectiveness=== {{fillin}} b904369f924b22ae0e077409a8fd036fd1e362d3 640 639 2013-02-19T07:33:19Z Vipul 2 /* History of operations and funding */ wikitext text/x-wiki ==Summary== ===Evaluation on key parameters=== '''Schistosomiasis Control Initiative''' ('''SCI''') is a recommended organization. {| class="sortable" border="1" ! Parameter !! One-word summary !! More details (section in this page) |- | Evidence of effectiveness || Strong || [[#Independent evidence of program effectiveness]] |- | Cost-effectiveness || Excellent || [[#Independent evidence of program effectiveness]] |- | Funding gap aka room for more funding || Significant || [[#Room for more funding]] |- | Transparency || ? || ? |- | Monitoring and evaluation || Strong || [[#Internal monitoring: large-scale programs]] |} ===Summary of who they are and what they do=== {| class="sortable" border="1" ! Item !! Value |- | Countries of operation || 8-10 African countries (which countries? Difficult to glean) |- | Headquarters || Imperial College, London, United Kingdom |- | Goals || Create or scale up [[mass drug administration program]]s for [[neglected tropical diseases]] (NTD), particularly [[schistosomiasis]] and [[soil-transmitted helminth]]s (STHs), in school-aged children and other groups determined to be at high risk. |- | Method summary || Advocate for the benefits of mass drug distributions to government officials.<br>Assist with planning and fund raising.<br>Deliver funding and drugs to governments.<br>Provide financial management and technical support.<br>Develop procedures for monitoring and evaluation. |- | Previous sources of funding || grant from [[Gates Foundation]] (private philanthropic foundation)<br>grants from [[USAID]] (US federal grantmaking agency)<br>grants from [[DFID]] (UK central grantmaking agency). More in the [[#History of operations and funding]] section. |} ===Review process=== The review of SCI began in 2009. Below are details of the kind of inputs used in the review process. {| class="sortable" border="1" ! What we sought to learn !! How we learned it |- | Effectiveness and other effects of these kinds of programs || Study of the published literature |- | Details and big picture of SCI's operations || Extensive communications with SCI Director Alan Fenwick to discuss SCI's methods and funding needs<br>meetings with other staff at SCI's headquarters in London<br>Requesting and reviewing SCI internal financial and organizational documents. |- | Feel for how SCI's programs are run on the ground || Visiting a national schistosomiasis control program meeting and demonstration mass drug administration in Malawi (site visit: October 2011) |} ==Operations and history== {{quotation|To understand this section better, follow the links to get more background information on these topics:<br>[[neglected tropical diseases]] (NTDs): diseases that currently afflict people largely in low-income tropical areas, particularly Africa, and are ''neglected'' by most research efforts in the developed world because they do not cause much trouble in developed world countries.<br>List of NTDs mentioned here: [[schistosomiasis]], [[soil-transmitted helminth]]s (acronym '''STH'''s), [[lymphatic filariasis]], [[onchocerciasis]], and [[trachoma]].<br>[[mass drug administration program]]: {{fillin}} }} ===Broad description of goals and methods=== SCI's goal is to create or scale up [[mass drug administration program]]s for [[neglected tropical diseases]] (NTD), particularly [[schistosomiasis]] and [[soil-transmitted helminth]]s (STHs), in school-aged children and other groups determined to be at high risk. They list the following five methods: * Advocate for the benefits of mass drug distributions to government officials in the African countries * Assist with planning and fund raising * Deliver funding and drugs to governments * Provide financial management and technical support * Develop procedures for monitoring and evaluation ===History of operations and funding=== Below are some of the important funders and details of the programs funded (the ?s reflect things that weren't clear from GiveWell's page but can be dug up from the original sources, I'm sure): {| class="sortable" border="1" ! Funder !! Year of funding !! Amount of funding !! Neglected Tropical Diseases (NTDs) tackled !! Countries tackled !! Expiry of grant !! Additional comments |- | [[Gates Foundation]] (private philanthropic foundation that makes grants for developing world health charities) || 2002 || $32 million || schistosomiasis, STHs || six African countries: {{fillin}} || unspecified || This was a founding grant. |- | [[USAID]] (US federal grantmaking agency) and [[Gates Foundation]] || 2006 || ? || schistosomiasis, STHs, lymphatic filariasis, onchocerciasis, and trachoma || eight African countries: {{fillin}} || Till 2011 (five year grant) || |- | ? || 2007 || ? || schistosomiasis, STHs, lymphatic filariasis, onchocerciasis, and trachoma || Rwanda and Burundi || Till 2011 || This was intended to expand the program from the eight current countries of operation to these two new countries. |- | [[DFID]] (UK central grantmaking agency) || 2010 || ? || schistosomiasis, STHs || eight African countries (same eight?) || Till 2015 (?), five year grant || Other NTDs are not covered by the grant, <toggledisplay>though DFID also provided funding to the Liverpool School of Tropical Medicine to integrate treatment for lymphatic filariasis with SCI-funded schistosomiasis and STH programs in six countries</toggledisplay> |- ! Total !! -- !! add up !! -- !! -- !! -- !! -- |} ===Other projects=== {{fillin}} -- long and tedious ===Overall spending breakdown=== {{fillin}} -- long and tedious ==Does it work?== ===Independent evidence of program effectiveness=== {{fillin}} e4ab4384aab3c6f08eb8bde84cb312c0fe36945e 641 640 2013-02-19T07:36:23Z Vipul 2 /* History of operations and funding */ wikitext text/x-wiki ==Summary== ===Evaluation on key parameters=== '''Schistosomiasis Control Initiative''' ('''SCI''') is a recommended organization. {| class="sortable" border="1" ! Parameter !! One-word summary !! More details (section in this page) |- | Evidence of effectiveness || Strong || [[#Independent evidence of program effectiveness]] |- | Cost-effectiveness || Excellent || [[#Independent evidence of program effectiveness]] |- | Funding gap aka room for more funding || Significant || [[#Room for more funding]] |- | Transparency || ? || ? |- | Monitoring and evaluation || Strong || [[#Internal monitoring: large-scale programs]] |} ===Summary of who they are and what they do=== {| class="sortable" border="1" ! Item !! Value |- | Countries of operation || 8-10 African countries (which countries? Difficult to glean) |- | Headquarters || Imperial College, London, United Kingdom |- | Goals || Create or scale up [[mass drug administration program]]s for [[neglected tropical diseases]] (NTD), particularly [[schistosomiasis]] and [[soil-transmitted helminth]]s (STHs), in school-aged children and other groups determined to be at high risk. |- | Method summary || Advocate for the benefits of mass drug distributions to government officials.<br>Assist with planning and fund raising.<br>Deliver funding and drugs to governments.<br>Provide financial management and technical support.<br>Develop procedures for monitoring and evaluation. |- | Previous sources of funding || grant from [[Gates Foundation]] (private philanthropic foundation)<br>grants from [[USAID]] (US federal grantmaking agency)<br>grants from [[DFID]] (UK central grantmaking agency). More in the [[#History of operations and funding]] section. |} ===Review process=== The review of SCI began in 2009. Below are details of the kind of inputs used in the review process. {| class="sortable" border="1" ! What we sought to learn !! How we learned it |- | Effectiveness and other effects of these kinds of programs || Study of the published literature |- | Details and big picture of SCI's operations || Extensive communications with SCI Director Alan Fenwick to discuss SCI's methods and funding needs<br>meetings with other staff at SCI's headquarters in London<br>Requesting and reviewing SCI internal financial and organizational documents. |- | Feel for how SCI's programs are run on the ground || Visiting a national schistosomiasis control program meeting and demonstration mass drug administration in Malawi (site visit: October 2011) |} ==Operations and history== {{quotation|To understand this section better, follow the links to get more background information on these topics:<br>[[neglected tropical diseases]] (NTDs): diseases that currently afflict people largely in low-income tropical areas, particularly Africa, and are ''neglected'' by most research efforts in the developed world because they do not cause much trouble in developed world countries.<br>List of NTDs mentioned here: [[schistosomiasis]], [[soil-transmitted helminth]]s (acronym '''STH'''s), [[lymphatic filariasis]], [[onchocerciasis]], and [[trachoma]].<br>[[mass drug administration program]]: {{fillin}} }} ===Broad description of goals and methods=== SCI's goal is to create or scale up [[mass drug administration program]]s for [[neglected tropical diseases]] (NTD), particularly [[schistosomiasis]] and [[soil-transmitted helminth]]s (STHs), in school-aged children and other groups determined to be at high risk. They list the following five methods: * Advocate for the benefits of mass drug distributions to government officials in the African countries * Assist with planning and fund raising * Deliver funding and drugs to governments * Provide financial management and technical support * Develop procedures for monitoring and evaluation ===History of operations and funding=== Below are some of the important funders and details of the programs funded (the ?s reflect things that weren't clear from GiveWell's page but can be dug up from the original sources, I'm sure): {| class="sortable" border="1" ! Funder !! Year of funding !! Amount of funding !! Neglected Tropical Diseases (NTDs) tackled !! Countries tackled !! Expiry of grant !! Additional comments |- | [[Gates Foundation]] (private philanthropic foundation that makes grants for developing world health charities) || 2002 || $32 million || schistosomiasis, STHs || six African countries: <toggledisplay>Uganda, Burkina Faso, Niger, Mali, Tanzania, Zambia</toggledisplay> || unspecified || This was a founding grant. |- | [[USAID]] (US federal grantmaking agency) and [[Gates Foundation]] || 2006 || ? || schistosomiasis, STHs, lymphatic filariasis, onchocerciasis, and trachoma || eight African countries (or six? inconsistences between footnote and data) || Till 2011 (five year grant) || |- | ? || 2007 || ? || schistosomiasis, STHs, lymphatic filariasis, onchocerciasis, and trachoma || Rwanda and Burundi || Till 2011 || This was intended to expand the program from the eight current countries of operation to these two new countries. |- | [[DFID]] (UK central grantmaking agency) || 2010 || ? || schistosomiasis, STHs || eight African countries (same eight?) || Till 2015 (?), five year grant || Other NTDs are not covered by the grant, <toggledisplay>though DFID also provided funding to the Liverpool School of Tropical Medicine to integrate treatment for lymphatic filariasis with SCI-funded schistosomiasis and STH programs in six countries</toggledisplay> |- ! Total !! -- !! add up !! -- !! -- !! -- !! -- |} ===Other projects=== {{fillin}} -- long and tedious ===Overall spending breakdown=== {{fillin}} -- long and tedious ==Does it work?== ===Independent evidence of program effectiveness=== {{fillin}} 8b7f5bec210181b4a1990ae01942675b06e98b05 642 641 2013-02-19T07:48:06Z Vipul 2 /* Summary of who they are and what they do */ wikitext text/x-wiki ==Summary== ===Evaluation on key parameters=== '''Schistosomiasis Control Initiative''' ('''SCI''') is a recommended organization. {| class="sortable" border="1" ! Parameter !! One-word summary !! More details (section in this page) |- | Evidence of effectiveness || Strong || [[#Independent evidence of program effectiveness]] |- | Cost-effectiveness || Excellent || [[#Independent evidence of program effectiveness]] |- | Funding gap aka room for more funding || Significant || [[#Room for more funding]] |- | Transparency || ? || ? |- | Monitoring and evaluation || Strong || [[#Internal monitoring: large-scale programs]] |} ===Summary of who they are and what they do=== {| class="sortable" border="1" ! Item !! Value |- | Countries of operation || 8-10 African countries: Uganda, Burkina Faso, Niger, Mali, Tanzania, Zambia, Rwanda, Burundi (any more?) |- | Headquarters || Imperial College, London, United Kingdom |- | Goals || Create or scale up [[mass drug administration program]]s for [[neglected tropical diseases]] (NTD), particularly [[schistosomiasis]] and [[soil-transmitted helminth]]s (STHs), in school-aged children and other groups determined to be at high risk. |- | Method summary || Advocate for the benefits of mass drug distributions to government officials.<br>Assist with planning and fund raising.<br>Deliver funding and drugs to governments.<br>Provide financial management and technical support.<br>Develop procedures for monitoring and evaluation. |- | Previous sources of funding || grant from [[Gates Foundation]] (private philanthropic foundation)<br>grants from [[USAID]] (US federal grantmaking agency)<br>grants from [[DFID]] (UK central grantmaking agency). More in the [[#History of operations and funding]] section. |} ===Review process=== The review of SCI began in 2009. Below are details of the kind of inputs used in the review process. {| class="sortable" border="1" ! What we sought to learn !! How we learned it |- | Effectiveness and other effects of these kinds of programs || Study of the published literature |- | Details and big picture of SCI's operations || Extensive communications with SCI Director Alan Fenwick to discuss SCI's methods and funding needs<br>meetings with other staff at SCI's headquarters in London<br>Requesting and reviewing SCI internal financial and organizational documents. |- | Feel for how SCI's programs are run on the ground || Visiting a national schistosomiasis control program meeting and demonstration mass drug administration in Malawi (site visit: October 2011) |} ==Operations and history== {{quotation|To understand this section better, follow the links to get more background information on these topics:<br>[[neglected tropical diseases]] (NTDs): diseases that currently afflict people largely in low-income tropical areas, particularly Africa, and are ''neglected'' by most research efforts in the developed world because they do not cause much trouble in developed world countries.<br>List of NTDs mentioned here: [[schistosomiasis]], [[soil-transmitted helminth]]s (acronym '''STH'''s), [[lymphatic filariasis]], [[onchocerciasis]], and [[trachoma]].<br>[[mass drug administration program]]: {{fillin}} }} ===Broad description of goals and methods=== SCI's goal is to create or scale up [[mass drug administration program]]s for [[neglected tropical diseases]] (NTD), particularly [[schistosomiasis]] and [[soil-transmitted helminth]]s (STHs), in school-aged children and other groups determined to be at high risk. They list the following five methods: * Advocate for the benefits of mass drug distributions to government officials in the African countries * Assist with planning and fund raising * Deliver funding and drugs to governments * Provide financial management and technical support * Develop procedures for monitoring and evaluation ===History of operations and funding=== Below are some of the important funders and details of the programs funded (the ?s reflect things that weren't clear from GiveWell's page but can be dug up from the original sources, I'm sure): {| class="sortable" border="1" ! Funder !! Year of funding !! Amount of funding !! Neglected Tropical Diseases (NTDs) tackled !! Countries tackled !! Expiry of grant !! Additional comments |- | [[Gates Foundation]] (private philanthropic foundation that makes grants for developing world health charities) || 2002 || $32 million || schistosomiasis, STHs || six African countries: <toggledisplay>Uganda, Burkina Faso, Niger, Mali, Tanzania, Zambia</toggledisplay> || unspecified || This was a founding grant. |- | [[USAID]] (US federal grantmaking agency) and [[Gates Foundation]] || 2006 || ? || schistosomiasis, STHs, lymphatic filariasis, onchocerciasis, and trachoma || eight African countries (or six? inconsistences between footnote and data) || Till 2011 (five year grant) || |- | ? || 2007 || ? || schistosomiasis, STHs, lymphatic filariasis, onchocerciasis, and trachoma || Rwanda and Burundi || Till 2011 || This was intended to expand the program from the eight current countries of operation to these two new countries. |- | [[DFID]] (UK central grantmaking agency) || 2010 || ? || schistosomiasis, STHs || eight African countries (same eight?) || Till 2015 (?), five year grant || Other NTDs are not covered by the grant, <toggledisplay>though DFID also provided funding to the Liverpool School of Tropical Medicine to integrate treatment for lymphatic filariasis with SCI-funded schistosomiasis and STH programs in six countries</toggledisplay> |- ! Total !! -- !! add up !! -- !! -- !! -- !! -- |} ===Other projects=== {{fillin}} -- long and tedious ===Overall spending breakdown=== {{fillin}} -- long and tedious ==Does it work?== ===Independent evidence of program effectiveness=== {{fillin}} c94700073780ccc511f3273dd2dd8cb476411fe1 8 21 616 241 2011-11-23T23:06:35Z Vipul 2 Blanked the page wikitext text/x-wiki da39a3ee5e6b4b0d3255bfef95601890afd80709 10 30 617 64 2011-11-23T23:08:59Z Vipul 2 wikitext text/x-wiki <blockquote class="toccolours" style="float:none; background:white; padding: 10px 15px 10px 15px; display:table;"> {{{1<noinclude>|&nbsp;&nbsp;{{Lorem}}</noinclude>}}}</blockquote><noinclude>''This template is copied from Wikipedia (source [[wp:Template:Quotation|here]]) and is released under the GFDL 1.2''</noinclude> 592f99431a24e07fa01c3349fa61613f7d7de818 4 133 628 258 2012-02-16T01:05:19Z Vipul 2 wikitext text/x-wiki This is a common copyright notice to all subject wikis. ==General license information== All content is put up under the [http://creativecommons.org/licenses/by-sa/3.0/ Creative Commons Attribute Share-Alike 3.0 license (unported)] (shorthand: CC-by-SA). 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In case copyright violations are detected, please email vipul.wikis@gmail.com and vipul@math.uchicago.edu to have the matter looked into immediately. 054360ee5bbbbff13953c11db639e62dbec13b9f 2 163 629 574 2012-04-14T23:59:31Z Jon Awbrey 3 update wikitext text/x-wiki <br> <p><font face="lucida calligraphy" size="7">Jon Awbrey</font></p> <br> __NOTOC__ ==Presently &hellip;== <center> [http://oeis.org/wiki/Riffs_and_Rotes R&R] [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems PERS] [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Information_%3D_Comprehension_%C3%97_Extension I = C &times; E] [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Peirce%27s_1870_Logic_Of_Relatives LOR 1870] [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositions_As_Types PAT Analogy] [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Analytic_Turing_Automata DATA Project] [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Cactus_Language Cactus Language] [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Inquiry Driven Systems] Logical Graphs : [http://inquiryintoinquiry.com/2008/07/29/logical-graphs-1/ One] [http://inquiryintoinquiry.com/2008/09/19/logical-graphs-2/ Two] [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Differential Logic : Introduction] [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Propositional_Calculus Differential Propositional Calculus] [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] </center> ==Recent Sightings== {| align="center" style="text-align:center" width="100%" |- | [http://mywikibiz.com/Inquiry_Live Inquiry Live] | [http://mywikibiz.com/Logic_Live Logic Live] |- | [http://stderr.org/cgi-bin/mailman/listinfo/inquiry Inquiry Project] | [http://stderr.org/pipermail/inquiry/ Inquiry Archive] |- | [http://inquiryintoinquiry.com/ Inquiry Into Inquiry] | [http://jonawbrey.wordpress.com/ Jon Awbrey • Blog] |- | [http://mywikibiz.com/Directory:Jon_Awbrey MyWikiBiz Directory] | [http://mywikibiz.com/Directory_talk:Jon_Awbrey MyWikiBiz Discussion] |- | [http://mywikibiz.com/User:Jon_Awbrey MyWikiBiz User Page] | [http://mywikibiz.com/User_talk:Jon_Awbrey MyWikiBiz Talk Page] |- | [http://planetmath.org/ PlanetMath Project] | [http://planetmath.org/?op=getuser&id=15246 PlanetMath Profile] |- | [http://planetphysics.us/ PlanetPhysics Project] | [http://planetphysics.us/?op=getuser&id=513 PlanetPhysics Profile] |- | [http://knol.google.com/ Google Knol Project] | [http://knol.google.com/k/jon-awbrey/jon-awbrey/3fkwvf69kridz/1 Google Knol Profile] |- | [http://mathforum.org/kb/ Math Forum Project] | [http://mathforum.org/kb/accountView.jspa?userID=99854 Math Forum Profile] |- | [http://list.seqfan.eu/cgi-bin/mailman/listinfo/seqfan Fantasia Sequentia] | [http://list.seqfan.eu/pipermail/seqfan/ SeqFan Archive] |- | [http://oeis.org/wiki/Welcome OEIS Land] | [http://oeis.org/search?q=Awbrey Bolgia Mia] |- | [http://oeis.org/wiki/User:Jon_Awbrey OEIS Wiki Page] | [http://oeis.org/wiki/User_talk:Jon_Awbrey OEIS Wiki Talk] |- | [http://intersci.ss.uci.edu/wiki/index.php/User:Jon_Awbrey InterSciWiki Page] | [http://intersci.ss.uci.edu/wiki/index.php/User_talk:Jon_Awbrey InterSciWiki Talk] |- | [http://www.mathweb.org/wiki/User:Jon_Awbrey MathWeb Page] | [http://www.mathweb.org/wiki/User_talk:Jon_Awbrey MathWeb Talk] |- | [http://www.proofwiki.org/wiki/User:Jon_Awbrey ProofWiki Page] | [http://www.proofwiki.org/wiki/User_talk:Jon_Awbrey ProofWiki Talk] |- | [http://mathoverflow.net/ MathOverFlow] | [http://mathoverflow.net/users/1636/jon-awbrey MOFler Profile] |- | [http://p2pfoundation.net/User:JonAwbrey P2P Wiki Page] | [http://p2pfoundation.net/User_talk:JonAwbrey P2P Wiki Talk] |- | [http://vectors.usc.edu/ Vectors Project] | [http://vectors.usc.edu/thoughtmesh/ ThoughtMesh] |- | [http://altheim.4java.ca/ceryle/wiki/ Ceryle Project] | [http://altheim.4java.ca/ceryle/wiki/Wiki.jsp?page=JonAwbrey Ceryle Profile] |- | [http://forum.wolframscience.com/ NKS Forum] | [http://forum.wolframscience.com/member.php?s=&action=getinfo&userid=336 NKS Profile] |- | [http://www.wikinfo.org/index.php/User:Jon_Awbrey WikInfo Page] | [http://www.wikinfo.org/index.php/User_talk:Jon_Awbrey WikInfo Talk] |- | [http://getwiki.net/-User:Jon_Awbrey GetWiki Page] | [http://getwiki.net/-UserTalk:Jon_Awbrey GetWiki Talk] |- | [http://ontolog.cim3.net/ OntoLog Project] | [http://ontolog.cim3.net/cgi-bin/wiki.pl?JonAwbrey OntoLog Profile] |- | [http://semanticweb.org/wiki/User:Jon_Awbrey SemanticWeb Page] | [http://semanticweb.org/wiki/User_talk:Jon_Awbrey SemanticWeb Talk] |- | [http://zh.wikipedia.org/wiki/User:Jon_Awbrey 维基百科 : 用户页面] | [http://zh.wikipedia.org/wiki/User_talk:Jon_Awbrey 维基百科 : 讨论] |} ==Presentations and Publications== * Awbrey, S.M., and Awbrey, J.L. (May 2001), &ldquo;Conceptual Barriers to Creating Integrative Universities&rdquo;, ''Organization : The Interdisciplinary Journal of Organization, Theory, and Society'' 8(2), Sage Publications, London, UK, pp. 269&ndash;284. [http://org.sagepub.com/cgi/content/abstract/8/2/269 Abstract]. * Awbrey, S.M., and Awbrey, J.L. (September 1999), &ldquo;Organizations of Learning or Learning Organizations : The Challenge of Creating Integrative Universities for the Next Century&rdquo;, ''Second International Conference of the Journal &lsquo;Organization&rsquo;'', ''Re-Organizing Knowledge, Trans-Forming Institutions : Knowing, Knowledge, and the University in the 21st Century'', University of Massachusetts, Amherst, MA. [http://cspeirce.com/menu/library/aboutcsp/awbrey/integrat.htm Online]. * Awbrey, J.L., and Awbrey, S.M. (Autumn 1995), &ldquo;Interpretation as Action : The Risk of Inquiry&rdquo;, ''Inquiry : Critical Thinking Across the Disciplines'' 15(1), pp. 40&ndash;52. [http://web.archive.org/web/19970626071826/http://chss.montclair.edu/inquiry/fall95/awbrey.html Online]. * Awbrey, J.L., and Awbrey, S.M. (June 1992), &ldquo;Interpretation as Action : The Risk of Inquiry&rdquo;, ''The Eleventh International Human Science Research Conference'', Oakland University, Rochester, Michigan. * Awbrey, S.M., and Awbrey, J.L. (May 1991), &ldquo;An Architecture for Inquiry : Building Computer Platforms for Discovery&rdquo;, ''Proceedings of the Eighth International Conference on Technology and Education'', Toronto, Canada, pp. 874&ndash;875. [http://abccommunity.org/tmp-a.html Online]. * Awbrey, J.L., and Awbrey, S.M. (January 1991), &ldquo;Exploring Research Data Interactively : Developing a Computer Architecture for Inquiry&rdquo;, Poster presented at the ''Annual Sigma Xi Research Forum'', University of Texas Medical Branch, Galveston, TX. * Awbrey, J.L., and Awbrey, S.M. (August 1990), &ldquo;Exploring Research Data Interactively. Theme One : A Program of Inquiry&rdquo;, ''Proceedings of the Sixth Annual Conference on Applications of Artificial Intelligence and CD-ROM in Education and Training'', Society for Applied Learning Technology, Washington, DC, pp. 9&ndash;15. ==Education== * 1993&ndash;2003. [http://www2.oakland.edu/secs/dispprofile.asp?Fname=Fatma&Lname=Mili Graduate Study], [http://www2.oakland.edu/secs/dispprofile.asp?Fname=Mohamed&Lname=Zohdy Systems Engineering], [http://meadowbrookhall.org/explore/history/meadowbrookhall Oakland University]. * '''1989. [http://web.archive.org/web/20001206101800/http://www.msu.edu/dig/msumap/psychology.html M.A. Psychology]''', [http://web.archive.org/web/20000902103838/http://www.msu.edu/dig/msumap/beaumont.html Michigan State University]. * 1985&ndash;1986. [http://quod.lib.umich.edu/cgi/i/image/image-idx?id=S-BHL-X-BL001808%5DBL001808 Graduate Study, Mathematics], [http://www.umich.edu/ University of Michigan]. * 1985. [http://web.archive.org/web/20061230174652/http://www.uiuc.edu/navigation/buildings/altgeld.top.html Graduate Study, Mathematics], [http://www.uiuc.edu/ University of Illinois at Urbana&ndash;Champaign]. * 1984. [http://www.psych.uiuc.edu/graduate/ Graduate Study, Psychology], [http://www.uiuc.edu/ University of Illinois at Champaign&ndash;Urbana]. * '''1980. [http://www.mth.msu.edu/images/wells_medium.jpg M.A. Mathematics]''', [http://www.msu.edu/~hvac/survey/BeaumontTower.html Michigan State University]. * '''1976. [http://web.archive.org/web/20001206050600/http://www.msu.edu/dig/msumap/phillips.html B.A. Mathematical and Philosophical Method]''', <br> [http://www.enolagaia.com/JMC.html Justin Morrill College], [http://www.msu.edu/ Michigan State University]. 88a71f640742818b42e42bfcc947c754dbde1b38 643 629 2013-10-03T14:10:49Z Jon Awbrey 3 update wikitext text/x-wiki <br> <p><font face="lucida calligraphy" size="7">Jon Awbrey</font></p> <br> __NOTOC__ ==Presently &hellip;== <center> [https://oeis.org/wiki/Riffs_and_Rotes Riffs and Rotes] [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] [http://intersci.ss.uci.edu/wiki/index.php/Theme_One_Program Theme One Program] [http://intersci.ss.uci.edu/wiki/index.php/Propositions_As_Types Propositions As Types] [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems] [http://intersci.ss.uci.edu/wiki/index.php/Peirce%27s_1870_Logic_Of_Relatives Peirce's Logic Of Relatives] Logical Graphs : [http://inquiryintoinquiry.com/2008/07/29/logical-graphs-1/ One] [http://inquiryintoinquiry.com/2008/09/19/logical-graphs-2/ Two] [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] [http://intersci.ss.uci.edu/wiki/index.php/Differential_Analytic_Turing_Automata Differential Analytic Turing Automata] [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] [http://intersci.ss.uci.edu/wiki/index.php/Information_%3D_Comprehension_%C3%97_Extension Information = Comprehension &times; Extension] [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] </center> ==Recent Sightings== {| align="center" style="text-align:center" width="100%" |- | [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Live Inquiry Live] | [http://intersci.ss.uci.edu/wiki/index.php/Logic_Live Logic Live] |- | [http://stderr.org/cgi-bin/mailman/listinfo/inquiry Inquiry Project] | [http://stderr.org/pipermail/inquiry/ Inquiry Archive] |- | [http://inquiryintoinquiry.com/ Inquiry Into Inquiry] | [http://jonawbrey.wordpress.com/ Jon Awbrey • Blog] |- | [http://intersci.ss.uci.edu/wiki/index.php/User:Jon_Awbrey InterSciWiki User Page] | [http://intersci.ss.uci.edu/wiki/index.php/User_talk:Jon_Awbrey InterSciWiki Talk Page] |- | [http://mywikibiz.com/Directory:Jon_Awbrey MyWikiBiz Directory] | [http://mywikibiz.com/Directory_talk:Jon_Awbrey MyWikiBiz Discussion] |- | [http://mywikibiz.com/User:Jon_Awbrey MyWikiBiz User Page] | [http://mywikibiz.com/User_talk:Jon_Awbrey MyWikiBiz Talk Page] |- | [http://planetmath.org/ PlanetMath Project] | [http://planetmath.org/?op=getuser&id=15246 PlanetMath Profile] |- | [http://planetphysics.us/ PlanetPhysics Project] | [http://planetphysics.us/?op=getuser&id=513 PlanetPhysics Profile] |- | [http://knol.google.com/ Google Knol Project] | [http://knol.google.com/k/jon-awbrey/jon-awbrey/3fkwvf69kridz/1 Google Knol Profile] |- | [http://mathforum.org/kb/ Math Forum Project] | [http://mathforum.org/kb/accountView.jspa?userID=99854 Math Forum Profile] |- | [http://list.seqfan.eu/cgi-bin/mailman/listinfo/seqfan Fantasia Sequentia] | [http://list.seqfan.eu/pipermail/seqfan/ SeqFan Archive] |- | [http://oeis.org/wiki/Welcome OEIS Land] | [http://oeis.org/search?q=Awbrey Bolgia Mia] |- | [http://oeis.org/wiki/User:Jon_Awbrey OEIS Wiki Page] | [http://oeis.org/wiki/User_talk:Jon_Awbrey OEIS Wiki Talk] |- | [http://www.mathweb.org/wiki/User:Jon_Awbrey MathWeb Page] | [http://www.mathweb.org/wiki/User_talk:Jon_Awbrey MathWeb Talk] |- | [http://www.proofwiki.org/wiki/User:Jon_Awbrey ProofWiki Page] | [http://www.proofwiki.org/wiki/User_talk:Jon_Awbrey ProofWiki Talk] |- | [http://mathoverflow.net/ MathOverFlow] | [http://mathoverflow.net/users/1636/jon-awbrey MOFler Profile] |- | [http://p2pfoundation.net/User:JonAwbrey P2P Wiki Page] | [http://p2pfoundation.net/User_talk:JonAwbrey P2P Wiki Talk] |- | [http://vectors.usc.edu/ Vectors Project] | [http://vectors.usc.edu/thoughtmesh/ ThoughtMesh] |- | [http://forum.wolframscience.com/ NKS Forum] | [http://forum.wolframscience.com/member.php?s=&action=getinfo&userid=336 NKS Profile] |- | [http://www.wikinfo.org/index.php/User:Jon_Awbrey WikInfo Page] | [http://www.wikinfo.org/index.php/User_talk:Jon_Awbrey WikInfo Talk] |- | [http://getwiki.net/-User:Jon_Awbrey GetWiki Page] | [http://getwiki.net/-UserTalk:Jon_Awbrey GetWiki Talk] |- | [http://ontolog.cim3.net/ OntoLog Project] | [http://ontolog.cim3.net/cgi-bin/wiki.pl?JonAwbrey OntoLog Profile] |- | [http://semanticweb.org/wiki/User:Jon_Awbrey SemanticWeb Page] | [http://semanticweb.org/wiki/User_talk:Jon_Awbrey SemanticWeb Talk] |- | [http://zh.wikipedia.org/wiki/User:Jon_Awbrey 维基百科 : 用户页面] | [http://zh.wikipedia.org/wiki/User_talk:Jon_Awbrey 维基百科 : 讨论] |} ==Presentations and Publications== * Awbrey, S.M., and Awbrey, J.L. (May 2001), &ldquo;Conceptual Barriers to Creating Integrative Universities&rdquo;, ''Organization : The Interdisciplinary Journal of Organization, Theory, and Society'' 8(2), Sage Publications, London, UK, pp. 269&ndash;284. [http://org.sagepub.com/cgi/content/abstract/8/2/269 Abstract]. * Awbrey, S.M., and Awbrey, J.L. (September 1999), &ldquo;Organizations of Learning or Learning Organizations : The Challenge of Creating Integrative Universities for the Next Century&rdquo;, ''Second International Conference of the Journal &lsquo;Organization&rsquo;'', ''Re-Organizing Knowledge, Trans-Forming Institutions : Knowing, Knowledge, and the University in the 21st Century'', University of Massachusetts, Amherst, MA. [http://cspeirce.com/menu/library/aboutcsp/awbrey/integrat.htm Online]. * Awbrey, J.L., and Awbrey, S.M. (Autumn 1995), &ldquo;Interpretation as Action : The Risk of Inquiry&rdquo;, ''Inquiry : Critical Thinking Across the Disciplines'' 15(1), pp. 40&ndash;52. [http://web.archive.org/web/19970626071826/http://chss.montclair.edu/inquiry/fall95/awbrey.html Archive]. [http://independent.academia.edu/JonAwbrey/Papers/1302117/Interpretation_as_Action_The_Risk_of_Inquiry Online]. * Awbrey, J.L., and Awbrey, S.M. (June 1992), &ldquo;Interpretation as Action : The Risk of Inquiry&rdquo;, ''The Eleventh International Human Science Research Conference'', Oakland University, Rochester, Michigan. * Awbrey, S.M., and Awbrey, J.L. (May 1991), &ldquo;An Architecture for Inquiry : Building Computer Platforms for Discovery&rdquo;, ''Proceedings of the Eighth International Conference on Technology and Education'', Toronto, Canada, pp. 874&ndash;875. [http://abccommunity.org/tmp-a.html Online]. * Awbrey, J.L., and Awbrey, S.M. (January 1991), &ldquo;Exploring Research Data Interactively : Developing a Computer Architecture for Inquiry&rdquo;, Poster presented at the ''Annual Sigma Xi Research Forum'', University of Texas Medical Branch, Galveston, TX. * Awbrey, J.L., and Awbrey, S.M. (August 1990), &ldquo;Exploring Research Data Interactively. Theme One : A Program of Inquiry&rdquo;, ''Proceedings of the Sixth Annual Conference on Applications of Artificial Intelligence and CD-ROM in Education and Training'', Society for Applied Learning Technology, Washington, DC, pp. 9&ndash;15. [http://academia.edu/1272839/Exploring_Research_Data_Interactively._Theme_One_A_Program_of_Inquiry Online]. ==Education== * 1993&ndash;2003. [http://web.archive.org/web/20120202222443/http://www2.oakland.edu/secs/dispprofile.asp?Fname=Fatma&Lname=Mili Graduate Study], [http://web.archive.org/web/20120203004703/http://www2.oakland.edu/secs/dispprofile.asp?Fname=Mohamed&Lname=Zohdy Systems Engineering], [http://meadowbrookhall.org/explore/history/meadowbrookhall Oakland University]. * '''1989. [http://web.archive.org/web/20001206101800/http://www.msu.edu/dig/msumap/psychology.html M.A. Psychology]''', [http://web.archive.org/web/20000902103838/http://www.msu.edu/dig/msumap/beaumont.html Michigan State University]. * 1985&ndash;1986. [http://quod.lib.umich.edu/cgi/i/image/image-idx?id=S-BHL-X-BL001808%5DBL001808 Graduate Study, Mathematics], [http://www.umich.edu/ University of Michigan]. * 1985. [http://web.archive.org/web/20061230174652/http://www.uiuc.edu/navigation/buildings/altgeld.top.html Graduate Study, Mathematics], [http://www.uiuc.edu/ University of Illinois at Urbana&ndash;Champaign]. * 1984. [http://www.psych.uiuc.edu/graduate/ Graduate Study, Psychology], [http://www.uiuc.edu/ University of Illinois at Champaign&ndash;Urbana]. * '''1980. [http://www.mth.msu.edu/images/wells_medium.jpg M.A. Mathematics]''', [http://www.msu.edu/~hvac/survey/BeaumontTower.html Michigan State University]. * '''1976. [http://web.archive.org/web/20001206050600/http://www.msu.edu/dig/msumap/phillips.html B.A. Mathematical and Philosophical Method]''', <br> [http://www.enolagaia.com/JMC.html Justin Morrill College], [http://www.msu.edu/ Michigan State University]. f4d272f4ea7a7a0d2745ad888e2075d509775438 650 643 2013-10-23T19:45:17Z Jon Awbrey 3 update wikitext text/x-wiki <br> <p><font face="lucida calligraphy" size="7">Jon Awbrey</font></p> <br> __NOTOC__ ==Presently &hellip;== <center> [https://oeis.org/wiki/Riffs_and_Rotes Riffs and Rotes] [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] [http://intersci.ss.uci.edu/wiki/index.php/Theme_One_Program Theme One Program] [http://intersci.ss.uci.edu/wiki/index.php/Propositions_As_Types Propositions As Types] [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems] [http://intersci.ss.uci.edu/wiki/index.php/Peirce%27s_1870_Logic_Of_Relatives Peirce's Logic Of Relatives] Logical Graphs : [http://inquiryintoinquiry.com/2008/07/29/logical-graphs-1/ One] [http://inquiryintoinquiry.com/2008/09/19/logical-graphs-2/ Two] [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] [http://intersci.ss.uci.edu/wiki/index.php/Differential_Analytic_Turing_Automata Differential Analytic Turing Automata] [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] [http://intersci.ss.uci.edu/wiki/index.php/Information_%3D_Comprehension_%C3%97_Extension Information = Comprehension &times; Extension] [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] </center> ==Recent Sightings== {| align="center" style="text-align:center" width="100%" |- | [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Live Inquiry Live] | [http://intersci.ss.uci.edu/wiki/index.php/Logic_Live Logic Live] |- | [http://stderr.org/cgi-bin/mailman/listinfo/inquiry Inquiry Project] | [http://stderr.org/pipermail/inquiry/ Inquiry Archive] |- | [http://inquiryintoinquiry.com/ Inquiry Into Inquiry] | [http://jonawbrey.wordpress.com/ Jon Awbrey • Blog] |- | [http://intersci.ss.uci.edu/wiki/index.php/User:Jon_Awbrey InterSciWiki User Page] | [http://intersci.ss.uci.edu/wiki/index.php/User_talk:Jon_Awbrey InterSciWiki Talk Page] |- | [http://mywikibiz.com/Directory:Jon_Awbrey MyWikiBiz Directory] | [http://mywikibiz.com/Directory_talk:Jon_Awbrey MyWikiBiz Discussion] |- | [http://mywikibiz.com/User:Jon_Awbrey MyWikiBiz User Page] | [http://mywikibiz.com/User_talk:Jon_Awbrey MyWikiBiz Talk Page] |- | [http://planetmath.org/ PlanetMath Project] | [http://planetmath.org/users/Jon-Awbrey PlanetMath Profile] |- | [http://forum.wolframscience.com/ NKS Forum] | [http://forum.wolframscience.com/member.php?s=&action=getinfo&userid=336 NKS Profile] |- | [http://oeis.org/wiki/Welcome OEIS Land] | [http://oeis.org/search?q=Awbrey Bolgia Mia] |- | [http://oeis.org/wiki/User:Jon_Awbrey OEIS Wiki Page] | [http://oeis.org/wiki/User_talk:Jon_Awbrey OEIS Wiki Talk] |- | [http://list.seqfan.eu/cgi-bin/mailman/listinfo/seqfan Fantasia Sequentia] | [http://list.seqfan.eu/pipermail/seqfan/ SeqFan Archive] |- | [http://mathforum.org/kb/ Math Forum Project] | [http://mathforum.org/kb/accountView.jspa?userID=99854 Math Forum Profile] |- | [http://www.mathweb.org/wiki/User:Jon_Awbrey MathWeb Page] | [http://www.mathweb.org/wiki/User_talk:Jon_Awbrey MathWeb Talk] |- | [http://www.proofwiki.org/wiki/User:Jon_Awbrey ProofWiki Page] | [http://www.proofwiki.org/wiki/User_talk:Jon_Awbrey ProofWiki Talk] |- | [http://mathoverflow.net/ MathOverFlow] | [http://mathoverflow.net/users/1636/jon-awbrey MOFler Profile] |- | [http://p2pfoundation.net/User:JonAwbrey P2P Wiki Page] | [http://p2pfoundation.net/User_talk:JonAwbrey P2P Wiki Talk] |- | [http://vectors.usc.edu/ Vectors Project] | [http://vectors.usc.edu/thoughtmesh/ ThoughtMesh] |- | [http://ontolog.cim3.net/ OntoLog Project] | [http://ontolog.cim3.net/cgi-bin/wiki.pl?JonAwbrey OntoLog Profile] |- | [http://semanticweb.org/wiki/User:Jon_Awbrey SemanticWeb Page] | [http://semanticweb.org/wiki/User_talk:Jon_Awbrey SemanticWeb Talk] |- | [http://zh.wikipedia.org/wiki/User:Jon_Awbrey 维基百科 : 用户页面] | [http://zh.wikipedia.org/wiki/User_talk:Jon_Awbrey 维基百科 : 讨论] |} ==Presentations and Publications== * Awbrey, S.M., and Awbrey, J.L. (May 2001), &ldquo;Conceptual Barriers to Creating Integrative Universities&rdquo;, ''Organization : The Interdisciplinary Journal of Organization, Theory, and Society'' 8(2), Sage Publications, London, UK, pp. 269&ndash;284. [http://org.sagepub.com/cgi/content/abstract/8/2/269 Abstract]. * Awbrey, S.M., and Awbrey, J.L. (September 1999), &ldquo;Organizations of Learning or Learning Organizations : The Challenge of Creating Integrative Universities for the Next Century&rdquo;, ''Second International Conference of the Journal &lsquo;Organization&rsquo;'', ''Re-Organizing Knowledge, Trans-Forming Institutions : Knowing, Knowledge, and the University in the 21st Century'', University of Massachusetts, Amherst, MA. [http://cspeirce.com/menu/library/aboutcsp/awbrey/integrat.htm Online]. * Awbrey, J.L., and Awbrey, S.M. (Autumn 1995), &ldquo;Interpretation as Action : The Risk of Inquiry&rdquo;, ''Inquiry : Critical Thinking Across the Disciplines'' 15(1), pp. 40&ndash;52. [http://web.archive.org/web/19970626071826/http://chss.montclair.edu/inquiry/fall95/awbrey.html Archive]. [http://independent.academia.edu/JonAwbrey/Papers/1302117/Interpretation_as_Action_The_Risk_of_Inquiry Online]. * Awbrey, J.L., and Awbrey, S.M. (June 1992), &ldquo;Interpretation as Action : The Risk of Inquiry&rdquo;, ''The Eleventh International Human Science Research Conference'', Oakland University, Rochester, Michigan. * Awbrey, S.M., and Awbrey, J.L. (May 1991), &ldquo;An Architecture for Inquiry : Building Computer Platforms for Discovery&rdquo;, ''Proceedings of the Eighth International Conference on Technology and Education'', Toronto, Canada, pp. 874&ndash;875. [http://www.academia.edu/1270327/An_Architecture_for_Inquiry_Building_Computer_Platforms_for_Discovery Online]. * Awbrey, J.L., and Awbrey, S.M. (January 1991), &ldquo;Exploring Research Data Interactively : Developing a Computer Architecture for Inquiry&rdquo;, Poster presented at the ''Annual Sigma Xi Research Forum'', University of Texas Medical Branch, Galveston, TX. * Awbrey, J.L., and Awbrey, S.M. (August 1990), &ldquo;Exploring Research Data Interactively. Theme One : A Program of Inquiry&rdquo;, ''Proceedings of the Sixth Annual Conference on Applications of Artificial Intelligence and CD-ROM in Education and Training'', Society for Applied Learning Technology, Washington, DC, pp. 9&ndash;15. [http://academia.edu/1272839/Exploring_Research_Data_Interactively._Theme_One_A_Program_of_Inquiry Online]. ==Education== * 1993&ndash;2003. [http://web.archive.org/web/20120202222443/http://www2.oakland.edu/secs/dispprofile.asp?Fname=Fatma&Lname=Mili Graduate Study], [http://web.archive.org/web/20120203004703/http://www2.oakland.edu/secs/dispprofile.asp?Fname=Mohamed&Lname=Zohdy Systems Engineering], [http://meadowbrookhall.org/explore/history/meadowbrookhall Oakland University]. * '''1989. [http://web.archive.org/web/20001206101800/http://www.msu.edu/dig/msumap/psychology.html M.A. Psychology]''', [http://web.archive.org/web/20000902103838/http://www.msu.edu/dig/msumap/beaumont.html Michigan State University]. * 1985&ndash;1986. [http://quod.lib.umich.edu/cgi/i/image/image-idx?id=S-BHL-X-BL001808%5DBL001808 Graduate Study, Mathematics], [http://www.umich.edu/ University of Michigan]. * 1985. [http://web.archive.org/web/20061230174652/http://www.uiuc.edu/navigation/buildings/altgeld.top.html Graduate Study, Mathematics], [http://www.uiuc.edu/ University of Illinois at Urbana&ndash;Champaign]. * 1984. [http://www.psych.uiuc.edu/graduate/ Graduate Study, Psychology], [http://www.uiuc.edu/ University of Illinois at Champaign&ndash;Urbana]. * '''1980. [http://www.mth.msu.edu/images/wells_medium.jpg M.A. Mathematics]''', [http://www.msu.edu/~hvac/survey/BeaumontTower.html Michigan State University]. * '''1976. [http://web.archive.org/web/20001206050600/http://www.msu.edu/dig/msumap/phillips.html B.A. Mathematical and Philosophical Method]''', <br> [http://www.enolagaia.com/JMC.html Justin Morrill College], [http://www.msu.edu/ Michigan State University]. 0eb91fb90a6c4e6811316ba7b9b99c877522d3d0 0 360 630 2012-05-14T04:02:00Z Jon Awbrey 3 place holder wikitext text/x-wiki place holder e7cb921567548ef08d7808344c540a524628c926 0 361 631 2012-05-14T04:05:30Z Jon Awbrey 3 place holder wikitext text/x-wiki place holder e7cb921567548ef08d7808344c540a524628c926 632 631 2012-05-14T04:08:37Z Jon Awbrey 3 + article wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. '''Exclusive disjunction''', also known as '''logical inequality''' or '''symmetric difference''', is an operation on two logical values, typically the values of two propositions, that produces a value of ''true'' just in case exactly one of its operands is true. The [[truth table]] of <math>p ~\operatorname{XOR}~ q</math> (also written as <math>p + q\!</math> or <math>p \ne q\!</math>) is as follows: <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:45%" |+ style="height:30px" | <math>\text{Exclusive Disjunction}\!</math> |- style="height:40px; background:#f0f0ff" | style="width:33%" | <math>p\!</math> | style="width:33%" | <math>q\!</math> | style="width:33%" | <math>p ~\operatorname{XOR}~ q</math> |- | <math>\operatorname{F}</math> || <math>\operatorname{F}</math> || <math>\operatorname{F}</math> |- | <math>\operatorname{F}</math> || <math>\operatorname{T}</math> || <math>\operatorname{T}</math> |- | <math>\operatorname{T}</math> || <math>\operatorname{F}</math> || <math>\operatorname{T}</math> |- | <math>\operatorname{T}</math> || <math>\operatorname{T}</math> || <math>\operatorname{F}</math> |} <br> The following equivalents may then be deduced: {| align="center" cellspacing="10" width="90%" | <math>\begin{matrix} p + q & = & (p \land \lnot q) & \lor & (\lnot p \land q) \\[6pt] & = & (p \lor q) & \land & (\lnot p \lor \lnot q) \\[6pt] & = & (p \lor q) & \land & \lnot (p \land q) \end{matrix}</math> |} ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://mywikibiz.com/Exclusive_disjunction Exclusive Disjunction @ MyWikiBiz] * [http://intersci.ss.uci.edu/wiki/index.php/Exclusive_disjunction Exclusive Disjunction @ InterSci Wiki] * [http://wiki.oercommons.org/mediawiki/index.php/Exclusive_disjunction Exclusive Disjunction @ OER Commons] {{col-break}} * [http://p2pfoundation.net/Exclusive_Disjunction Exclusive Disjunction @ P2P Foundation] * [http://ref.subwiki.org/wiki/Exclusive_disjunction Exclusive Disjunction @ Subject Wikis] * [http://beta.wikiversity.org/wiki/Exclusive_disjunction Exclusive Disjunction @ Wikiversity Beta] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;] * [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;] ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Exclusive_disjunction Exclusive Disjunction], [http://mywikibiz.com/ MyWikiBiz] * [http://beta.wikiversity.org/wiki/Exclusive_disjunction Exclusive Disjunction], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://getwiki.net/-Exclusive_Disjunction Exclusive Disjunction], [http://getwiki.net/ GetWiki] {{col-break}} * [http://wikinfo.org/index.php/Exclusive_disjunction Exclusive Disjunction], [http://wikinfo.org/ Wikinfo] * [http://textop.org/wiki/index.php?title=Exclusive_disjunction Exclusive Disjunction], [http://textop.org/wiki/ Textop Wiki] * [http://en.wikipedia.org/w/index.php?title=Exclusive_disjunction&oldid=75153068 Exclusive Disjunction], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Charles Sanders Peirce]] [[Category:Computer Science]] [[Category:Formal Languages]] [[Category:Formal Sciences]] [[Category:Formal Systems]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Philosophy]] [[Category:Semiotics]] 139ba813c1e9ea42d14d03a9943adb059f7c39fa 3 362 633 2012-05-14T04:14:56Z Jon Awbrey 3 Problems with LaTeX wikitext text/x-wiki ==Problems with LaTeX== Hi, it looks like LaTeX is not working. : For example, see [http://ref.subwiki.org/wiki/Exclusive_disjunction this article]. : It should look like [http://mywikibiz.com/Exclusive_disjunction this]. Thanks, [[User:Jon Awbrey|Jon Awbrey]] 21:14, 13 May 2012 (PDT) 530be28d62946d4314b6ac7d8feb1ecf3c594c12 0 181 634 401 2012-05-14T04:20:22Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page serves as a '''focal node''' for a collection of related resources. ==Participants== * Interested parties may add their names on [[Logic Live/Participants|this page]]. ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://mywikibiz.com/Logic_Live Logic Live @ MyWikiBiz] * [http://intersci.ss.uci.edu/wiki/index.php/Logic_Live Logic Live @ InterSciWiki] * [http://wiki.oercommons.org/mediawiki/index.php/Logic_Live Logic Live @ OER Commons] {{col-break}} * [http://p2pfoundation.net/Logic_Live Logic Live @ P2P Foundation] * [http://ref.subwiki.org/wiki/Logic_Live Logic Live @ Subject Wikis] * [http://beta.wikiversity.org/wiki/Logic_Live Logic Live @ Wikiversity Beta] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;] * [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;] [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] d434b5cba51e0f2a9f1905c6795d2be3507b3ec6 644 634 2013-10-03T14:40:09Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page serves as a '''focal node''' for a collection of related resources. ==Participants== * Interested parties may add their names on [[Logic Live/Participants|this page]]. ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Logic_Live Logic Live @ InterSciWiki] * [http://mywikibiz.com/Logic_Live Logic Live @ MyWikiBiz] {{col-break}} * [http://beta.wikiversity.org/wiki/Logic_Live Logic Live @ Wikiversity Beta] * [http://ref.subwiki.org/wiki/Logic_Live Logic Live @ Subject Wikis] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] e201a1ff14fe9557cf46fad4564f1853f68dcfd4 0 180 635 400 2012-05-14T04:22:08Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page serves as a '''focal node''' for a collection of related resources. ==Participants== Interested parties may add their names on [[Inquiry Live/Participants|this page]]. ==Rudiments of organization== ===Focal nodes=== '''Focal nodes''' serve as hubs for collections of related resources, in particular, activity sites and article contents. {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== '''Peer nodes''' are roughly parallel pages on different sites that are not necessarily identical in content &mdash; especially as they develop in time across different environments through interaction with diverse populations &mdash; but they should preserve enough information to reproduce each other, more or less, in case of damage or loss. For example, the peer nodes of the present page are listed here: {{col-begin}} {{col-break}} * [http://mywikibiz.com/Inquiry_Live Inquiry Live @ MyWikiBiz] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Live Inquiry Live @ InterSciWiki] * [http://wiki.oercommons.org/mediawiki/index.php/Inquiry_Live Inquiry Live @ OER Commons] {{col-break}} * [http://p2pfoundation.net/Inquiry_Live Inquiry Live @ P2P Foundation] * [http://ref.subwiki.org/wiki/Inquiry_Live Inquiry Live @ Subject Wikis] * [http://beta.wikiversity.org/wiki/Inquiry_Live Inquiry Live @ Wikiversity Beta] {{col-end}} [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] 016e174b191e785e8901ff5ae6aac1457fcaba39 645 635 2013-10-03T14:42:21Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page serves as a '''focal node''' for a collection of related resources. ==Participants== Interested parties may add their names on [[Inquiry Live/Participants|this page]]. ==Rudiments of organization== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} '''Focal nodes''' serve as hubs for collections of related resources, in particular, activity sites and article contents. ===Peer nodes=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Live Inquiry Live @ InterSciWiki] * [http://mywikibiz.com/Inquiry_Live Inquiry Live @ MyWikiBiz] {{col-break}} * [http://beta.wikiversity.org/wiki/Inquiry_Live Inquiry Live @ Wikiversity Beta] * [http://ref.subwiki.org/wiki/Inquiry_Live Inquiry Live @ Subject Wikis] {{col-end}} '''Peer nodes''' are roughly parallel pages on different sites that are not necessarily identical in content &mdash; especially as they develop in time across different environments through interaction with diverse populations &mdash; but they should preserve enough information to reproduce each other, more or less, in case of damage or loss. [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] 158cc8e23291f5f1b2c796fcd9bf49378859aba8 0 315 636 536 2012-05-21T14:20:05Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''minimal negation operator''' <math>(\texttt{Mno})</math> is a logical connective that says “just one false” of its logical arguments. If the list of arguments is empty, as expressed in the form <math>\texttt{Mno}(),</math> then it cannot be true that exactly one of the arguments is false, so <math>\texttt{Mno}() = \texttt{False}.</math> If <math>p\!</math> is the only argument, then <math>\texttt{Mno}(p)</math> says that <math>p\!</math> is false, so <math>\texttt{Mno}(p)</math> expresses the logical negation of the proposition <math>p.\!</math> Wrtten in several different notations, <math>\texttt{Mno}(p) = \texttt{Not}(p) = \lnot p = \tilde{p} = p^\prime.</math> If <math>p\!</math> and <math>q\!</math> are the only two arguments, then <math>\texttt{Mno}(p, q)</math> says that exactly one of <math>p, q\!</math> is false, so <math>\texttt{Mno}(p, q)</math> says the same thing as <math>p \neq q.\!</math> Expressing <math>\texttt{Mno}(p, q)</math> in terms of ands <math>(\cdot),</math> ors <math>(\lor),</math> and nots <math>(\tilde{~})</math> gives the following form. {| align="center" cellpadding="8" | <math>\texttt{Mno}(p, q) = \tilde{p} \cdot q \lor p \cdot \tilde{q}.</math> |} As usual, one drops the dots <math>(\cdot)</math> in contexts where they are understood, giving the following form. {| align="center" cellpadding="8" | <math>\texttt{Mno}(p, q) = \tilde{p}q \lor p\tilde{q}.</math> |} The venn diagram for <math>\texttt{Mno}(p, q)</math> is shown in Figure&nbsp;1. {| align="center" cellpadding="8" style="text-align:center" | <p>[[Image:Venn Diagram (P,Q).jpg|500px]]</p> <p><math>\text{Figure 1.}~~\texttt{Mno}(p, q)</math></p> |} The venn diagram for <math>\texttt{Mno}(p, q, r)</math> is shown in Figure&nbsp;2. {| align="center" cellpadding="8" style="text-align:center" | <p>[[Image:Venn Diagram (P,Q,R).jpg|500px]]</p> <p><math>\text{Figure 2.}~~\texttt{Mno}(p, q, r)</math></p> |} The center cell is the region where all three arguments <math>p, q, r\!</math> hold true, so <math>\texttt{Mno}(p, q, r)</math> holds true in just the three neighboring cells. In other words, <math>\texttt{Mno}(p, q, r) = \tilde{p}qr \lor p\tilde{q}r \lor pq\tilde{r}.</math> ==Initial definition== The '''minimal negation operator''' <math>\nu\!</math> is a [[multigrade operator]] <math>(\nu_k)_{k \in \mathbb{N}}</math> where each <math>\nu_k\!</math> is a <math>k\!</math>-ary [[boolean function]] defined in such a way that <math>\nu_k (x_1, \ldots , x_k) = 1</math> in just those cases where exactly one of the arguments <math>x_j\!</math> is <math>0.\!</math> In contexts where the initial letter <math>\nu\!</math> is understood, the minimal negation operators can be indicated by argument lists in parentheses. In the following text a distinctive typeface will be used for logical expressions based on minimal negation operators, for example, <math>\texttt{(x, y, z)}</math> = <math>\nu (x, y, z).\!</math> The first four members of this family of operators are shown below, with paraphrases in a couple of other notations, where tildes and primes, respectively, indicate logical negation. {| align="center" cellpadding="8" style="text-align:center" | <math>\begin{matrix} \texttt{()} & = & \nu_0 & = & 0 & = & \operatorname{false} \\[6pt] \texttt{(x)} & = & \nu_1 (x) & = & \tilde{x} & = & x^\prime \\[6pt] \texttt{(x, y)} & = & \nu_2 (x, y) & = & \tilde{x}y \lor x\tilde{y} & = & x^\prime y \lor x y^\prime \\[6pt] \texttt{(x, y, z)} & = & \nu_3 (x, y, z) & = & \tilde{x}yz \lor x\tilde{y}z \lor xy\tilde{z} & = & x^\prime y z \lor x y^\prime z \lor x y z^\prime \end{matrix}</math> |} ==Formal definition== To express the general case of <math>\nu_k\!</math> in terms of familiar operations, it helps to introduce an intermediary concept: '''Definition.''' Let the function <math>\lnot_j : \mathbb{B}^k \to \mathbb{B}</math> be defined for each integer <math>j\!</math> in the interval <math>[1, k]\!</math> by the following equation: {| align="center" cellpadding="8" | <math>\lnot_j (x_1, \ldots, x_j, \ldots, x_k) ~=~ x_1 \land \ldots \land x_{j-1} \land \lnot x_j \land x_{j+1} \land \ldots \land x_k.</math> |} Then <math>\nu_k : \mathbb{B}^k \to \mathbb{B}</math> is defined by the following equation: {| align="center" cellpadding="8" | <math>\nu_k (x_1, \ldots, x_k) ~=~ \lnot_1 (x_1, \ldots, x_k) \lor \ldots \lor \lnot_j (x_1, \ldots, x_k) \lor \ldots \lor \lnot_k (x_1, \ldots, x_k).</math> |} If we think of the point <math>x = (x_1, \ldots, x_k) \in \mathbb{B}^k</math> as indicated by the boolean product <math>x_1 \cdot \ldots \cdot x_k</math> or the logical conjunction <math>x_1 \land \ldots \land x_k,</math> then the minimal negation <math>\texttt{(} x_1, \ldots, x_k \texttt{)}</math> indicates the set of points in <math>\mathbb{B}^k</math> that differ from <math>x\!</math> in exactly one coordinate. This makes <math>\texttt{(} x_1, \ldots, x_k \texttt{)}</math> a discrete functional analogue of a ''point omitted neighborhood'' in analysis, more exactly, a ''point omitted distance one neighborhood''. In this light, the minimal negation operator can be recognized as a differential construction, an observation that opens a very wide field. It also serves to explain a variety of other names for the same concept, for example, ''logical boundary operator'', ''limen operator'', ''least action operator'', or ''hedge operator'', to name but a few. The rationale for these names is visible in the venn diagrams of the corresponding operations on sets. The remainder of this discussion proceeds on the ''algebraic boolean convention'' that the plus sign <math>(+)\!</math> and the summation symbol <math>(\textstyle\sum)</math> both refer to addition modulo 2. Unless otherwise noted, the boolean domain <math>\mathbb{B} = \{ 0, 1 \}</math> is interpreted so that <math>0 = \operatorname{false}</math> and <math>1 = \operatorname{true}.</math> This has the following consequences: {| align="center" cellpadding="4" width="90%" | valign="top" | <big>&bull;</big> | The operation <math>x + y\!</math> is a function equivalent to the exclusive disjunction of <math>x\!</math> and <math>y,\!</math> while its fiber of 1 is the relation of inequality between <math>x\!</math> and <math>y.\!</math> |- | valign="top" | <big>&bull;</big> | The operation <math>\textstyle\sum_{j=1}^k x_j</math> maps the bit sequence <math>(x_1, \ldots, x_k)\!</math> to its ''parity''. |} The following properties of the minimal negation operators <math>\nu_k : \mathbb{B}^k \to \mathbb{B}</math> may be noted: {| align="center" cellpadding="4" width="90%" | valign="top" | <big>&bull;</big> | The function <math>\texttt{(x, y)}</math> is the same as that associated with the operation <math>x + y\!</math> and the relation <math>x \ne y.</math> |- | valign="top" | <big>&bull;</big> | In contrast, <math>\texttt{(x, y, z)}</math> is not identical to <math>x + y + z.\!</math> |- | valign="top" | <big>&bull;</big> | More generally, the function <math>\nu_k (x_1, \dots, x_k)</math> for <math>k > 2\!</math> is not identical to the boolean sum <math>\textstyle\sum_{j=1}^k x_j.</math> |- | valign="top" | <big>&bull;</big> | The inclusive disjunctions indicated for the <math>\nu_k\!</math> of more than one argument may be replaced with exclusive disjunctions without affecting the meaning, since the terms disjoined are already disjoint. |} ==Truth tables== Table&nbsp;3 is a [[truth table]] for the sixteen boolean functions of type <math>f : \mathbb{B}^3 \to \mathbb{B}</math> whose fibers of 1 are either the boundaries of points in <math>\mathbb{B}^3</math> or the complements of those boundaries. <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ <math>\text{Table 3.}~~\text{Logical Boundaries and Their Complements}</math> |- style="background:#f0f0ff" | <math>\mathcal{L}_1</math> | <math>\mathcal{L}_2</math> | <math>\mathcal{L}_3</math> | <math>\mathcal{L}_4</math> |- style="background:#f0f0ff" | &nbsp; | align="right" | <math>p\colon\!</math> | <math>1~1~1~1~0~0~0~0</math> | &nbsp; |- style="background:#f0f0ff" | &nbsp; | align="right" | <math>q\colon\!</math> | <math>1~1~0~0~1~1~0~0</math> | &nbsp; |- style="background:#f0f0ff" | &nbsp; | align="right" | <math>r\colon\!</math> | <math>1~0~1~0~1~0~1~0</math> | &nbsp; |- | <math>\begin{matrix} f_{104} \\[4pt] f_{148} \\[4pt] f_{146} \\[4pt] f_{97} \\[4pt] f_{134} \\[4pt] f_{73} \\[4pt] f_{41} \\[4pt] f_{22} \end{matrix}</math> | <math>\begin{matrix} f_{01101000} \\[4pt] f_{10010100} \\[4pt] f_{10010010} \\[4pt] f_{01100001} \\[4pt] f_{10000110} \\[4pt] f_{01001001} \\[4pt] f_{00101001} \\[4pt] f_{00010110} \end{matrix}</math> | <math>\begin{matrix} 0~1~1~0~1~0~0~0 \\[4pt] 1~0~0~1~0~1~0~0 \\[4pt] 1~0~0~1~0~0~1~0 \\[4pt] 0~1~1~0~0~0~0~1 \\[4pt] 1~0~0~0~0~1~1~0 \\[4pt] 0~1~0~0~1~0~0~1 \\[4pt] 0~0~1~0~1~0~0~1 \\[4pt] 0~0~0~1~0~1~1~0 \end{matrix}</math> | <math>\begin{matrix} \texttt{(~p~,~q~,~r~)} \\[4pt] \texttt{(~p~,~q~,(r))} \\[4pt] \texttt{(~p~,(q),~r~)} \\[4pt] \texttt{(~p~,(q),(r))} \\[4pt] \texttt{((p),~q~,~r~)} \\[4pt] \texttt{((p),~q~,(r))} \\[4pt] \texttt{((p),(q),~r~)} \\[4pt] \texttt{((p),(q),(r))} \end{matrix}</math> |- | <math>\begin{matrix} f_{233} \\[4pt] f_{214} \\[4pt] f_{182} \\[4pt] f_{121} \\[4pt] f_{158} \\[4pt] f_{109} \\[4pt] f_{107} \\[4pt] f_{151} \end{matrix}</math> | <math>\begin{matrix} f_{11101001} \\[4pt] f_{11010110} \\[4pt] f_{10110110} \\[4pt] f_{01111001} \\[4pt] f_{10011110} \\[4pt] f_{01101101} \\[4pt] f_{01101011} \\[4pt] f_{10010111} \end{matrix}</math> | <math>\begin{matrix} 1~1~1~0~1~0~0~1 \\[4pt] 1~1~0~1~0~1~1~0 \\[4pt] 1~0~1~1~0~1~1~0 \\[4pt] 0~1~1~1~1~0~0~1 \\[4pt] 1~0~0~1~1~1~1~0 \\[4pt] 0~1~1~0~1~1~0~1 \\[4pt] 0~1~1~0~1~0~1~1 \\[4pt] 1~0~0~1~0~1~1~1 \end{matrix}</math> | <math>\begin{matrix} \texttt{(((p),(q),(r)))} \\[4pt] \texttt{(((p),(q),~r~))} \\[4pt] \texttt{(((p),~q~,(r)))} \\[4pt] \texttt{(((p),~q~,~r~))} \\[4pt] \texttt{((~p~,(q),(r)))} \\[4pt] \texttt{((~p~,(q),~r~))} \\[4pt] \texttt{((~p~,~q~,(r)))} \\[4pt] \texttt{((~p~,~q~,~r~))} \end{matrix}</math> |} <br> ==Charts and graphs== This Section focuses on visual representations of minimal negation operators. A few bits of terminology are useful in describing the pictures, but the formal details are tedious reading, and may be familiar to many readers, so the full definitions of the terms marked in ''italics'' are relegated to a Glossary at the end of the article. Two ways of visualizing the space <math>\mathbb{B}^k</math> of <math>2^k\!</math> points are the [[hypercube]] picture and the [[venn diagram]] picture. The hypercube picture associates each point of <math>\mathbb{B}^k</math> with a unique point of the <math>k\!</math>-dimensional hypercube. The venn diagram picture associates each point of <math>\mathbb{B}^k</math> with a unique "cell" of the venn diagram on <math>k\!</math> "circles". In addition, each point of <math>\mathbb{B}^k</math> is the unique point in the ''[[fiber (mathematics)|fiber]] of truth'' <math>[|s|]\!</math> of a ''singular proposition'' <math>s : \mathbb{B}^k \to \mathbb{B},</math> and thus it is the unique point where a ''singular conjunction'' of <math>k\!</math> ''literals'' is <math>1.\!</math> For example, consider two cases at opposite vertices of the cube: {| align="center" cellpadding="4" width="90%" | valign="top" | <big>&bull;</big> | The point <math>(1, 1, \ldots , 1, 1)</math> with all 1's as coordinates is the point where the conjunction of all posited variables evaluates to <math>1,\!</math> namely, the point where: |- | &nbsp; | align="center" | <math>x_1 ~ x_2 ~\ldots~ x_{n-1} ~ x_n ~=~ 1.</math> |- | valign="top" | <big>&bull;</big> | The point <math>(0, 0, \ldots , 0, 0)</math> with all 0's as coordinates is the point where the conjunction of all negated variables evaluates to <math>1,\!</math> namely, the point where: |- | &nbsp; | align="center" | <math>\texttt{(} x_1 \texttt{)(} x_2 \texttt{)} \ldots \texttt{(} x_{n-1} \texttt{)(} x_n \texttt{)} ~=~ 1.</math> |} To pass from these limiting examples to the general case, observe that a singular proposition <math>s : \mathbb{B}^k \to \mathbb{B}</math> can be given canonical expression as a conjunction of literals, <math>s = e_1 e_2 \ldots e_{k-1} e_k</math>. Then the proposition <math>\nu (e_1, e_2, \ldots, e_{k-1}, e_k)</math> is <math>1\!</math> on the points adjacent to the point where <math>s\!</math> is <math>1,\!</math> and 0 everywhere else on the cube. For example, consider the case where <math>k = 3.\!</math> Then the minimal negation operation <math>\nu (p, q, r)\!</math> &mdash; written more simply as <math>\texttt{(p, q, r)}</math> &mdash; has the following venn diagram: {| align="center" cellpadding="8" style="text-align:center" | <p>[[Image:Venn Diagram (P,Q,R).jpg|500px]]</p> <p><math>\text{Figure 4.}~~\texttt{(p, q, r)}</math></p> |} For a contrasting example, the boolean function expressed by the form <math>\texttt{((p),(q),(r))}</math> has the following venn diagram: {| align="center" cellpadding="8" style="text-align:center" | <p>[[Image:Venn Diagram ((P),(Q),(R)).jpg|500px]]</p> <p><math>\text{Figure 5.}~~\texttt{((p),(q),(r))}</math></p> |} ==Glossary of basic terms== ; Boolean domain : A ''[[boolean domain]]'' <math>\mathbb{B}</math> is a generic 2-element set, for example, <math>\mathbb{B} = \{ 0, 1 \},</math> whose elements are interpreted as logical values, usually but not invariably with <math>0 = \operatorname{false}</math> and <math>1 = \operatorname{true}.</math> ; Boolean variable : A ''[[boolean variable]]'' <math>x\!</math> is a variable that takes its value from a boolean domain, as <math>x \in \mathbb{B}.</math> ; Proposition : In situations where boolean values are interpreted as logical values, a [[boolean-valued function]] <math>f : X \to \mathbb{B}</math> or a [[boolean function]] <math>g : \mathbb{B}^k \to \mathbb{B}</math> is frequently called a ''[[proposition]]''. ; Basis element, Coordinate projection : Given a sequence of <math>k\!</math> boolean variables, <math>x_1, \ldots, x_k,</math> each variable <math>x_j\!</math> may be treated either as a ''basis element'' of the space <math>\mathbb{B}^k</math> or as a ''coordinate projection'' <math>x_j : \mathbb{B}^k \to \mathbb{B}.</math> ; Basic proposition : This means that the set of objects <math>\{ x_j : 1 \le j \le k \}</math> is a set of boolean functions <math>\{ x_j : \mathbb{B}^k \to \mathbb{B} \}</math> subject to logical interpretation as a set of ''basic propositions'' that collectively generate the complete set of <math>2^{2^k}</math> propositions over <math>\mathbb{B}^k.</math> ; Literal : A ''literal'' is one of the <math>2k\!</math> propositions <math>x_1, \ldots, x_k, \texttt{(} x_1 \texttt{)}, \ldots, \texttt{(} x_k \texttt{)},</math> in other words, either a ''posited'' basic proposition <math>x_j\!</math> or a ''negated'' basic proposition <math>\texttt{(} x_j \texttt{)},</math> for some <math>j = 1 ~\text{to}~ k.</math> ; Fiber : In mathematics generally, the ''[[fiber (mathematics)|fiber]]'' of a point <math>y \in Y</math> under a function <math>f : X \to Y</math> is defined as the inverse image <math>f^{-1}(y) \subseteq X.</math> : In the case of a boolean function <math>f : \mathbb{B}^k \to \mathbb{B},</math> there are just two fibers: : The fiber of <math>0\!</math> under <math>f,\!</math> defined as <math>f^{-1}(0),\!</math> is the set of points where the value of <math>f\!</math> is <math>0.\!</math> : The fiber of <math>1\!</math> under <math>f,\!</math> defined as <math>f^{-1}(1),\!</math> is the set of points where the value of <math>f\!</math> is <math>1.\!</math> ; Fiber of truth : When <math>1\!</math> is interpreted as the logical value <math>\operatorname{true},</math> then <math>f^{-1}(1)\!</math> is called the ''fiber of truth'' in the proposition <math>f.\!</math> Frequent mention of this fiber makes it useful to have a shorter way of referring to it. This leads to the definition of the notation <math>[|f|] = f^{-1}(1)\!</math> for the fiber of truth in the proposition <math>f.\!</math> ; Singular boolean function : A ''singular boolean function'' <math>s : \mathbb{B}^k \to \mathbb{B}</math> is a boolean function whose fiber of <math>1\!</math> is a single point of <math>\mathbb{B}^k.</math> ; Singular proposition : In the interpretation where <math>1\!</math> equals <math>\operatorname{true},</math> a singular boolean function is called a ''singular proposition''. : Singular boolean functions and singular propositions serve as functional or logical representatives of the points in <math>\mathbb{B}^k.</math> ; Singular conjunction : A ''singular conjunction'' in <math>\mathbb{B}^k \to \mathbb{B}</math> is a conjunction of <math>k\!</math> literals that includes just one conjunct of the pair <math>\{ x_j, ~\nu(x_j) \}</math> for each <math>j = 1 ~\text{to}~ k.</math> : A singular proposition <math>s : \mathbb{B}^k \to \mathbb{B}</math> can be expressed as a singular conjunction: {| align="center" cellspacing"10" width="90%" | height="36" | <math>s ~=~ e_1 e_2 \ldots e_{k-1} e_k</math>, |- | <math>\begin{array}{llll} \text{where} & e_j & = & x_j \\[6pt] \text{or} & e_j & = & \nu (x_j), \\[6pt] \text{for} & j & = & 1 ~\text{to}~ k. \end{array}</math> |} ==Resources== * [http://planetmath.org/encyclopedia/MinimalNegationOperator.html Minimal Negation Operator] @ [http://planetmath.org/ PlanetMath] * [http://planetphysics.us/encyclopedia/MinimalNegationOperator.html Minimal Negation Operator] @ [http://planetphysics.us/ PlanetPhysics] * [http://www.proofwiki.org/wiki/Definition:Minimal_Negation_Operator Minimal Negation Operator] @ [http://www.proofwiki.org/ ProofWiki] * [http://atlas.wolfram.com/ Wolfram Atlas of Simple Programs] ** [http://atlas.wolfram.com/01/01/ Elementary Cellular Automata Rules (ECARs)] ** [http://atlas.wolfram.com/01/01/rulelist.html ECAR Index] ** [http://atlas.wolfram.com/01/01/views/3/TableView.html ECAR Icons] ** [http://atlas.wolfram.com/01/01/views/87/TableView.html ECAR Examples] ** [http://atlas.wolfram.com/01/01/views/172/TableView.html ECAR Formulas] ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://mywikibiz.com/Minimal_negation_operator Minimal Negation Operator @ MyWikiBiz] * [http://intersci.ss.uci.edu/wiki/index.php/Minimal_negation_operator Minimal Negation Operator @ InterSciWiki] * [http://wiki.oercommons.org/mediawiki/index.php/Minimal_negation_operator Minimal Negation Operator @ OER Commons] {{col-break}} * [http://p2pfoundation.net/Minimal_Negation_Operator Minimal Negation Operator @ P2P Foundation] * [http://ref.subwiki.org/wiki/Minimal_negation_operator Minimal Negation Operator @ Subject Wikis] * [http://beta.wikiversity.org/wiki/Minimal_negation_operator Minimal Negation Operator @ Wikiversity Beta] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;] * [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;] ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Minimal_negation_operator Minimal Negation Operator], [http://mywikibiz.com/ MyWikiBiz] * [http://planetmath.org/encyclopedia/MinimalNegationOperator.html Minimal Negation Operator], [http://planetmath.org/ PlanetMath] * [http://proofwiki.org/wiki/Definition:Minimal_Negation_Operator Minimal Negation Operator], [http://proofwiki.org/ ProofWiki] * [http://beta.wikiversity.org/wiki/Minimal_negation_operator Minimal Negation Operator], [http://beta.wikiversity.org/ Wikiversity Beta] {{col-break}} * [http://getwiki.net/-Minimal_Negation_Operator Minimal Negation Operator], [http://getwiki.net/ GetWiki] * [http://wikinfo.org/index.php/Minimal_negation_operator Minimal Negation Operator], [http://wikinfo.org/ Wikinfo] * [http://textop.org/wiki/index.php?title=Minimal_negation_operator Minimal Negation Operator], [http://textop.org/wiki/ Textop Wiki] * [http://en.wikipedia.org/w/index.php?title=Minimal_negation_operator&oldid=75156728 Minimal Negation Operator], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Automata Theory]] [[Category:Charles Sanders Peirce]] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Differential Logic]] [[Category:Equational Reasoning]] [[Category:Formal Languages]] [[Category:Formal Sciences]] [[Category:Formal Systems]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Neural Networks]] [[Category:Philosophy]] [[Category:Semiotics]] 99ea53b160024d295ebd1c490ebb0c8e9e9ec144 646 636 2013-10-03T14:46:56Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''minimal negation operator''' <math>(\texttt{Mno})</math> is a logical connective that says &ldquo;just one false&rdquo; of its logical arguments. If the list of arguments is empty, as expressed in the form <math>\texttt{Mno}(),</math> then it cannot be true that exactly one of the arguments is false, so <math>\texttt{Mno}() = \texttt{False}.</math> If <math>p\!</math> is the only argument, then <math>\texttt{Mno}(p)</math> says that <math>p\!</math> is false, so <math>\texttt{Mno}(p)</math> expresses the logical negation of the proposition <math>p.\!</math> Wrtten in several different notations, <math>\texttt{Mno}(p) = \texttt{Not}(p) = \lnot p = \tilde{p} = p^\prime.</math> If <math>p\!</math> and <math>q\!</math> are the only two arguments, then <math>\texttt{Mno}(p, q)</math> says that exactly one of <math>p, q\!</math> is false, so <math>\texttt{Mno}(p, q)</math> says the same thing as <math>p \neq q.\!</math> Expressing <math>\texttt{Mno}(p, q)</math> in terms of ands <math>(\cdot),</math> ors <math>(\lor),</math> and nots <math>(\tilde{~})</math> gives the following form. {| align="center" cellpadding="8" | <math>\texttt{Mno}(p, q) = \tilde{p} \cdot q \lor p \cdot \tilde{q}.</math> |} As usual, one drops the dots <math>(\cdot)</math> in contexts where they are understood, giving the following form. {| align="center" cellpadding="8" | <math>\texttt{Mno}(p, q) = \tilde{p}q \lor p\tilde{q}.</math> |} The venn diagram for <math>\texttt{Mno}(p, q)</math> is shown in Figure&nbsp;1. {| align="center" cellpadding="8" style="text-align:center" | <p>[[Image:Venn Diagram (P,Q).jpg|500px]]</p> <p><math>\text{Figure 1.}~~\texttt{Mno}(p, q)</math></p> |} The venn diagram for <math>\texttt{Mno}(p, q, r)</math> is shown in Figure&nbsp;2. {| align="center" cellpadding="8" style="text-align:center" | <p>[[Image:Venn Diagram (P,Q,R).jpg|500px]]</p> <p><math>\text{Figure 2.}~~\texttt{Mno}(p, q, r)</math></p> |} The center cell is the region where all three arguments <math>p, q, r\!</math> hold true, so <math>\texttt{Mno}(p, q, r)</math> holds true in just the three neighboring cells. In other words, <math>\texttt{Mno}(p, q, r) = \tilde{p}qr \lor p\tilde{q}r \lor pq\tilde{r}.</math> ==Initial definition== The '''minimal negation operator''' <math>\nu\!</math> is a [[multigrade operator]] <math>(\nu_k)_{k \in \mathbb{N}}</math> where each <math>\nu_k\!</math> is a <math>k\!</math>-ary [[boolean function]] defined in such a way that <math>\nu_k (x_1, \ldots , x_k) = 1</math> in just those cases where exactly one of the arguments <math>x_j\!</math> is <math>0.\!</math> In contexts where the initial letter <math>\nu\!</math> is understood, the minimal negation operators can be indicated by argument lists in parentheses. In the following text a distinctive typeface will be used for logical expressions based on minimal negation operators, for example, <math>\texttt{(x, y, z)}</math> = <math>\nu (x, y, z).\!</math> The first four members of this family of operators are shown below, with paraphrases in a couple of other notations, where tildes and primes, respectively, indicate logical negation. {| align="center" cellpadding="8" style="text-align:center" | <math>\begin{matrix} \texttt{()} & = & \nu_0 & = & 0 & = & \operatorname{false} \\[6pt] \texttt{(x)} & = & \nu_1 (x) & = & \tilde{x} & = & x^\prime \\[6pt] \texttt{(x, y)} & = & \nu_2 (x, y) & = & \tilde{x}y \lor x\tilde{y} & = & x^\prime y \lor x y^\prime \\[6pt] \texttt{(x, y, z)} & = & \nu_3 (x, y, z) & = & \tilde{x}yz \lor x\tilde{y}z \lor xy\tilde{z} & = & x^\prime y z \lor x y^\prime z \lor x y z^\prime \end{matrix}</math> |} ==Formal definition== To express the general case of <math>\nu_k\!</math> in terms of familiar operations, it helps to introduce an intermediary concept: '''Definition.''' Let the function <math>\lnot_j : \mathbb{B}^k \to \mathbb{B}</math> be defined for each integer <math>j\!</math> in the interval <math>[1, k]\!</math> by the following equation: {| align="center" cellpadding="8" | <math>\lnot_j (x_1, \ldots, x_j, \ldots, x_k) ~=~ x_1 \land \ldots \land x_{j-1} \land \lnot x_j \land x_{j+1} \land \ldots \land x_k.</math> |} Then <math>\nu_k : \mathbb{B}^k \to \mathbb{B}</math> is defined by the following equation: {| align="center" cellpadding="8" | <math>\nu_k (x_1, \ldots, x_k) ~=~ \lnot_1 (x_1, \ldots, x_k) \lor \ldots \lor \lnot_j (x_1, \ldots, x_k) \lor \ldots \lor \lnot_k (x_1, \ldots, x_k).</math> |} If we think of the point <math>x = (x_1, \ldots, x_k) \in \mathbb{B}^k</math> as indicated by the boolean product <math>x_1 \cdot \ldots \cdot x_k</math> or the logical conjunction <math>x_1 \land \ldots \land x_k,</math> then the minimal negation <math>\texttt{(} x_1, \ldots, x_k \texttt{)}</math> indicates the set of points in <math>\mathbb{B}^k</math> that differ from <math>x\!</math> in exactly one coordinate. This makes <math>\texttt{(} x_1, \ldots, x_k \texttt{)}</math> a discrete functional analogue of a ''point omitted neighborhood'' in analysis, more exactly, a ''point omitted distance one neighborhood''. In this light, the minimal negation operator can be recognized as a differential construction, an observation that opens a very wide field. It also serves to explain a variety of other names for the same concept, for example, ''logical boundary operator'', ''limen operator'', ''least action operator'', or ''hedge operator'', to name but a few. The rationale for these names is visible in the venn diagrams of the corresponding operations on sets. The remainder of this discussion proceeds on the ''algebraic boolean convention'' that the plus sign <math>(+)\!</math> and the summation symbol <math>(\textstyle\sum)</math> both refer to addition modulo 2. Unless otherwise noted, the boolean domain <math>\mathbb{B} = \{ 0, 1 \}</math> is interpreted so that <math>0 = \operatorname{false}</math> and <math>1 = \operatorname{true}.</math> This has the following consequences: {| align="center" cellpadding="4" width="90%" | valign="top" | <big>&bull;</big> | The operation <math>x + y\!</math> is a function equivalent to the exclusive disjunction of <math>x\!</math> and <math>y,\!</math> while its fiber of 1 is the relation of inequality between <math>x\!</math> and <math>y.\!</math> |- | valign="top" | <big>&bull;</big> | The operation <math>\textstyle\sum_{j=1}^k x_j</math> maps the bit sequence <math>(x_1, \ldots, x_k)\!</math> to its ''parity''. |} The following properties of the minimal negation operators <math>\nu_k : \mathbb{B}^k \to \mathbb{B}</math> may be noted: {| align="center" cellpadding="4" width="90%" | valign="top" | <big>&bull;</big> | The function <math>\texttt{(x, y)}</math> is the same as that associated with the operation <math>x + y\!</math> and the relation <math>x \ne y.</math> |- | valign="top" | <big>&bull;</big> | In contrast, <math>\texttt{(x, y, z)}</math> is not identical to <math>x + y + z.\!</math> |- | valign="top" | <big>&bull;</big> | More generally, the function <math>\nu_k (x_1, \dots, x_k)</math> for <math>k > 2\!</math> is not identical to the boolean sum <math>\textstyle\sum_{j=1}^k x_j.</math> |- | valign="top" | <big>&bull;</big> | The inclusive disjunctions indicated for the <math>\nu_k\!</math> of more than one argument may be replaced with exclusive disjunctions without affecting the meaning, since the terms disjoined are already disjoint. |} ==Truth tables== Table&nbsp;3 is a [[truth table]] for the sixteen boolean functions of type <math>f : \mathbb{B}^3 \to \mathbb{B}</math> whose fibers of 1 are either the boundaries of points in <math>\mathbb{B}^3</math> or the complements of those boundaries. <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ <math>\text{Table 3.}~~\text{Logical Boundaries and Their Complements}</math> |- style="background:#f0f0ff" | <math>\mathcal{L}_1</math> | <math>\mathcal{L}_2</math> | <math>\mathcal{L}_3</math> | <math>\mathcal{L}_4</math> |- style="background:#f0f0ff" | &nbsp; | align="right" | <math>p\colon\!</math> | <math>1~1~1~1~0~0~0~0</math> | &nbsp; |- style="background:#f0f0ff" | &nbsp; | align="right" | <math>q\colon\!</math> | <math>1~1~0~0~1~1~0~0</math> | &nbsp; |- style="background:#f0f0ff" | &nbsp; | align="right" | <math>r\colon\!</math> | <math>1~0~1~0~1~0~1~0</math> | &nbsp; |- | <math>\begin{matrix} f_{104} \\[4pt] f_{148} \\[4pt] f_{146} \\[4pt] f_{97} \\[4pt] f_{134} \\[4pt] f_{73} \\[4pt] f_{41} \\[4pt] f_{22} \end{matrix}</math> | <math>\begin{matrix} f_{01101000} \\[4pt] f_{10010100} \\[4pt] f_{10010010} \\[4pt] f_{01100001} \\[4pt] f_{10000110} \\[4pt] f_{01001001} \\[4pt] f_{00101001} \\[4pt] f_{00010110} \end{matrix}</math> | <math>\begin{matrix} 0~1~1~0~1~0~0~0 \\[4pt] 1~0~0~1~0~1~0~0 \\[4pt] 1~0~0~1~0~0~1~0 \\[4pt] 0~1~1~0~0~0~0~1 \\[4pt] 1~0~0~0~0~1~1~0 \\[4pt] 0~1~0~0~1~0~0~1 \\[4pt] 0~0~1~0~1~0~0~1 \\[4pt] 0~0~0~1~0~1~1~0 \end{matrix}</math> | <math>\begin{matrix} \texttt{(~p~,~q~,~r~)} \\[4pt] \texttt{(~p~,~q~,(r))} \\[4pt] \texttt{(~p~,(q),~r~)} \\[4pt] \texttt{(~p~,(q),(r))} \\[4pt] \texttt{((p),~q~,~r~)} \\[4pt] \texttt{((p),~q~,(r))} \\[4pt] \texttt{((p),(q),~r~)} \\[4pt] \texttt{((p),(q),(r))} \end{matrix}</math> |- | <math>\begin{matrix} f_{233} \\[4pt] f_{214} \\[4pt] f_{182} \\[4pt] f_{121} \\[4pt] f_{158} \\[4pt] f_{109} \\[4pt] f_{107} \\[4pt] f_{151} \end{matrix}</math> | <math>\begin{matrix} f_{11101001} \\[4pt] f_{11010110} \\[4pt] f_{10110110} \\[4pt] f_{01111001} \\[4pt] f_{10011110} \\[4pt] f_{01101101} \\[4pt] f_{01101011} \\[4pt] f_{10010111} \end{matrix}</math> | <math>\begin{matrix} 1~1~1~0~1~0~0~1 \\[4pt] 1~1~0~1~0~1~1~0 \\[4pt] 1~0~1~1~0~1~1~0 \\[4pt] 0~1~1~1~1~0~0~1 \\[4pt] 1~0~0~1~1~1~1~0 \\[4pt] 0~1~1~0~1~1~0~1 \\[4pt] 0~1~1~0~1~0~1~1 \\[4pt] 1~0~0~1~0~1~1~1 \end{matrix}</math> | <math>\begin{matrix} \texttt{(((p),(q),(r)))} \\[4pt] \texttt{(((p),(q),~r~))} \\[4pt] \texttt{(((p),~q~,(r)))} \\[4pt] \texttt{(((p),~q~,~r~))} \\[4pt] \texttt{((~p~,(q),(r)))} \\[4pt] \texttt{((~p~,(q),~r~))} \\[4pt] \texttt{((~p~,~q~,(r)))} \\[4pt] \texttt{((~p~,~q~,~r~))} \end{matrix}</math> |} <br> ==Charts and graphs== This Section focuses on visual representations of minimal negation operators. A few bits of terminology are useful in describing the pictures, but the formal details are tedious reading, and may be familiar to many readers, so the full definitions of the terms marked in ''italics'' are relegated to a Glossary at the end of the article. Two ways of visualizing the space <math>\mathbb{B}^k</math> of <math>2^k\!</math> points are the [[hypercube]] picture and the [[venn diagram]] picture. The hypercube picture associates each point of <math>\mathbb{B}^k</math> with a unique point of the <math>k\!</math>-dimensional hypercube. The venn diagram picture associates each point of <math>\mathbb{B}^k</math> with a unique "cell" of the venn diagram on <math>k\!</math> "circles". In addition, each point of <math>\mathbb{B}^k</math> is the unique point in the ''[[fiber (mathematics)|fiber]] of truth'' <math>[|s|]\!</math> of a ''singular proposition'' <math>s : \mathbb{B}^k \to \mathbb{B},</math> and thus it is the unique point where a ''singular conjunction'' of <math>k\!</math> ''literals'' is <math>1.\!</math> For example, consider two cases at opposite vertices of the cube: {| align="center" cellpadding="4" width="90%" | valign="top" | <big>&bull;</big> | The point <math>(1, 1, \ldots , 1, 1)</math> with all 1's as coordinates is the point where the conjunction of all posited variables evaluates to <math>1,\!</math> namely, the point where: |- | &nbsp; | align="center" | <math>x_1 ~ x_2 ~\ldots~ x_{n-1} ~ x_n ~=~ 1.</math> |- | valign="top" | <big>&bull;</big> | The point <math>(0, 0, \ldots , 0, 0)</math> with all 0's as coordinates is the point where the conjunction of all negated variables evaluates to <math>1,\!</math> namely, the point where: |- | &nbsp; | align="center" | <math>\texttt{(} x_1 \texttt{)(} x_2 \texttt{)} \ldots \texttt{(} x_{n-1} \texttt{)(} x_n \texttt{)} ~=~ 1.</math> |} To pass from these limiting examples to the general case, observe that a singular proposition <math>s : \mathbb{B}^k \to \mathbb{B}</math> can be given canonical expression as a conjunction of literals, <math>s = e_1 e_2 \ldots e_{k-1} e_k</math>. Then the proposition <math>\nu (e_1, e_2, \ldots, e_{k-1}, e_k)</math> is <math>1\!</math> on the points adjacent to the point where <math>s\!</math> is <math>1,\!</math> and 0 everywhere else on the cube. For example, consider the case where <math>k = 3.\!</math> Then the minimal negation operation <math>\nu (p, q, r)\!</math> &mdash; written more simply as <math>\texttt{(p, q, r)}</math> &mdash; has the following venn diagram: {| align="center" cellpadding="8" style="text-align:center" | <p>[[Image:Venn Diagram (P,Q,R).jpg|500px]]</p> <p><math>\text{Figure 4.}~~\texttt{(p, q, r)}</math></p> |} For a contrasting example, the boolean function expressed by the form <math>\texttt{((p),(q),(r))}</math> has the following venn diagram: {| align="center" cellpadding="8" style="text-align:center" | <p>[[Image:Venn Diagram ((P),(Q),(R)).jpg|500px]]</p> <p><math>\text{Figure 5.}~~\texttt{((p),(q),(r))}</math></p> |} ==Glossary of basic terms== ; Boolean domain : A ''[[boolean domain]]'' <math>\mathbb{B}</math> is a generic 2-element set, for example, <math>\mathbb{B} = \{ 0, 1 \},</math> whose elements are interpreted as logical values, usually but not invariably with <math>0 = \operatorname{false}</math> and <math>1 = \operatorname{true}.</math> ; Boolean variable : A ''[[boolean variable]]'' <math>x\!</math> is a variable that takes its value from a boolean domain, as <math>x \in \mathbb{B}.</math> ; Proposition : In situations where boolean values are interpreted as logical values, a [[boolean-valued function]] <math>f : X \to \mathbb{B}</math> or a [[boolean function]] <math>g : \mathbb{B}^k \to \mathbb{B}</math> is frequently called a ''[[proposition]]''. ; Basis element, Coordinate projection : Given a sequence of <math>k\!</math> boolean variables, <math>x_1, \ldots, x_k,</math> each variable <math>x_j\!</math> may be treated either as a ''basis element'' of the space <math>\mathbb{B}^k</math> or as a ''coordinate projection'' <math>x_j : \mathbb{B}^k \to \mathbb{B}.</math> ; Basic proposition : This means that the set of objects <math>\{ x_j : 1 \le j \le k \}</math> is a set of boolean functions <math>\{ x_j : \mathbb{B}^k \to \mathbb{B} \}</math> subject to logical interpretation as a set of ''basic propositions'' that collectively generate the complete set of <math>2^{2^k}</math> propositions over <math>\mathbb{B}^k.</math> ; Literal : A ''literal'' is one of the <math>2k\!</math> propositions <math>x_1, \ldots, x_k, \texttt{(} x_1 \texttt{)}, \ldots, \texttt{(} x_k \texttt{)},</math> in other words, either a ''posited'' basic proposition <math>x_j\!</math> or a ''negated'' basic proposition <math>\texttt{(} x_j \texttt{)},</math> for some <math>j = 1 ~\text{to}~ k.</math> ; Fiber : In mathematics generally, the ''[[fiber (mathematics)|fiber]]'' of a point <math>y \in Y</math> under a function <math>f : X \to Y</math> is defined as the inverse image <math>f^{-1}(y) \subseteq X.</math> : In the case of a boolean function <math>f : \mathbb{B}^k \to \mathbb{B},</math> there are just two fibers: : The fiber of <math>0\!</math> under <math>f,\!</math> defined as <math>f^{-1}(0),\!</math> is the set of points where the value of <math>f\!</math> is <math>0.\!</math> : The fiber of <math>1\!</math> under <math>f,\!</math> defined as <math>f^{-1}(1),\!</math> is the set of points where the value of <math>f\!</math> is <math>1.\!</math> ; Fiber of truth : When <math>1\!</math> is interpreted as the logical value <math>\operatorname{true},</math> then <math>f^{-1}(1)\!</math> is called the ''fiber of truth'' in the proposition <math>f.\!</math> Frequent mention of this fiber makes it useful to have a shorter way of referring to it. This leads to the definition of the notation <math>[|f|] = f^{-1}(1)\!</math> for the fiber of truth in the proposition <math>f.\!</math> ; Singular boolean function : A ''singular boolean function'' <math>s : \mathbb{B}^k \to \mathbb{B}</math> is a boolean function whose fiber of <math>1\!</math> is a single point of <math>\mathbb{B}^k.</math> ; Singular proposition : In the interpretation where <math>1\!</math> equals <math>\operatorname{true},</math> a singular boolean function is called a ''singular proposition''. : Singular boolean functions and singular propositions serve as functional or logical representatives of the points in <math>\mathbb{B}^k.</math> ; Singular conjunction : A ''singular conjunction'' in <math>\mathbb{B}^k \to \mathbb{B}</math> is a conjunction of <math>k\!</math> literals that includes just one conjunct of the pair <math>\{ x_j, ~\nu(x_j) \}</math> for each <math>j = 1 ~\text{to}~ k.</math> : A singular proposition <math>s : \mathbb{B}^k \to \mathbb{B}</math> can be expressed as a singular conjunction: {| align="center" cellspacing"10" width="90%" | height="36" | <math>s ~=~ e_1 e_2 \ldots e_{k-1} e_k</math>, |- | <math>\begin{array}{llll} \text{where} & e_j & = & x_j \\[6pt] \text{or} & e_j & = & \nu (x_j), \\[6pt] \text{for} & j & = & 1 ~\text{to}~ k. \end{array}</math> |} ==Resources== * [http://planetmath.org/MinimalNegationOperator Minimal Negation Operator] @ [http://planetmath.org/ PlanetMath] * [http://planetphysics.us/encyclopedia/MinimalNegationOperator.html Minimal Negation Operator] @ [http://planetphysics.us/ PlanetPhysics] * [http://www.proofwiki.org/wiki/Definition:Minimal_Negation_Operator Minimal Negation Operator] @ [http://www.proofwiki.org/ ProofWiki] * [http://atlas.wolfram.com/ Wolfram Atlas of Simple Programs] ** [http://atlas.wolfram.com/01/01/ Elementary Cellular Automata Rules (ECARs)] ** [http://atlas.wolfram.com/01/01/rulelist.html ECAR Index] ** [http://atlas.wolfram.com/01/01/views/3/TableView.html ECAR Icons] ** [http://atlas.wolfram.com/01/01/views/87/TableView.html ECAR Examples] ** [http://atlas.wolfram.com/01/01/views/172/TableView.html ECAR Formulas] ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Minimal_negation_operator Minimal Negation Operator @ InterSciWiki] * [http://mywikibiz.com/Minimal_negation_operator Minimal Negation Operator @ MyWikiBiz] {{col-break}} * [http://beta.wikiversity.org/wiki/Minimal_negation_operator Minimal Negation Operator @ Wikiversity Beta] * [http://ref.subwiki.org/wiki/Minimal_negation_operator Minimal Negation Operator @ Subject Wikis] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Minimal_negation_operator Minimal Negation Operator], [http://mywikibiz.com/ MyWikiBiz] * [http://planetmath.org/MinimalNegationOperator Minimal Negation Operator], [http://planetmath.org/ PlanetMath] * [http://proofwiki.org/wiki/Definition:Minimal_Negation_Operator Minimal Negation Operator], [http://proofwiki.org/ ProofWiki] * [http://beta.wikiversity.org/wiki/Minimal_negation_operator Minimal Negation Operator], [http://beta.wikiversity.org/ Wikiversity Beta] {{col-break}} * [http://getwiki.net/-Minimal_Negation_Operator Minimal Negation Operator], [http://getwiki.net/ GetWiki] * [http://wikinfo.org/index.php/Minimal_negation_operator Minimal Negation Operator], [http://wikinfo.org/ Wikinfo] * [http://textop.org/wiki/index.php?title=Minimal_negation_operator Minimal Negation Operator], [http://textop.org/wiki/ Textop Wiki] * [http://en.wikipedia.org/w/index.php?title=Minimal_negation_operator&oldid=75156728 Minimal Negation Operator], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Automata Theory]] [[Category:Charles Sanders Peirce]] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Differential Logic]] [[Category:Equational Reasoning]] [[Category:Formal Languages]] [[Category:Formal Sciences]] [[Category:Formal Systems]] [[Category:Inquiry]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Neural Networks]] [[Category:Philosophy]] [[Category:Semiotics]] 06257c55f475ca21b4fca3bf476e51096e8ceb06 1 363 637 2012-05-21T14:22:21Z Jon Awbrey 3 Created page with "Hi, it seems that something is wrong with the LaTeX implementation. For example, see [[Minimal negation operator]]. ~~~~" wikitext text/x-wiki Hi, it seems that something is wrong with the LaTeX implementation. For example, see [[Minimal negation operator]]. [[User:Jon Awbrey|Jon Awbrey]] 07:22, 21 May 2012 (PDT) e7378adffcf673775b744fdaa06a260ca015a899 0 333 638 555 2012-05-22T03:00:24Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''logical matrix''', in the finite dimensional case, is a <math>k\!</math>-dimensional array with entries from the [[boolean domain]] <math>\mathbb{B} = \{ 0,1 \}.</math> Such a matrix affords a matrix representation of a <math>k\!</math>-adic [[relation (mathematics)|relation]]. ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://mywikibiz.com/Logical_matrix Logical Matrix @ MyWikiBiz] * [http://intersci.ss.uci.edu/wiki/index.php/Logical_matrix Logical Matrix @ InterSciWiki] * [http://wiki.oercommons.org/mediawiki/index.php/Logical_matrix Logical Matrix @ OER Commons] {{col-break}} * [http://p2pfoundation.net/Logical_Matrix Logical Matrix @ P2P Foundation] * [http://ref.subwiki.org/wiki/Logical_matrix Logical Matrix @ Subject Wikis] * [http://beta.wikiversity.org/wiki/Logical_matrix Logical Matrix @ Wikiversity Beta] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;] * [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;] ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Logical_matrix Logical Matrix], [http://mywikibiz.com/ MyWikiBiz] * [http://planetmath.org/encyclopedia/LogicalMatrix.html Logical Matrix], [http://planetmath.org/ PlanetMath] * [http://beta.wikiversity.org/wiki/Logical_matrix Logical Matrix], [http://beta.wikiversity.org/ Wikiversity Beta] {{col-break}} * [http://wikinfo.org/index.php/Logical_matrix Logical Matrix], [http://wikinfo.org/ Wikinfo] * [http://textop.org/wiki/index.php?title=Logical_matrix Logical Matrix], [http://textop.org/wiki/ Textop Wiki] * [http://en.wikipedia.org/w/index.php?title=Logical_matrix&oldid=43606082 Logical Matrix], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Charles Sanders Peirce]] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Discrete Mathematics]] [[Category:Logic]] [[Category:Mathematics]] f0b13ea41fd79ee7592fbc6534c83a209f4524d2 6 364 647 2013-10-03T14:50:27Z Jon Awbrey 3 wikitext text/x-wiki da39a3ee5e6b4b0d3255bfef95601890afd80709 0 312 648 533 2013-10-09T05:00:23Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. This article treats relations from the perspective of combinatorics, in other words, as a subject matter in discrete mathematics, with special attention to finite structures and concrete set-theoretic constructions, many of which arise quite naturally in applications. This approach to relation theory, or the theory of relations, is distinguished from, though closely related to, its study from the perspectives of abstract algebra on the one hand and formal logic on the other. ==Preliminaries== Two definitions of the relation concept are common in the literature. Although it is usually clear in context which definition is being used at a given time, it tends to become less clear as contexts collide, or as discussion moves from one context to another. The same sort of ambiguity arose in the development of the function concept and it may save some effort to follow the pattern of resolution that worked itself out there. When we speak of a function <math>f : X \to Y\!</math> we are thinking of a mathematical object whose articulation requires three pieces of data, specifying the set <math>X,\!</math> the set <math>Y,\!</math> and a particular subset of their cartesian product <math>{X \times Y}.\!</math> So far so good. Let us write <math>f = (\mathrm{obj_1}f, \mathrm{obj_2}f, \mathrm{obj_{12}}f)\!</math> to express what has been said so far. When it comes to parsing the notation <math>{}^{\backprime\backprime} f : X \to Y {}^{\prime\prime},\!</math> everyone takes the part <math>{}^{\backprime\backprime} X \to Y {}^{\prime\prime}\!</math> to specify the \textit{type} of the function, that is, the pair <math>(\mathrm{obj_1}f, \mathrm{obj_2}f),\!</math> but <math>{}^{\backprime\backprime} f {}^{\prime\prime}\!</math> is used equivocally to denote both the triple and the subset <math>\mathrm{obj_{12}}f\!</math> that forms one part of it. One way to resolve the ambiguity is to formalize a distinction between a function and its \textit{graph}, letting <math>\mathrm{graph}(f) := \mathrm{obj_{12}}f.\!</math> Another tactic treats the whole notation <math>{}^{\backprime\backprime} f : X \to Y {}^{\prime\prime}\!</math> as sufficient denotation for the triple, letting <math>{}^{\backprime\backprime} f {}^{\prime\prime}\!</math> denote <math>\mathrm{graph}(f).\!</math> In categorical and computational contexts, at least initially, the type is regarded as an essential attribute or an integral part of the function itself. In other contexts it may be desirable to use a more abstract concept of function, treating a function as a mathematical object that appears in connection with many different types. Following the pattern of the functional case, let the notation <math>{}^{\backprime\backprime} L \subseteq X \times Y {}^{\prime\prime}\!</math> bring to mind a mathematical object that is specified by three pieces of data, the set <math>X,\!</math> the set <math>Y,\!</math> and a particular subset of their cartesian product <math>{X \times Y}.\!</math> As before we have two choices, either let <math>L = (X, Y, \mathrm{graph}(L))\!</math> or let <math>{}^{\backprime\backprime} L {}^{\prime\prime}\!</math> denote <math>\mathrm{graph}(L)\!</math> and choose another name for the triple. ==Definition== It is convenient to begin with the definition of a ''<math>k\!</math>-place relation'', where <math>k\!</math> is a positive integer. '''Definition.''' A ''<math>k\!</math>-place relation'' <math>L \subseteq X_1 \times \ldots \times X_k\!</math> over the nonempty sets <math>X_1, \ldots, X_k\!</math> is a <math>(k+1)\!</math>-tuple <math>(X_1, \ldots, X_k, L)\!</math> where <math>L\!</math> is a subset of the cartesian product <math>X_1 \times \ldots \times X_k.\!</math> ==Remarks== Though usage varies as usage will, there are several bits of optional language that are frequently useful in discussing relations. The sets <math>X_1, \ldots, X_k\!</math> are called the ''domains'' of the relation <math>L \subseteq X_1 \times \ldots \times X_k,\!</math> with <math>{X_j}\!</math> being the <math>j^\text{th}\!</math> domain. If all of the <math>{X_j}\!</math> are the same set <math>X\!</math> then <math>L \subseteq X_1 \times \ldots \times X_k\!</math> is more simply described as a <math>k\!</math>-place relation over <math>X.\!</math> The set <math>L\!</math> is called the ''graph'' of the relation <math>L \subseteq X_1 \times \ldots \times X_k,\!</math> on analogy with the graph of a function. If the sequence of sets <math>X_1, \ldots, X_k\!</math> is constant throughout a given discussion or is otherwise determinate in context, then the relation <math>L \subseteq X_1 \times \ldots \times X_k\!</math> is determined by its graph <math>L,\!</math> making it acceptable to denote the relation by referring to its graph. Other synonyms for the adjective ''<math>k\!</math>-place'' are ''<math>k\!</math>-adic'' and ''<math>k\!</math>-ary'', all of which leads to the integer <math>k\!</math> being called the ''dimension'', ''adicity'', or ''arity'' of the relation <math>L.\!</math> ==Local incidence properties== A ''local incidence property'' (LIP) of a relation <math>L\!</math> is a property that depends in turn on the properties of special subsets of <math>L\!</math> that are known as its ''local flags''. The local flags of a relation are defined in the following way: Let <math>L\!</math> be a <math>k\!</math>-place relation <math>L \subseteq X_1 \times \ldots \times X_k.\!</math> Select a relational domain <math>{X_j}\!</math> and one of its elements <math>x.\!</math> Then <math>L_{x \operatorname{at} j}\!</math> is a subset of <math>L\!</math> that is referred to as the ''flag'' of <math>L\!</math> with <math>x\!</math> at <math>{j},\!</math> or the ''<math>x \operatorname{at} j\!</math>-flag'' of <math>L,\!</math> an object that has the following definition: {| align="center" cellpadding="8" style="text-align:center" | <math>L_{x \operatorname{at} j} ~=~ \{ (x_1, \ldots, x_j, \ldots, x_k) \in L ~:~ x_j = x \}.\!</math> |} Any property <math>C\!</math> of the local flag <math>L_{x \operatorname{at} j} \subseteq L\!</math> is said to be a ''local incidence property'' of <math>L\!</math> with respect to the ''locus'' <math>x \operatorname{at} j.\!</math> A <math>k\!</math>-adic relation <math>L \subseteq X_1 \times \ldots \times X_k\!</math> is said to be ''<math>C\!</math>-regular'' at <math>j\!</math> if and only if every flag of <math>L\!</math> with <math>x\!</math> at <math>j\!</math> has the property <math>C,\!</math> where <math>x\!</math> is taken to vary over the ''theme'' of the fixed domain <math>X_j.\!</math> Expressed in symbols, <math>L\!</math> is ''<math>C\!</math>-regular'' at <math>j\!</math> if and only if <math>C(L_{x \operatorname{at} j})\!</math> is true for all <math>x\!</math> in <math>X_j.\!</math> ==Regional incidence properties== The definition of a local flag can be broadened from a point <math>x\!</math> in <math>{X_j}\!</math> to a subset <math>M\!</math> of <math>X_j,\!</math> arriving at the definition of a ''regional flag'' in the following way: Suppose that <math>L \subseteq X_1 \times \ldots \times X_k,\!</math> and choose a subset <math>M \subseteq X_j.\!</math> Then <math>L_{M \operatorname{at} j}\!</math> is a subset of <math>L\!</math> that is said to be the ''flag'' of <math>L\!</math> with <math>M\!</math> at <math>{j},\!</math> or the ''<math>M \operatorname{at} j\!</math>-flag'' of <math>L,\!</math> an object which has the following definition: {| align="center" cellpadding="8" style="text-align:center" | <math>L_{M \operatorname{at} j} ~=~ \{ (x_1, \ldots, x_j, \ldots, x_k) \in L ~:~ x_j \in M \}.\!</math> |} ==Numerical incidence properties== A ''numerical incidence property'' (NIP) of a relation is a local incidence property that depends on the cardinalities of its local flags. For example, <math>L\!</math> is said to be ''<math>c\!</math>-regular at <math>j\!</math>'' if and only if the cardinality of the local flag <math>L_{x \operatorname{at} j}\!</math> is <math>c\!</math> for all <math>x\!</math> in <math>{X_j},\!</math> or, to write it in symbols, if and only if <math>|L_{x \operatorname{at} j}| = c\!</math> for all <math>{x \in X_j}.\!</math> In a similar fashion, one can define the NIPs, ''<math>(<\!c)\!</math>-regular at <math>{j},\!</math>'' ''<math>(>\!c)\!</math>-regular at <math>{j},\!</math>'' and so on. For ease of reference, a few of these definitions are recorded here: {| align="center" cellpadding="8" style="text-align:center" | <math>\begin{matrix} L & \text{is} & c\textit{-regular} & \text{at}\ j & \text{if and only if} & |L_{x \operatorname{at} j}| & = & c & \text{for all}\ x \in X_j. \\[4pt] L & \text{is} & (<\!c)\textit{-regular} & \text{at}\ j & \text{if and only if} & |L_{x \operatorname{at} j}| & < & c & \text{for all}\ x \in X_j. \\[4pt] L & \text{is} & (>\!c)\textit{-regular} & \text{at}\ j & \text{if and only if} & |L_{x \operatorname{at} j}| & > & c & \text{for all}\ x \in X_j. \end{matrix}</math> |} ==Species of 2-adic relations== Returning to 2-adic relations, it is useful to describe some familiar classes of objects in terms of their local and numerical incidence properties. Let <math>L \subseteq S \times T\!</math> be an arbitrary 2-adic relation. The following properties of <math>L\!</math> can be defined: {| align="center" cellpadding="8" style="text-align:center" | <math>\begin{matrix} L & \text{is} & \textit{total} & \text{at}~ S & \text{if and only if} & L & \text{is} & (\ge 1)\text{-regular} & \text{at}~ S. \\[4pt] L & \text{is} & \textit{total} & \text{at}~ T & \text{if and only if} & L & \text{is} & (\ge 1)\text{-regular} & \text{at}~ T. \\[4pt] L & \text{is} & \textit{tubular} & \text{at}~ S & \text{if and only if} & L & \text{is} & (\le 1)\text{-regular} & \text{at}\ S. \\[4pt] L & \text{is} & \textit{tubular} & \text{at}~ T & \text{if and only if} & L & \text{is} & (\le 1)\text{-regular} & \text{at}~ T. \end{matrix}</math> |} If <math>L \subseteq S \times T\!</math> is tubular at <math>S\!</math> then <math>L\!</math> is called a ''partial function'' or a ''prefunction'' from <math>S\!</math> to <math>T.\!</math> This is sometimes indicated by giving <math>L\!</math> an alternate name, say, <math>{}^{\backprime\backprime} p {}^{\prime\prime},~\!</math> and writing <math>L = p : S \rightharpoonup T.\!</math> Just by way of formalizing the definition: {| align="center" cellpadding="8" style="text-align:center" | <math>\begin{matrix} L & = & p : S \rightharpoonup T & \text{if and only if} & L & \text{is} & \text{tubular} & \text{at}~ S. \end{matrix}</math> |} If <math>L\!</math> is a prefunction <math>p : S \rightharpoonup T\!</math> that happens to be total at <math>S,\!</math> then <math>L\!</math> is called a ''function'' from <math>S\!</math> to <math>T,\!</math> indicated by writing <math>L = f : S \to T.\!</math> To say that a relation <math>L \subseteq S \times T\!</math> is ''totally tubular'' at <math>S\!</math> is to say that it is <math>1\!</math>-regular at <math>S.\!</math> Thus, we may formalize the following definition: {| align="center" cellpadding="8" style="text-align:center" | <math>\begin{matrix} L & = & f : S \to T & \text{if and only if} & L & \text{is} & 1\text{-regular} & \text{at}~ S. \end{matrix}</math> |} In the case of a function <math>f : S \to T,\!</math> one has the following additional definitions: {| align="center" cellpadding="8" style="text-align:center" | <math>\begin{matrix} f & \text{is} & \textit{surjective} & \text{if and only if} & f & \text{is} & \text{total} & \text{at}~ T. \\[4pt] f & \text{is} & \textit{injective} & \text{if and only if} & f & \text{is} & \text{tubular} & \text{at}~ T. \\[4pt] f & \text{is} & \textit{bijective} & \text{if and only if} & f & \text{is} & 1\text{-regular} & \text{at}~ T. \end{matrix}</math> |} ==Variations== Because the concept of a relation has been developed quite literally from the beginnings of logic and mathematics, and because it has incorporated contributions from a diversity of thinkers from many different times and intellectual climes, there is a wide variety of terminology that the reader may run across in connection with the subject. One dimension of variation is reflected in the names that are given to <math>k\!</math>-place relations, for <math>k = 0, 1, 2, 3, \ldots,\!</math> with some writers using the Greek forms, ''medadic'', ''monadic'', ''dyadic'', ''triadic'', ''<math>k\!</math>-adic'', and other writers using the Latin forms, ''nullary'', ''unary'', ''binary'', ''ternary'', ''<math>k\!</math>-ary''. The number of relational domains may be referred to as the ''adicity'', ''arity'', or ''dimension'' of the relation. Accordingly, one finds a relation on a finite number of domains described as a ''polyadic'' relation or a ''finitary'' relation, but others count infinitary relations among the polyadic. If the number of domains is finite, say equal to <math>k,\!</math> then the relation may be described as a ''<math>k\!</math>-adic'' relation, a ''<math>k\!</math>-ary'' relation, or a ''<math>k\!</math>-dimensional'' relation, respectively. A more conceptual than nominal variation depends on whether one uses terms like ''predicate'', ''relation'', and even ''term'' to refer to the formal object proper or else to the allied syntactic items that are used to denote them. Compounded with this variation is still another, frequently associated with philosophical differences over the status in reality accorded formal objects. Among those who speak of numbers, functions, properties, relations, and sets as being real, that is to say, as having objective properties, there are divergences as to whether some things are more real than others, especially whether particulars or properties are equally real or else one is derivative in relationship to the other. Historically speaking, just about every combination of modalities has been used by one school of thought or another, but it suffices here merely to indicate how the options are generated. ==Examples== : ''See the articles on [[relation (mathematics)|relations]], [[relation composition]], [[relation reduction]], [[sign relation]]s, and [[triadic relation]]s for concrete examples of relations.'' Many relations of the greatest interest in mathematics are triadic relations, but this fact is somewhat disguised by the circumstance that many of them are referred to as ''binary operations'', and because the most familiar of these have very specific properties that are dictated by their axioms. This makes it practical to study these operations for quite some time by focusing on their dyadic aspects before being forced to consider their proper characters as triadic relations. ==References== * Peirce, C.S., &ldquo;Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic&rdquo;, ''Memoirs of the American Academy of Arts and Sciences'', 9, 317&ndash;378, 1870. Reprinted, ''Collected Papers'' CP 3.45&ndash;149, ''Chronological Edition'' CE 2, 359&ndash;429. * Ulam, S.M. and Bednarek, A.R., &ldquo;On the Theory of Relational Structures and Schemata for Parallel Computation&rdquo;, pp. 477&ndash;508 in A.R. Bednarek and Françoise Ulam (eds.), ''Analogies Between Analogies : The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators'', University of California Press, Berkeley, CA, 1990. ==Bibliography== * Barr, Michael, and Wells, Charles (1990), ''Category Theory for Computing Science'', Prentice Hall, Hemel Hempstead, UK. * Bourbaki, Nicolas (1994), ''Elements of the History of Mathematics'', John Meldrum (trans.), Springer-Verlag, Berlin, Germany. * Carnap, Rudolf (1958), ''Introduction to Symbolic Logic with Applications'', Dover Publications, New York, NY. * Chang, C.C., and Keisler, H.J. (1973), ''Model Theory'', North-Holland, Amsterdam, Netherlands. * van Dalen, Dirk (1980), \text{Logic and Structure'', 2nd edition, Springer-Verlag, Berlin, Germany. * Devlin, Keith J. (1993), ''The Joy of Sets : Fundamentals of Contemporary Set Theory'', 2nd edition, Springer-Verlag, New York, NY. * Halmos, Paul Richard (1960), ''Naive Set Theory'', D. 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(1977), ''A Course in Mathematical Logic'', Neal Koblitz (trans.), Springer-Verlag, New York, NY. * Mathematical Society of Japan (1993), ''Encyclopedic Dictionary of Mathematics'', 2nd edition, 2 vols., Kiyosi Itô (ed.), MIT Press, Cambridge, MA. * Mili, A., Desharnais, J., Mili, F., with Frappier, M. (1994), ''Computer Program Construction'', Oxford University Press, New York, NY. (Introduction to Tarskian relation theory and relational programming.) * Mitchell, John C. (1996), ''Foundations for Programming Languages'', MIT Press, Cambridge, MA. * Peirce, Charles Sanders (1870), ``Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic", ''Memoirs of the American Academy of Arts and Sciences'' 9 (1870), 317&ndash;378. Reprinted (CP 3.45&ndash;149), (CE 2, 359&ndash;429). * Peirce, Charles Sanders (1931&ndash;1935, 1958), ''Collected Papers of Charles Sanders Peirce'', vols. 1&ndash;6, Charles Hartshorne and Paul Weiss (eds.), vols. 7&ndash;8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA. Cited as (CP volume.paragraph). * Peirce, Charles Sanders (1981&ndash;), ''Writings of Charles S. Peirce : A Chronological Edition'', Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN. Cited as (CE volume, page). * Poizat, Bruno (2000), ''A Course in Model Theory : An Introduction to Contemporary Mathematical Logic'', Moses Klein (trans.), Springer-Verlag, New York, NY. * Quine, Willard Van Orman (1940/1981), ''Mathematical Logic'', 1940. Revised edition, Harvard University Press, Cambridge, MA, 1951. New preface, 1981. * Royce, Josiah (1961), ''The Principles of Logic'', Philosophical Library, New York, NY. * Runes, Dagobert D. (ed., 1962), ''Dictionary of Philosophy'', Littlefield, Adams, and Company, Totowa, NJ. * Shoenfield, Joseph R. (1967), ''Mathematical Logic'', Addison-Wesley, Reading, MA. * Styazhkin, N.I. (1969), ''History of Mathematical Logic from Leibniz to Peano'', MIT Press, Cambridge, MA. * Suppes, Patrick (1957/1999), ''Introduction to Logic'', 1st published 1957. Reprinted, Dover Publications, New York, NY, 1999. * Suppes, Patrick (1960/1972), ''Axiomatic Set Theory'', 1st published 1960. Reprinted, Dover Publications, New York, NY, 1972. * Tarski, Alfred (1956/1983), ''Logic, Semantics, Metamathematics : Papers from 1923 to 1938'', J.H. Woodger (trans.), Oxford University Press, 1956. 2nd edition, J. Corcoran (ed.), Hackett Publishing, Indianapolis, IN, 1983. * Ulam, Stanislaw Marcin; and Bednarek, A.R. (1977), &ldquo;On the Theory of Relational Structures and Schemata for Parallel Computation&rdquo;, pp. 477&ndash;508 in A.R. Bednarek and Françoise Ulam (eds.), ''Analogies Between Analogies : The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators'', University of California Press, Berkeley, CA, 1990. * Ulam, Stanislaw Marcin (1990), ''Analogies Between Analogies : The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators'', A.R. Bednarek and Françoise Ulam (eds.), University of California Press, Berkeley, CA. * Ullman, Jeffrey D. (1980), ''Principles of Database Systems'', Computer Science Press, Rockville, MD. * Venetus, Paulus (1472/1984), ''Logica Parva : Translation of the 1472 Edition with Introduction and Notes'', Alan R. Perreiah (trans.), Philosophia Verlag, Munich, Germany. ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Relation_theory Relation Theory @ InterSciWiki] * [http://mywikibiz.com/Relation_theory Relation Theory @ MyWikiBiz] {{col-break}} * [http://beta.wikiversity.org/wiki/Relation_theory Relation Theory @ Wikiversity Beta] * [http://ref.subwiki.org/wiki/Relation_theory Relation Theory @ Subject Wikis] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * 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[[Category:Set Theory]] 07fafa3955496e3ac4005f28ef05f584d4f73f48 649 648 2013-10-09T15:02:29Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. This article treats relations from the perspective of combinatorics, in other words, as a subject matter in discrete mathematics, with special attention to finite structures and concrete set-theoretic constructions, many of which arise quite naturally in applications. This approach to relation theory, or the theory of relations, is distinguished from, though closely related to, its study from the perspectives of abstract algebra on the one hand and formal logic on the other. ==Preliminaries== Two definitions of the relation concept are common in the literature. Although it is usually clear in context which definition is being used at a given time, it tends to become less clear as contexts collide, or as discussion moves from one context to another. The same sort of ambiguity arose in the development of the function concept and it may save some effort to follow the pattern of resolution that worked itself out there. When we speak of a function <math>f : X \to Y\!</math> we are thinking of a mathematical object whose articulation requires three pieces of data, specifying the set <math>X,\!</math> the set <math>Y,\!</math> and a particular subset of their cartesian product <math>{X \times Y}.\!</math> So far so good. Let us write <math>f = (\mathrm{obj_1}f, \mathrm{obj_2}f, \mathrm{obj_{12}}f)\!</math> to express what has been said so far. When it comes to parsing the notation <math>{}^{\backprime\backprime} f : X \to Y {}^{\prime\prime},\!</math> everyone takes the part <math>{}^{\backprime\backprime} X \to Y {}^{\prime\prime}\!</math> to specify the ''type'' of the function, that is, the pair <math>(\mathrm{obj_1}f, \mathrm{obj_2}f),\!</math> but <math>{}^{\backprime\backprime} f {}^{\prime\prime}\!</math> is used equivocally to denote both the triple and the subset <math>\mathrm{obj_{12}}f\!</math> that forms one part of it. One way to resolve the ambiguity is to formalize a distinction between a function and its ''graph'', letting <math>\mathrm{graph}(f) := \mathrm{obj_{12}}f.\!</math> Another tactic treats the whole notation <math>{}^{\backprime\backprime} f : X \to Y {}^{\prime\prime}\!</math> as sufficient denotation for the triple, letting <math>{}^{\backprime\backprime} f {}^{\prime\prime}\!</math> denote <math>\mathrm{graph}(f).\!</math> In categorical and computational contexts, at least initially, the type is regarded as an essential attribute or an integral part of the function itself. In other contexts it may be desirable to use a more abstract concept of function, treating a function as a mathematical object that appears in connection with many different types. Following the pattern of the functional case, let the notation <math>{}^{\backprime\backprime} L \subseteq X \times Y {}^{\prime\prime}\!</math> bring to mind a mathematical object that is specified by three pieces of data, the set <math>X,\!</math> the set <math>Y,\!</math> and a particular subset of their cartesian product <math>{X \times Y}.\!</math> As before we have two choices, either let <math>L = (X, Y, \mathrm{graph}(L))\!</math> or let <math>{}^{\backprime\backprime} L {}^{\prime\prime}\!</math> denote <math>\mathrm{graph}(L)\!</math> and choose another name for the triple. ==Definition== It is convenient to begin with the definition of a ''<math>k\!</math>-place relation'', where <math>k\!</math> is a positive integer. '''Definition.''' A ''<math>k\!</math>-place relation'' <math>L \subseteq X_1 \times \ldots \times X_k\!</math> over the nonempty sets <math>X_1, \ldots, X_k\!</math> is a <math>(k+1)\!</math>-tuple <math>(X_1, \ldots, X_k, L)\!</math> where <math>L\!</math> is a subset of the cartesian product <math>X_1 \times \ldots \times X_k.\!</math> ==Remarks== Though usage varies as usage will, there are several bits of optional language that are frequently useful in discussing relations. The sets <math>X_1, \ldots, X_k\!</math> are called the ''domains'' of the relation <math>L \subseteq X_1 \times \ldots \times X_k,\!</math> with <math>{X_j}\!</math> being the <math>j^\text{th}\!</math> domain. If all of the <math>{X_j}\!</math> are the same set <math>X\!</math> then <math>L \subseteq X_1 \times \ldots \times X_k\!</math> is more simply described as a <math>k\!</math>-place relation over <math>X.\!</math> The set <math>L\!</math> is called the ''graph'' of the relation <math>L \subseteq X_1 \times \ldots \times X_k,\!</math> on analogy with the graph of a function. If the sequence of sets <math>X_1, \ldots, X_k\!</math> is constant throughout a given discussion or is otherwise determinate in context, then the relation <math>L \subseteq X_1 \times \ldots \times X_k\!</math> is determined by its graph <math>L,\!</math> making it acceptable to denote the relation by referring to its graph. Other synonyms for the adjective ''<math>k\!</math>-place'' are ''<math>k\!</math>-adic'' and ''<math>k\!</math>-ary'', all of which leads to the integer <math>k\!</math> being called the ''dimension'', ''adicity'', or ''arity'' of the relation <math>L.\!</math> ==Local incidence properties== A ''local incidence property'' (LIP) of a relation <math>L\!</math> is a property that depends in turn on the properties of special subsets of <math>L\!</math> that are known as its ''local flags''. The local flags of a relation are defined in the following way: Let <math>L\!</math> be a <math>k\!</math>-place relation <math>L \subseteq X_1 \times \ldots \times X_k.\!</math> Select a relational domain <math>{X_j}\!</math> and one of its elements <math>x.\!</math> Then <math>L_{x \operatorname{at} j}\!</math> is a subset of <math>L\!</math> that is referred to as the ''flag'' of <math>L\!</math> with <math>x\!</math> at <math>{j},\!</math> or the ''<math>x \operatorname{at} j\!</math>-flag'' of <math>L,\!</math> an object that has the following definition: {| align="center" cellpadding="8" style="text-align:center" | <math>L_{x \operatorname{at} j} ~=~ \{ (x_1, \ldots, x_j, \ldots, x_k) \in L ~:~ x_j = x \}.\!</math> |} Any property <math>C\!</math> of the local flag <math>L_{x \operatorname{at} j} \subseteq L\!</math> is said to be a ''local incidence property'' of <math>L\!</math> with respect to the ''locus'' <math>x \operatorname{at} j.\!</math> A <math>k\!</math>-adic relation <math>L \subseteq X_1 \times \ldots \times X_k\!</math> is said to be ''<math>C\!</math>-regular'' at <math>j\!</math> if and only if every flag of <math>L\!</math> with <math>x\!</math> at <math>j\!</math> has the property <math>C,\!</math> where <math>x\!</math> is taken to vary over the ''theme'' of the fixed domain <math>X_j.\!</math> Expressed in symbols, <math>L\!</math> is ''<math>C\!</math>-regular'' at <math>j\!</math> if and only if <math>C(L_{x \operatorname{at} j})\!</math> is true for all <math>x\!</math> in <math>X_j.\!</math> ==Regional incidence properties== The definition of a local flag can be broadened from a point <math>x\!</math> in <math>{X_j}\!</math> to a subset <math>M\!</math> of <math>X_j,\!</math> arriving at the definition of a ''regional flag'' in the following way: Suppose that <math>L \subseteq X_1 \times \ldots \times X_k,\!</math> and choose a subset <math>M \subseteq X_j.\!</math> Then <math>L_{M \operatorname{at} j}\!</math> is a subset of <math>L\!</math> that is said to be the ''flag'' of <math>L\!</math> with <math>M\!</math> at <math>{j},\!</math> or the ''<math>M \operatorname{at} j\!</math>-flag'' of <math>L,\!</math> an object which has the following definition: {| align="center" cellpadding="8" style="text-align:center" | <math>L_{M \operatorname{at} j} ~=~ \{ (x_1, \ldots, x_j, \ldots, x_k) \in L ~:~ x_j \in M \}.\!</math> |} ==Numerical incidence properties== A ''numerical incidence property'' (NIP) of a relation is a local incidence property that depends on the cardinalities of its local flags. For example, <math>L\!</math> is said to be ''<math>c\!</math>-regular at <math>j\!</math>'' if and only if the cardinality of the local flag <math>L_{x \operatorname{at} j}\!</math> is <math>c\!</math> for all <math>x\!</math> in <math>{X_j},\!</math> or, to write it in symbols, if and only if <math>|L_{x \operatorname{at} j}| = c\!</math> for all <math>{x \in X_j}.\!</math> In a similar fashion, one can define the NIPs, ''<math>(<\!c)\!</math>-regular at <math>{j},\!</math>'' ''<math>(>\!c)\!</math>-regular at <math>{j},\!</math>'' and so on. For ease of reference, a few of these definitions are recorded here: {| align="center" cellpadding="8" style="text-align:center" | <math>\begin{matrix} L & \text{is} & c\textit{-regular} & \text{at}\ j & \text{if and only if} & |L_{x \operatorname{at} j}| & = & c & \text{for all}\ x \in X_j. \\[4pt] L & \text{is} & (<\!c)\textit{-regular} & \text{at}\ j & \text{if and only if} & |L_{x \operatorname{at} j}| & < & c & \text{for all}\ x \in X_j. \\[4pt] L & \text{is} & (>\!c)\textit{-regular} & \text{at}\ j & \text{if and only if} & |L_{x \operatorname{at} j}| & > & c & \text{for all}\ x \in X_j. \end{matrix}</math> |} ==Species of 2-adic relations== Returning to 2-adic relations, it is useful to describe some familiar classes of objects in terms of their local and numerical incidence properties. Let <math>L \subseteq S \times T\!</math> be an arbitrary 2-adic relation. The following properties of <math>L\!</math> can be defined: {| align="center" cellpadding="8" style="text-align:center" | <math>\begin{matrix} L & \text{is} & \textit{total} & \text{at}~ S & \text{if and only if} & L & \text{is} & (\ge 1)\text{-regular} & \text{at}~ S. \\[4pt] L & \text{is} & \textit{total} & \text{at}~ T & \text{if and only if} & L & \text{is} & (\ge 1)\text{-regular} & \text{at}~ T. \\[4pt] L & \text{is} & \textit{tubular} & \text{at}~ S & \text{if and only if} & L & \text{is} & (\le 1)\text{-regular} & \text{at}\ S. \\[4pt] L & \text{is} & \textit{tubular} & \text{at}~ T & \text{if and only if} & L & \text{is} & (\le 1)\text{-regular} & \text{at}~ T. \end{matrix}</math> |} If <math>L \subseteq S \times T\!</math> is tubular at <math>S\!</math> then <math>L\!</math> is called a ''partial function'' or a ''prefunction'' from <math>S\!</math> to <math>T.\!</math> This is sometimes indicated by giving <math>L\!</math> an alternate name, say, <math>{}^{\backprime\backprime} p {}^{\prime\prime},~\!</math> and writing <math>L = p : S \rightharpoonup T.\!</math> Just by way of formalizing the definition: {| align="center" cellpadding="8" style="text-align:center" | <math>\begin{matrix} L & = & p : S \rightharpoonup T & \text{if and only if} & L & \text{is} & \text{tubular} & \text{at}~ S. \end{matrix}</math> |} If <math>L\!</math> is a prefunction <math>p : S \rightharpoonup T\!</math> that happens to be total at <math>S,\!</math> then <math>L\!</math> is called a ''function'' from <math>S\!</math> to <math>T,\!</math> indicated by writing <math>L = f : S \to T.\!</math> To say that a relation <math>L \subseteq S \times T\!</math> is ''totally tubular'' at <math>S\!</math> is to say that it is <math>1\!</math>-regular at <math>S.\!</math> Thus, we may formalize the following definition: {| align="center" cellpadding="8" style="text-align:center" | <math>\begin{matrix} L & = & f : S \to T & \text{if and only if} & L & \text{is} & 1\text{-regular} & \text{at}~ S. \end{matrix}</math> |} In the case of a function <math>f : S \to T,\!</math> one has the following additional definitions: {| align="center" cellpadding="8" style="text-align:center" | <math>\begin{matrix} f & \text{is} & \textit{surjective} & \text{if and only if} & f & \text{is} & \text{total} & \text{at}~ T. \\[4pt] f & \text{is} & \textit{injective} & \text{if and only if} & f & \text{is} & \text{tubular} & \text{at}~ T. \\[4pt] f & \text{is} & \textit{bijective} & \text{if and only if} & f & \text{is} & 1\text{-regular} & \text{at}~ T. \end{matrix}</math> |} ==Variations== Because the concept of a relation has been developed quite literally from the beginnings of logic and mathematics, and because it has incorporated contributions from a diversity of thinkers from many different times and intellectual climes, there is a wide variety of terminology that the reader may run across in connection with the subject. One dimension of variation is reflected in the names that are given to <math>k\!</math>-place relations, for <math>k = 0, 1, 2, 3, \ldots,\!</math> with some writers using the Greek forms, ''medadic'', ''monadic'', ''dyadic'', ''triadic'', ''<math>k\!</math>-adic'', and other writers using the Latin forms, ''nullary'', ''unary'', ''binary'', ''ternary'', ''<math>k\!</math>-ary''. The number of relational domains may be referred to as the ''adicity'', ''arity'', or ''dimension'' of the relation. Accordingly, one finds a relation on a finite number of domains described as a ''polyadic'' relation or a ''finitary'' relation, but others count infinitary relations among the polyadic. If the number of domains is finite, say equal to <math>k,\!</math> then the relation may be described as a ''<math>k\!</math>-adic'' relation, a ''<math>k\!</math>-ary'' relation, or a ''<math>k\!</math>-dimensional'' relation, respectively. A more conceptual than nominal variation depends on whether one uses terms like ''predicate'', ''relation'', and even ''term'' to refer to the formal object proper or else to the allied syntactic items that are used to denote them. Compounded with this variation is still another, frequently associated with philosophical differences over the status in reality accorded formal objects. Among those who speak of numbers, functions, properties, relations, and sets as being real, that is to say, as having objective properties, there are divergences as to whether some things are more real than others, especially whether particulars or properties are equally real or else one is derivative in relationship to the other. Historically speaking, just about every combination of modalities has been used by one school of thought or another, but it suffices here merely to indicate how the options are generated. ==Examples== : ''See the articles on [[relation (mathematics)|relations]], [[relation composition]], [[relation reduction]], [[sign relation]]s, and [[triadic relation]]s for concrete examples of relations.'' Many relations of the greatest interest in mathematics are triadic relations, but this fact is somewhat disguised by the circumstance that many of them are referred to as ''binary operations'', and because the most familiar of these have very specific properties that are dictated by their axioms. This makes it practical to study these operations for quite some time by focusing on their dyadic aspects before being forced to consider their proper characters as triadic relations. ==References== * Peirce, C.S., &ldquo;Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic&rdquo;, ''Memoirs of the American Academy of Arts and Sciences'', 9, 317&ndash;378, 1870. Reprinted, ''Collected Papers'' CP 3.45&ndash;149, ''Chronological Edition'' CE 2, 359&ndash;429. * Ulam, S.M. and Bednarek, A.R., &ldquo;On the Theory of Relational Structures and Schemata for Parallel Computation&rdquo;, pp. 477&ndash;508 in A.R. Bednarek and Françoise Ulam (eds.), ''Analogies Between Analogies : The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators'', University of California Press, Berkeley, CA, 1990. ==Bibliography== * Barr, Michael, and Wells, Charles (1990), ''Category Theory for Computing Science'', Prentice Hall, Hemel Hempstead, UK. * Bourbaki, Nicolas (1994), ''Elements of the History of Mathematics'', John Meldrum (trans.), Springer-Verlag, Berlin, Germany. * Carnap, Rudolf (1958), ''Introduction to Symbolic Logic with Applications'', Dover Publications, New York, NY. * Chang, C.C., and Keisler, H.J. (1973), ''Model Theory'', North-Holland, Amsterdam, Netherlands. * van Dalen, Dirk (1980), \text{Logic and Structure'', 2nd edition, Springer-Verlag, Berlin, Germany. * Devlin, Keith J. (1993), ''The Joy of Sets : Fundamentals of Contemporary Set Theory'', 2nd edition, Springer-Verlag, New York, NY. * Halmos, Paul Richard (1960), ''Naive Set Theory'', D. Van Nostrand Company, Princeton, NJ. * van Heijenoort, Jean (1967/1977), ''From Frege to Gödel : A Source Book in Mathematical Logic, 1879&ndash;1931'', Harvard University Press, Cambridge, MA, 1967. Reprinted with corrections, 1977. * Kelley, John L. (1955), ''General Topology'', Van Nostrand Reinhold, New York, NY. * Kneale, William; and Kneale, Martha (1962/1975), ''The Development of Logic'', Oxford University Press, Oxford, UK, 1962. Reprinted with corrections, 1975. * Lawvere, Francis William; and Rosebrugh, Robert (2003), ''Sets for Mathematics'', Cambridge University Press, Cambridge, UK. * Lawvere, Francis William; and Schanuel, Stephen H. (1997/2000), ''Conceptual Mathematics : A First Introduction to Categories'', Cambridge University Press, Cambridge, UK, 1997. Reprinted with corrections, 2000. * Manin, Yu. I. (1977), ''A Course in Mathematical Logic'', Neal Koblitz (trans.), Springer-Verlag, New York, NY. * Mathematical Society of Japan (1993), ''Encyclopedic Dictionary of Mathematics'', 2nd edition, 2 vols., Kiyosi Itô (ed.), MIT Press, Cambridge, MA. * Mili, A., Desharnais, J., Mili, F., with Frappier, M. (1994), ''Computer Program Construction'', Oxford University Press, New York, NY. (Introduction to Tarskian relation theory and relational programming.) * Mitchell, John C. (1996), ''Foundations for Programming Languages'', MIT Press, Cambridge, MA. * Peirce, Charles Sanders (1870), ``Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic", ''Memoirs of the American Academy of Arts and Sciences'' 9 (1870), 317&ndash;378. Reprinted (CP 3.45&ndash;149), (CE 2, 359&ndash;429). * Peirce, Charles Sanders (1931&ndash;1935, 1958), ''Collected Papers of Charles Sanders Peirce'', vols. 1&ndash;6, Charles Hartshorne and Paul Weiss (eds.), vols. 7&ndash;8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA. Cited as (CP volume.paragraph). * Peirce, Charles Sanders (1981&ndash;), ''Writings of Charles S. Peirce : A Chronological Edition'', Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN. Cited as (CE volume, page). * Poizat, Bruno (2000), ''A Course in Model Theory : An Introduction to Contemporary Mathematical Logic'', Moses Klein (trans.), Springer-Verlag, New York, NY. * Quine, Willard Van Orman (1940/1981), ''Mathematical Logic'', 1940. Revised edition, Harvard University Press, Cambridge, MA, 1951. New preface, 1981. * Royce, Josiah (1961), ''The Principles of Logic'', Philosophical Library, New York, NY. * Runes, Dagobert D. (ed., 1962), ''Dictionary of Philosophy'', Littlefield, Adams, and Company, Totowa, NJ. * Shoenfield, Joseph R. (1967), ''Mathematical Logic'', Addison-Wesley, Reading, MA. * Styazhkin, N.I. (1969), ''History of Mathematical Logic from Leibniz to Peano'', MIT Press, Cambridge, MA. * Suppes, Patrick (1957/1999), ''Introduction to Logic'', 1st published 1957. Reprinted, Dover Publications, New York, NY, 1999. * Suppes, Patrick (1960/1972), ''Axiomatic Set Theory'', 1st published 1960. Reprinted, Dover Publications, New York, NY, 1972. * Tarski, Alfred (1956/1983), ''Logic, Semantics, Metamathematics : Papers from 1923 to 1938'', J.H. Woodger (trans.), Oxford University Press, 1956. 2nd edition, J. Corcoran (ed.), Hackett Publishing, Indianapolis, IN, 1983. * Ulam, Stanislaw Marcin; and Bednarek, A.R. (1977), &ldquo;On the Theory of Relational Structures and Schemata for Parallel Computation&rdquo;, pp. 477&ndash;508 in A.R. Bednarek and Françoise Ulam (eds.), ''Analogies Between Analogies : The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators'', University of California Press, Berkeley, CA, 1990. * Ulam, Stanislaw Marcin (1990), ''Analogies Between Analogies : The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators'', A.R. Bednarek and Françoise Ulam (eds.), University of California Press, Berkeley, CA. * Ullman, Jeffrey D. (1980), ''Principles of Database Systems'', Computer Science Press, Rockville, MD. * Venetus, Paulus (1472/1984), ''Logica Parva : Translation of the 1472 Edition with Introduction and Notes'', Alan R. Perreiah (trans.), Philosophia Verlag, Munich, Germany. ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Relation_theory Relation Theory @ InterSciWiki] * [http://mywikibiz.com/Relation_theory Relation Theory @ MyWikiBiz] {{col-break}} * [http://beta.wikiversity.org/wiki/Relation_theory Relation Theory @ Wikiversity Beta] * [http://ref.subwiki.org/wiki/Relation_theory Relation Theory @ Subject Wikis] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Relation_theory Relation Theory], [http://mywikibiz.com/ MyWikiBiz] * [http://mathweb.org/wiki/Relation_theory Relation Theory], [http://mathweb.org/wiki/ MathWeb Wiki] * [http://netknowledge.org/wiki/Relation_theory Relation Theory], [http://netknowledge.org/ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Relation_theory Relation Theory], [http://wiki.oercommons.org/ OER Commons] {{col-break}} * [http://p2pfoundation.net/Relation_Theory Relation Theory], [http://p2pfoundation.net/ P2P Foundation] * [http://semanticweb.org/wiki/Relation_theory Relation Theory], [http://semanticweb.org/ 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[[Category:Set Theory]] e789b2aad1cd6e845aa4888287be3068db1cc438 0 187 651 575 2013-10-24T02:06:24Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''logical graph''' is a graph-theoretic structure in one of the systems of graphical syntax that Charles Sanders Peirce developed for logic. In his papers on ''qualitative logic'', ''entitative graphs'', and ''existential graphs'', Peirce developed several versions of a graphical formalism, or a graph-theoretic formal language, designed to be interpreted for logic. In the century since Peirce initiated this line of development, a variety of formal systems have branched out from what is abstractly the same formal base of graph-theoretic structures. This article examines the common basis of these formal systems from a bird's eye view, focusing on those aspects of form that are shared by the entire family of algebras, calculi, or languages, however they happen to be viewed in a given application. ==Abstract point of view== {| width="100%" cellpadding="2" cellspacing="0" | width="60%" | &nbsp; | width="40%" | ''Wollust ward dem Wurm gegeben &hellip;'' |- | &nbsp; | align="right" | &mdash; Friedrich Schiller, ''An die Freude'' |} The bird's eye view in question is more formally known as the perspective of formal equivalence, from which remove one cannot see many distinctions that appear momentous from lower levels of abstraction. In particular, expressions of different formalisms whose syntactic structures are isomorphic from the standpoint of algebra or topology are not recognized as being different from each other in any significant sense. Though we may note in passing such historical details as the circumstance that Charles Sanders Peirce used a ''streamer-cross symbol'' where George Spencer Brown used a ''carpenter's square marker'', the theme of principal interest at the abstract level of form is neutral with regard to variations of that order. ==In lieu of a beginning== Consider the formal equations indicated in Figures&nbsp;1 and 2. {| align="center" border="0" cellpadding="10" cellspacing="0" | [[Image:Logical_Graph_Figure_1_Visible_Frame.jpg|500px]] || (1) |- | [[Image:Logical_Graph_Figure_2_Visible_Frame.jpg|500px]] || (2) |} For the time being these two forms of transformation may be referred to as ''[[axioms]]'' or ''initial equations''. ==Duality : logical and topological== There are two types of duality that have to be kept separately mind in the use of logical graphs &mdash; logical duality and topological duality. There is a standard way that graphs of the order that Peirce considered, those embedded in a continuous [[manifold]] like that commonly represented by a plane sheet of paper &mdash; with or without the paper bridges that Peirce used to augment its topological genus &mdash; can be represented in linear text as what are called ''parse strings'' or ''traversal strings'' and parsed into ''pointer structures'' in computer memory. A blank sheet of paper can be represented in linear text as a blank space, but that way of doing it tends to be confusing unless the logical expression under consideration is set off in a separate display. For example, consider the axiom or initial equation that is shown below: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_3_Visible_Frame.jpg|500px]] || (3) |} This can be written inline as <math>{}^{\backprime\backprime} \texttt{(~(~)~)} = \quad {}^{\prime\prime}</math> or set off in a text display as follows: {| align="center" cellpadding="10" | width="33%" | <math>\texttt{(~(~)~)}</math> | width="34%" | <math>=\!</math> | width="33%" | &nbsp; |} When we turn to representing the corresponding expressions in computer memory, where they can be manipulated with utmost facility, we begin by transforming the planar graphs into their topological duals. The planar regions of the original graph correspond to nodes (or points) of the [[dual graph]], and the boundaries between planar regions in the original graph correspond to edges (or lines) between the nodes of the dual graph. For example, overlaying the corresponding dual graphs on the plane-embedded graphs shown above, we get the following composite picture: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_4_Visible_Frame.jpg|500px]] || (4) |} Though it's not really there in the most abstract topology of the matter, for all sorts of pragmatic reasons we find ourselves compelled to single out the outermost region of the plane in a distinctive way and to mark it as the ''[[root node]]'' of the corresponding dual graph. In the present style of Figure the root nodes are marked by horizontal strike-throughs. Extracting the dual graphs from their composite matrices, we get this picture: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_5_Visible_Frame.jpg|500px]] || (5) |} It is easy to see the relationship between the parenthetical expressions of Peirce's logical graphs, that somewhat clippedly picture the ordered containments of their formal contents, and the associated dual graphs, that constitute the species of [[rooted tree]]s here to be described. In the case of our last example, a moment's contemplation of the following picture will lead us to see that we can get the corresponding parenthesis string by starting at the root of the tree, climbing up the left side of the tree until we reach the top, then climbing back down the right side of the tree until we return to the root, all the while reading off the symbols, in this case either <math>{}^{\backprime\backprime} \texttt{(} {}^{\prime\prime}</math> or <math>{}^{\backprime\backprime} \texttt{)} {}^{\prime\prime},</math> that we happen to encounter in our travels. {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_6_Visible_Frame.jpg|500px]] || (6) |} This ritual is called ''[[traversing]]'' the tree, and the string read off is called the ''[[traversal string]]'' of the tree. The reverse ritual, that passes from the string to the tree, is called ''[[parsing]]'' the string, and the tree constructed is called the ''[[parse graph]]'' of the string. The speakers thereof tend to be a bit loose in this language, often using ''[[parse string]]'' to mean the string that gets parsed into the associated graph. We have treated in some detail various forms of the initial equation or logical axiom that is formulated in string form as <math>{}^{\backprime\backprime} \texttt{(~(~)~)} = \quad {}^{\prime\prime}.</math> For the sake of comparison, let's record the plane-embedded and topological dual forms of the axiom that is formulated in string form as <math>{}^{\backprime\backprime} \texttt{(~)(~)} = \texttt{(~)} {}^{\prime\prime}.</math> First the plane-embedded maps: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_7_Visible_Frame.jpg|500px]] || (7) |} Next the plane-embedded maps and their dual trees superimposed: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_8_Visible_Frame.jpg|500px]] || (8) |} Finally the dual trees by themselves: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_9_Visible_Frame.jpg|500px]] || (9) |} And here are the parse trees with their traversal strings indicated: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_10_Visible_Frame.jpg|500px]] || (10) |} We have at this point enough material to begin thinking about the forms of [[analogy]], [[iconicity]], [[metaphor]], [[morphism]], whatever you want to call them, that are pertinent to the use of logical graphs in their various logical interpretations, for instance, those that Peirce described as ''[[entitative graph]]s'' and ''[[existential graph]]s''. ==Computational representation== The parse graphs that we've been looking at so far bring us one step closer to the pointer graphs that it takes to make these maps and trees live in computer memory, but they are still a couple of steps too abstract to detail the concrete species of dynamic data structures that we need. The time has come to flesh out the skeletons that we've drawn up to this point. Nodes in a graph represent ''records'' in computer memory. A record is a collection of data that can be conceived to reside at a specific ''address''. The address of a record is analogous to a demonstrative pronoun, on which account programmers commonly describe it as a ''pointer'' and semioticians recognize it as a type of sign called an ''index''. At the next level of concreteness, a pointer-record structure is represented as follows: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_11_Visible_Frame.jpg|500px]] || (11) |} This portrays the pointer <math>\mathit{index}_0\!</math> as the address of a record that contains the following data: {| align="center" cellpadding="10" | <math>\mathit{datum}_1, \mathit{datum}_2, \mathit{datum}_3, \ldots,\!</math> and so on. |} What makes it possible to represent graph-theoretical structures as data structures in computer memory is the fact that an address is just another datum, and so we may have a state of affairs like the following: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_12_Visible_Frame.jpg|500px]] || (12) |} Returning to the abstract level, it takes three nodes to represent the three data records illustrated above: one root node connected to a couple of adjacent nodes. The items of data that do not point any further up the tree are then treated as labels on the record-nodes where they reside, as shown below: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_13_Visible_Frame.jpg|500px]] || (13) |} Notice that drawing the arrows is optional with rooted trees like these, since singling out a unique node as the root induces a unique orientation on all the edges of the tree, with ''up'' being the same direction as ''away from the root''. ==Quick tour of the neighborhood== This much preparation allows us to take up the founding axioms or initial equations that determine the entire system of logical graphs. ===Primary arithmetic as semiotic system=== Though it may not seem too exciting, logically speaking, there are many reasons to make oneself at home with the system of forms that is represented indifferently, topologically speaking, by rooted trees, balanced strings of parentheses, or finite sets of non-intersecting simple closed curves in the plane. :* One reason is that it gives us a respectable example of a sign domain on which to cut our semiotic teeth, non-trivial in the sense that it contains a [[countable]] [[infinity]] of signs. :* Another reason is that it allows us to study a simple form of [[computation]] that is recognizable as a species of ''[[semiosis]]'', or sign-transforming process. This space of forms, along with the two axioms that induce its [[partition of a set|partition]] into exactly two [[equivalence class]]es, is what [[George Spencer Brown]] called the ''primary arithmetic''. The axioms of the primary arithmetic are shown below, as they appear in both graph and string forms, along with pairs of names that come in handy for referring to the two opposing directions of applying the axioms. {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_14_Banner.jpg|500px]] || (14) |- | [[Image:Logical_Graph_Figure_15_Banner.jpg|500px]] || (15) |} Let <math>S\!</math> be the set of rooted trees and let <math>S_0\!</math> be the 2-element subset of <math>S\!</math> that consists of a rooted node and a rooted edge. {| align="center" cellpadding="10" style="text-align:center" | <math>S\!</math> | <math>=\!</math> | <math>\{ \text{rooted trees} \}\!</math> |- | <math>S_0\!</math> | <math>=\!</math> | <math>\{ \ominus, \vert \} = \{</math>[[Image:Rooted Node.jpg|16px]], [[Image:Rooted Edge.jpg|12px]]<math>\}\!</math> |} Simple intuition, or a simple inductive proof, assures us that any rooted tree can be reduced by way of the arithmetic initials either to a root node [[Image:Rooted Node.jpg|16px]] or else to a rooted edge [[Image:Rooted Edge.jpg|12px]]&nbsp;. For example, consider the reduction that proceeds as follows: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_16.jpg|500px]] || (16) |} Regarded as a semiotic process, this amounts to a sequence of signs, every one after the first serving as the ''[[interpretant]]'' of its predecessor, ending in a final sign that may be taken as the canonical sign for their common object, in the upshot being the result of the computation process. Simple as it is, this exhibits the main features of any computation, namely, a semiotic process that proceeds from an obscure sign to a clear sign of the same object, being in its aim and effect an action on behalf of clarification. ===Primary algebra as pattern calculus=== Experience teaches that complex objects are best approached in a gradual, laminar, [[module|modular]] fashion, one step, one layer, one piece at a time, and it's just as much the case when the complexity of the object is irreducible, that is, when the articulations of the representation are necessarily at joints that are cloven disjointedly from nature, with some assembly required in the synthetic integrity of the intuition. That's one good reason for spending so much time on the first half of [[zeroth order logic]], represented here by the primary arithmetic, a level of formal structure that C.S. Peirce verged on intuiting at numerous points and times in his work on logical graphs, and that Spencer Brown named and brought more completely to life. There is one other reason for lingering a while longer in these primitive forests, and this is that an acquaintance with "bare trees", those as yet unadorned with literal or numerical labels, will provide a firm basis for understanding what's really at issue in such problems as the "ontological status of variables". It is probably best to illustrate this theme in the setting of a concrete case, which we can do by reviewing the previous example of reductive evaluation shown in Figure&nbsp;16. The observation of several ''semioses'', or sign-transformations, of roughly this shape will most likely lead an observer with any observational facility whatever to notice that it doesn't really matter what sorts of branches happen to sprout from the side of the root aside from the lone edge that also grows there &mdash; the end result will always be the same. Our observer might think to summarize the results of many such observations by introducing a label or variable to signify any shape of branch whatever, writing something like the following: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_17.jpg|500px]] || (17) |} Observations like that, made about an arithmetic of any variety, germinated by their summarizations, are the root of all algebra. Speaking of algebra, and having encountered already one example of an algebraic law, we might as well introduce the axioms of the ''primary algebra'', once again deriving their substance and their name from the works of [[Charles Sanders Peirce]] and [[George Spencer Brown]], respectively. {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_18.jpg|500px]] || (18) |- | [[Image:Logical_Graph_Figure_19.jpg|500px]] || (19) |} The choice of axioms for any formal system is to some degree a matter of aesthetics, as it is commonly the case that many different selections of formal rules will serve as axioms to derive all the rest as theorems. As it happens, the example of an algebraic law that we noticed first, <math>a(~) = (~),</math> as simple as it appears, proves to be provable as a theorem on the grounds of the foregoing axioms. We might also notice at this point a subtle difference between the primary arithmetic and the primary algebra with respect to the grounds of justification that we have naturally if tacitly adopted for their respective sets of axioms. The arithmetic axioms were introduced by fiat, in a quasi-[[apriori]] fashion, though of course it is only long prior experience with the practical uses of comparably developed generations of formal systems that would actually induce us to such a quasi-primal move. The algebraic axioms, in contrast, can be seen to derive their motive and their justice from the observation and summarization of patterns that are visible in the arithmetic spectrum. ==Formal development== What precedes this point is intended as an informal introduction to the axioms of the primary arithmetic and primary algebra, and hopefully provides the reader with an intuitive sense of their motivation and rationale. The next order of business is to give the exact forms of the axioms that are used in the following more formal development, devolving from Peirce's various systems of logical graphs via Spencer-Brown's ''Laws of Form'' (LOF). In formal proofs, a variation of the annotation scheme from LOF will be used to mark each step of the proof according to which axiom, or ''initial'', is being invoked to justify the corresponding step of syntactic transformation, whether it applies to graphs or to strings. ===Axioms=== The axioms are just four in number, divided into the ''arithmetic initials'', <math>I_1\!</math> and <math>I_2,\!</math> and the ''algebraic initials'', <math>J_1\!</math> and <math>J_2.\!</math> {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_20.jpg|500px]] || (20) |- | [[Image:Logical_Graph_Figure_21.jpg|500px]] || (21) |- | [[Image:Logical_Graph_Figure_22.jpg|500px]] || (22) |- | [[Image:Logical_Graph_Figure_23.jpg|500px]] || (23) |} One way of assigning logical meaning to the initial equations is known as the ''entitative interpretation'' (EN). Under EN, the axioms read as follows: {| align="center" cellpadding="10" | <math>\begin{array}{ccccc} I_1 & : & \operatorname{true}\ \operatorname{or}\ \operatorname{true} & = & \operatorname{true} \\ I_2 & : & \operatorname{not}\ \operatorname{true}\ & = & \operatorname{false} \\ J_1 & : & a\ \operatorname{or}\ \operatorname{not}\ a & = & \operatorname{true} \\ J_2 & : & (a\ \operatorname{or}\ b)\ \operatorname{and}\ (a\ \operatorname{or}\ c) & = & a\ \operatorname{or}\ (b\ \operatorname{and}\ c) \\ \end{array}</math> |} Another way of assigning logical meaning to the initial equations is known as the ''existential interpretation'' (EX). Under EX, the axioms read as follows: {| align="center" cellpadding="10" | <math>\begin{array}{ccccc} I_1 & : & \operatorname{false}\ \operatorname{and}\ \operatorname{false} & = & \operatorname{false} \\ I_2 & : & \operatorname{not}\ \operatorname{false} & = & \operatorname{true} \\ J_1 & : & a\ \operatorname{and}\ \operatorname{not}\ a & = & \operatorname{false} \\ J_2 & : & (a\ \operatorname{and}\ b)\ \operatorname{or}\ (a\ \operatorname{and}\ c) & = & a\ \operatorname{and}\ (b\ \operatorname{or}\ c) \\ \end{array}</math> |} All of the axioms in this set have the form of equations. This means that all of the inference steps that they allow are reversible. The proof annotation scheme employed below makes use of a double bar <math>\overline{\underline{~~~~~~}}</math> to mark this fact, although it will often be left to the reader to decide which of the two possible directions is the one required for applying the indicated axiom. ===Frequently used theorems=== The actual business of proof is a far more strategic affair than the simple cranking of inference rules might suggest. Part of the reason for this lies in the circumstance that the usual brands of inference rules combine the moving forward of a state of inquiry with the losing of information along the way that doesn't appear to be immediately relevant, at least, not as viewed in the local focus and the short run of the moment to moment proceedings of the proof in question. Over the long haul, this has the pernicious side-effect that one is forever strategically required to reconstruct much of the information that one had strategically thought to forget in earlier stages of the proof, if "before the proof started" can be counted as an earlier stage of the proof in view. For this reason, among others, it is very instructive to study equational inference rules of the sort that our axioms have just provided. Although equational forms of reasoning are paramount in mathematics, they are less familiar to the student of conventional logic textbooks, who may find a few surprises here. By way of gaining a minimal experience with how equational proofs look in the present forms of syntax, let us examine the proofs of a few essential theorems in the primary algebra. ====C₁. Double negation==== The first theorem goes under the names of ''Consequence&nbsp;1'' <math>(C_1)\!</math>, the ''double negation theorem'' (DNT), or ''Reflection''. {| align="center" cellpadding="10" | [[Image:Double Negation 1.0 Splash Page.png|500px]] || (24) |} The proof that follows is adapted from the one that was given by [[George Spencer Brown]] in his book ''Laws of Form'' (LOF) and credited to two of his students, John Dawes and D.A. Utting. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Double Negation 1.0 Marquee Title.png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 1.png|500px]] |- | [[Image:Equational Inference I2 Elicit (( )).png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 2.png|500px]] |- | [[Image:Equational Inference J1 Insert (a).png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 3.png|500px]] |- | [[Image:Equational Inference J2 Distribute ((a)).png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 4.png|500px]] |- | [[Image:Equational Inference J1 Delete (a).png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 5.png|500px]] |- | [[Image:Equational Inference J1 Insert a.png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 6.png|500px]] |- | [[Image:Equational Inference J2 Collect a.png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 7.png|500px]] |- | [[Image:Equational Inference J1 Delete ((a)).png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 8.png|500px]] |- | [[Image:Equational Inference I2 Cancel (( )).png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 9.png|500px]] |- | [[Image:Equational Inference Marquee QED.png|500px]] |} | (25) |} The steps of this proof are replayed in the following animation. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Double Negation 2.0 Animation.gif]] |} | (26) |} ====C₂. Generation theorem==== One theorem of frequent use goes under the nickname of the ''weed and seed theorem'' (WAST). The proof is just an exercise in mathematical induction, once a suitable basis is laid down, and it will be left as an exercise for the reader. What the WAST says is that a label can be freely distributed or freely erased anywhere in a subtree whose root is labeled with that label. The second in our list of frequently used theorems is in fact the base case of this weed and seed theorem. In LOF, it goes by the names of ''Consequence&nbsp;2'' <math>(C_2)\!</math> or ''Generation''. {| align="center" cellpadding="10" | [[Image:Generation Theorem 1.0 Splash Page.png|500px]] || (27) |} Here is a proof of the Generation Theorem. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Generation Theorem 1.0 Marquee Title.png|500px]] |- | [[Image:Generation Theorem 1.0 Storyboard 1.png|500px]] |- | [[Image:Equational Inference C1 Reflect a(b).png|500px]] |- | [[Image:Generation Theorem 1.0 Storyboard 2.png|500px]] |- | [[Image:Equational Inference I2 Elicit (( )).png|500px]] |- | [[Image:Generation Theorem 1.0 Storyboard 3.png|500px]] |- | [[Image:Equational Inference J1 Insert a.png|500px]] |- | [[Image:Generation Theorem 1.0 Storyboard 4.png|500px]] |- | [[Image:Equational Inference J2 Collect a.png|500px]] |- | [[Image:Generation Theorem 1.0 Storyboard 5.png|500px]] |- | [[Image:Equational Inference C1 Reflect a, b.png|500px]] |- | [[Image:Generation Theorem 1.0 Storyboard 6.png|500px]] |- | [[Image:Equational Inference Marquee QED.png|500px]] |} | (28) |} The steps of this proof are replayed in the following animation. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Generation Theorem 2.0 Animation.gif]] |} | (29) |} ====C₃. Dominant form theorem==== The third of the frequently used theorems of service to this survey is one that Spencer-Brown annotates as ''Consequence&nbsp;3'' <math>(C_3)\!</math> or ''Integration''. A better mnemonic might be ''dominance and recession theorem'' (DART), but perhaps the brevity of ''dominant form theorem'' (DFT) is sufficient reminder of its double-edged role in proofs. {| align="center" cellpadding="10" | [[Image:Dominant Form 1.0 Splash Page.png|500px]] || (30) |} Here is a proof of the Dominant Form Theorem. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Dominant Form 1.0 Marquee Title.png|500px]] |- | [[Image:Dominant Form 1.0 Storyboard 1.png|500px]] |- | [[Image:Equational Inference C2 Regenerate a.png|500px]] |- | [[Image:Dominant Form 1.0 Storyboard 2.png|500px]] |- | [[Image:Equational Inference J1 Delete a.png|500px]] |- | [[Image:Dominant Form 1.0 Storyboard 3.png|500px]] |- | [[Image:Equational Inference Marquee QED.png|500px]] |} | (31) |} The following animation provides an instant re*play. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Dominant Form 2.0 Animation.gif]] |} | (32) |} ===Exemplary proofs=== Based on the axioms given at the outest, and aided by the theorems recorded so far, it is possible to prove a multitude of much more complex theorems. A couple of all-time favorites are given next. ====Peirce's law==== : ''Main article'' : [[Peirce's law]] Peirce's law is commonly written in the following form: {| align="center" cellpadding="10" | <math>((p \Rightarrow q) \Rightarrow p) \Rightarrow p</math> |} The existential graph representation of Peirce's law is shown in Figure&nbsp;33. {| align="center" cellpadding="10" | [[Image:Peirce's Law 1.0 Splash Page.png|500px]] || (33) |} A graphical proof of Peirce's law is shown in Figure&nbsp;34. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Peirce's Law 1.0 Marquee Title.png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 1.png|500px]] |- | [[Image:Equational Inference Band Collect p.png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 2.png|500px]] |- | [[Image:Equational Inference Band Quit ((q)).png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 3.png|500px]] |- | [[Image:Equational Inference Band Cancel (( )).png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 4.png|500px]] |- | [[Image:Equational Inference Band Delete p.png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 5.png|500px]] |- | [[Image:Equational Inference Band Cancel (( )).png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 6.png|500px]] |- | [[Image:Equational Inference Marquee QED.png|500px]] |} | (34) |} The following animation replays the steps of the proof. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Peirce's Law 2.0 Animation.gif]] |} | (35) |} ====Praeclarum theorema==== An illustrious example of a propositional theorem is the ''praeclarum theorema'', the ''admirable'', ''shining'', or ''splendid'' theorem of [[Leibniz]]. {| align="center" cellspacing="6" width="90%" <!--QUOTE--> | <p>If ''a'' is ''b'' and ''d'' is ''c'', then ''ad'' will be ''bc''.</p> <p>This is a fine theorem, which is proved in this way:</p> <p>''a'' is ''b'', therefore ''ad'' is ''bd'' (by what precedes),</p> <p>''d'' is ''c'', therefore ''bd'' is ''bc'' (again by what precedes),</p> <p>''ad'' is ''bd'', and ''bd'' is ''bc'', therefore ''ad'' is ''bc''. Q.E.D.</p> <p>(Leibniz, ''Logical Papers'', p. 41).</p> |} Under the existential interpretation, the praeclarum theorema is represented by means of the following logical graph. {| align="center" cellpadding="10" | [[Image:Praeclarum Theorema 1.0 Splash Page.png|500px]] || (36) |} And here's a neat proof of that nice theorem. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Praeclarum Theorema 1.0 Marquee Title.png|500px]] |- | [[Image:Praeclarum Theorema 1.0 Storyboard 1.png|500px]] |- | [[Image:Equational Inference Rule Reflect ad(bc).png|500px]] |- | [[Image:Praeclarum Theorema 1.0 Storyboard 2.png|500px]] |- | [[Image:Equational Inference Rule Weed a, d.png|500px]] |- | [[Image:Praeclarum Theorema 1.0 Storyboard 3.png|500px]] |- | [[Image:Equational Inference Rule Reflect b, c.png|500px]] |- | [[Image:Praeclarum Theorema 1.0 Storyboard 4.png|500px]] |- | [[Image:Equational Inference Rule Weed bc.png|500px]] |- | [[Image:Praeclarum Theorema 1.0 Storyboard 5.png|500px]] |- | [[Image:Equational Inference Rule Quit abcd.png|500px]] |- | [[Image:Praeclarum Theorema 1.0 Storyboard 6.png|500px]] |- | [[Image:Equational Inference Rule Cancel (( )).png|500px]] |- | [[Image:Praeclarum Theorema 1.0 Storyboard 7.png|500px]] |- | [[Image:Equational Inference Marquee QED.png|500px]] |} | (37) |} The steps of the proof are replayed in the following animation. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Praeclarum Theorema 2.0 Animation.gif]] |} | (38) |} ====Two-thirds majority function==== Consider the following equation in boolean algebra, posted as a [http://mathoverflow.net/questions/9292/newbie-boolean-algebra-question problem for proof] at [http://mathoverflow.net/ MathOverFlow]. {| align="center" cellpadding="20" | <math>\begin{matrix} a b \bar{c} + a \bar{b} c + \bar{a} b c + a b c \\[6pt] \iff \\[6pt] a b + a c + b c \end{matrix}</math> | &nbsp;&nbsp;&nbsp;&nbsp; |} The required equation can be proven in the medium of logical graphs as shown in the following Figure. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Two-Thirds Majority Eq 1 Pf 1 Banner Title.png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 1.png|500px]] |- | [[Image:Equational Inference Bar Reflect ab, ac, bc.png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 2.png|500px]] |- | [[Image:Equational Inference Bar Distribute (abc).png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 3.png|500px]] |- | [[Image:Equational Inference Bar Collect ab, ac, bc.png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 4.png|500px]] |- | [[Image:Equational Inference Bar Quit (a), (b), (c).png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 5.png|500px]] |- | [[Image:Equational Inference Bar Cancel (( )).png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 6.png|500px]] |- | [[Image:Equational Inference Bar Weed ab, ac, bc.png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 7.png|500px]] |- | [[Image:Equational Inference Bar Delete a, b, c.png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 8.png|500px]] |- | [[Image:Equational Inference Bar Cancel (( )).png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 9.png|500px]] |- | [[Image:Equational Inference Banner QED.png|500px]] |} | (39) |} Here's an animated recap of the graphical transformations that occur in the above proof: {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Two-Thirds Majority Function 500 x 250 Animation.gif]] |} | (40) |} ==Bibliography== * [[Gottfried Leibniz|Leibniz, G.W.]] (1679&ndash;1686 ?), "Addenda to the Specimen of the Universal Calculus", pp. 40&ndash;46 in G.H.R. Parkinson (ed. and trans., 1966), ''Leibniz : Logical Papers'', Oxford University Press, London, UK. * [[Charles Peirce (Bibliography)|Peirce, C.S., Bibliography]]. * [[Charles Peirce|Peirce, C.S.]] (1931&ndash;1935, 1958), ''Collected Papers of Charles Sanders Peirce'', vols. 1&ndash;6, [[Charles Hartshorne]] and [[Paul Weiss]] (eds.), vols. 7&ndash;8, [[Arthur W. Burks]] (ed.), Harvard University Press, Cambridge, MA. Cited as (CP&nbsp;volume.paragraph). * Peirce, C.S. (1981&ndash;), ''Writings of Charles S. Peirce : A Chronological Edition'', [[Peirce Edition Project]] (eds.), Indiana University Press, Bloomington and Indianopolis, IN. Cited as (CE&nbsp;volume, page). * Peirce, C.S. (1885), "On the Algebra of Logic : A Contribution to the Philosophy of Notation", ''American Journal of Mathematics'' 7 (1885), 180&ndash;202. Reprinted as CP&nbsp;3.359&ndash;403 and CE&nbsp;5, 162&ndash;190. * Peirce, C.S. (''c.'' 1886), "Qualitative Logic", MS 736. Published as pp. 101&ndash;115 in Carolyn Eisele (ed., 1976), ''The New Elements of Mathematics by Charles S. Peirce, Volume 4, Mathematical Philosophy'', Mouton, The Hague. * Peirce, C.S. (1886 a), "Qualitative Logic", MS 582. Published as pp. 323&ndash;371 in ''Writings of Charles S. Peirce : A Chronological Edition, Volume 5, 1884&ndash;1886'', Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1993. * Peirce, C.S. (1886 b), "The Logic of Relatives : Qualitative and Quantitative", MS 584. Published as pp. 372&ndash;378 in ''Writings of Charles S. Peirce : A Chronological Edition, Volume 5, 1884&ndash;1886'', Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1993. * [[George Spencer Brown|Spencer Brown, George]] (1969), ''[[Laws of Form]]'', George Allen and Unwin, London, UK. ==Resources== * [http://planetmath.org/ PlanetMath] ** [http://planetmath.org/LogicalGraphIntroduction Logical Graph : Introduction] ** [http://planetmath.org/LogicalGraphFormalDevelopment Logical Graph : Formal Development] * Bergman and Paavola (eds.), [http://www.helsinki.fi/science/commens/dictionary.html Commens Dictionary of Peirce's Terms] ** [http://www.helsinki.fi/science/commens/terms/graphexis.html Existential Graph] ** [http://www.helsinki.fi/science/commens/terms/graphlogi.html Logical Graph] * [http://dr-dau.net/index.shtml Dau, Frithjof] ** [http://dr-dau.net/eg_readings.shtml Peirce's Existential Graphs : Readings and Links] ** [http://web.archive.org/web/20070706192257/http://dr-dau.net/pc.shtml Existential Graphs as Moving Pictures of Thought] &mdash; Computer Animated Proof of Leibniz's Praeclarum Theorema * [http://www.math.uic.edu/~kauffman/ Kauffman, Louis H.] ** [http://www.math.uic.edu/~kauffman/Arithmetic.htm Box Algebra, Boundary Mathematics, Logic, and Laws of Form] * [http://mathworld.wolfram.com/ MathWorld : A Wolfram Web Resource] ** [http://mathworld.wolfram.com/about/author.html Weisstein, Eric W.], [http://mathworld.wolfram.com/Spencer-BrownForm.html Spencer-Brown Form] * Shoup, Richard (ed.), [http://www.lawsofform.org/ Laws of Form Web Site] ** Spencer-Brown, George (1973), [http://www.lawsofform.org/aum/session1.html Transcript Session One], [http://www.lawsofform.org/aum/ AUM Conference], Esalen, CA. ==Translations== * [http://pt.wikipedia.org/wiki/Grafo_l%C3%B3gico Grafo lógico], [http://pt.wikipedia.org/ Portuguese Wikipedia]. ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Logical_graph Logical Graph @ InterSciWiki] * [http://mywikibiz.com/Logical_graph Logical Graph @ MyWikiBiz] {{col-break}} * [http://beta.wikiversity.org/wiki/Logical_graph Logical Graph @ Wikiversity Beta] * [http://ref.subwiki.org/wiki/Logical_graph Logical Graph @ Subject Wikis] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Logical_graph Logical Graph], [http://mywikibiz.com/ MyWikiBiz] * [http://mathweb.org/wiki/Logical_graph Logical Graph], [http://mathweb.org/wiki/ MathWeb Wiki] * [http://netknowledge.org/wiki/Logical_graph Logical Graph], [http://netknowledge.org/ NetKnowledge] * [http://p2pfoundation.net/Logical_Graph Logical Graph], [http://p2pfoundation.net/ P2P Foundation] * [http://semanticweb.org/wiki/Logical_graph Logical Graph], [http://semanticweb.org/ Semantic Web] {{col-break}} * [http://proofwiki.org/wiki/Definition:Logical_Graph Logical Graph], [http://proofwiki.org/ ProofWiki] * [http://planetmath.org/encyclopedia/LogicalGraph.html Logical Graph : 1], [http://planetmath.org/ PlanetMath] * [http://planetmath.org/encyclopedia/LogicalGraphFormalDevelopment.html Logical Graph : 2], [http://planetmath.org/ PlanetMath] * [http://knol.google.com/k/logical-graphs-1 Logical Graph : 1], [http://knol.google.com/ Google Knol] * [http://knol.google.com/k/logical-graphs-2 Logical Graph : 2], [http://knol.google.com/ Google Knol] {{col-break}} * [http://beta.wikiversity.org/wiki/Logical_graph Logical Graph], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://getwiki.net/-Logical_Graph Logical Graph], [http://getwiki.net/ GetWiki] * [http://wikinfo.org/index.php/Logical_graph Logical Graph], [http://wikinfo.org/ Wikinfo] * [http://textop.org/wiki/index.php?title=Logical_graph Logical Graph], [http://textop.org/wiki/ Textop Wiki] * [http://en.wikipedia.org/wiki/Logical_graph Logical Graph], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Artificial Intelligence]] [[Category:Boolean Functions]] [[Category:Charles Sanders Peirce]] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Cybernetics]] [[Category:Equational Reasoning]] [[Category:Formal Languages]] [[Category:Formal Systems]] [[Category:Graph Theory]] [[Category:History of Logic]] [[Category:History of Mathematics]] [[Category:Inquiry]] [[Category:Knowledge Representation]] [[Category:Logic]] [[Category:Logical Graphs]] [[Category:Mathematics]] [[Category:Philosophy]] [[Category:Propositional Calculus]] [[Category:Semiotics]] [[Category:Visualization]] d0186d07c70fdad68bb8de65f8b904aed3f5118b 0 313 652 534 2013-10-25T19:40:03Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. In logic, mathematics, and semiotics, a '''triadic relation''' is an important special case of a [[relation (mathematics)|polyadic or finitary relation]], one in which the number of places in the relation is three. In other language that is often used, a triadic relation is called a '''ternary relation'''. One may also see the adjectives ''3-adic'', ''3-ary'', ''3-dimensional'', or ''3-place'' being used to describe these relations. Mathematics is positively rife with examples of 3-adic relations, and a [[sign relation]], the arch-idea of the whole field of semiotics, is a special case of a 3-adic relation. Therefore it will be useful to consider a few concrete examples from each of these two realms. ==Examples from mathematics== For the sake of topics to be taken up later, it is useful to examine a pair of 3-adic relations in tandem, <math>L_0\!</math> and <math>L_1,\!</math> that can be described in the following manner. The first order of business is to define the space in which the relations <math>L_0\!</math> and <math>L_1\!</math> take up residence. This space is constructed as a 3-fold [[cartesian power]] in the following way. The ''[[boolean domain]]'' is the set <math>\mathbb{B} = \{ 0, 1 \}.\!</math> The ''plus sign'' <math>{}^{\backprime\backprime} + {}^{\prime\prime},\!</math> used in the context of the boolean domain <math>\mathbb{B},\!</math> denotes addition modulo 2. Interpreted for logic, the plus sign can be used to indicate either the boolean operation of ''[[exclusive disjunction]]'', <math>\mathrm{XOR} : \mathbb{B} \times \mathbb{B} \to \mathbb{B},\!</math> or the boolean relation of ''logical inequality'', <math>\mathrm{NEQ} \subseteq \mathbb{B} \times \mathbb{B}.\!</math> The third cartesian power of <math>\mathbb{B}\!</math> is the set <math>\mathbb{B}^3 = \mathbb{B} \times \mathbb{B} \times \mathbb{B} = \{ (x_1, x_2, x_3) : x_j \in \mathbb{B} ~\text{for}~ j = 1, 2, 3 \}.\!</math> In what follows, the space <math>X \times Y \times Z\!</math> is isomorphic to <math>\mathbb{B} \times \mathbb{B} \times \mathbb{B} ~=~ \mathbb{B}^3.\!</math> The relation <math>L_0\!</math> is defined as follows: {| align="center" cellpadding="6" width="90%" | <math>L_0 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 0 \}.~\!</math> |} The relation <math>L_0\!</math> is the set of four triples enumerated here: {| align="center" cellpadding="6" width="90%" | <math>L_0 ~=~ \{ (0, 0, 0), (0, 1, 1), (1, 0, 1), (1, 1, 0) \}.\!</math> |} The relation <math>L_1\!</math> is defined as follows: {| align="center" cellpadding="6" width="90%" | <math>L_1 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 1 \}.~\!</math> |} The relation <math>L_1\!</math> is the set of four triples enumerated here: {| align="center" cellpadding="6" width="90%" | <math>L_1 ~=~ \{ (0, 0, 1), (0, 1, 0), (1, 0, 0), (1, 1, 1) \}.\!</math> |} The triples that make up the relations <math>L_0\!</math> and <math>L_1\!</math> are conveniently arranged in the form of ''relational data tables'', as follows: <br> {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%" |+ style="height:30px" | <math>L_0 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 0 \}\!</math> |- style="height:40px; background:ghostwhite" | <math>X\!</math> || <math>Y\!</math> || <math>Z\!</math> |- | <math>0\!</math> || <math>0\!</math> || <math>0\!</math> |- | <math>0\!</math> || <math>1\!</math> || <math>1\!</math> |- | <math>1\!</math> || <math>0\!</math> || <math>1\!</math> |- | <math>1\!</math> || <math>1\!</math> || <math>0\!</math> |} <br> {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%" |+ style="height:30px" | <math>L_1 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 1 \}\!</math> |- style="height:40px; background:ghostwhite" | <math>X\!</math> || <math>Y\!</math> || <math>Z\!</math> |- | <math>0\!</math> || <math>0\!</math> || <math>1\!</math> |- | <math>0\!</math> || <math>1\!</math> || <math>0\!</math> |- | <math>1\!</math> || <math>0\!</math> || <math>0\!</math> |- | <math>1\!</math> || <math>1\!</math> || <math>1\!</math> |} <br> ==Examples from semiotics== The study of signs &mdash; the full variety of significant forms of expression &mdash; in relation to the things that signs are significant ''of'', and in relation to the beings that signs are significant ''to'', is known as ''[[semiotics]]'' or the ''theory of signs''. As just described, semiotics treats of a 3-place relation among ''signs'', their ''objects'', and their ''interpreters''. The term ''[[semiosis]]'' refers to any activity or process that involves signs. Studies of semiosis that deal with its more abstract form are not concerned with every concrete detail of the entities that act as signs, as objects, or as agents of semiosis, but only with the most salient patterns of relationship among these three roles. In particular, the formal theory of signs does not consider all of the properties of the interpretive agent but only the more striking features of the impressions that signs make on a representative interpreter. In its formal aspects, that impact or influence may be treated as just another sign, called the ''interpretant sign'', or the ''interpretant'' for short. Such a 3-adic relation, among objects, signs, and interpretants, is called a ''[[sign relation]]''. For example, consider the aspects of sign use that concern two people &mdash; let us say <math>\mathrm{Ann}\!</math> and <math>\mathrm{Bob}\!</math> &mdash; in using their own proper names, <math>{}^{\backprime\backprime} \mathrm{Ann} {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} \mathrm{Bob} {}^{\prime\prime},\!</math> together with the pronouns, <math>{}^{\backprime\backprime} \mathrm{I} {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} \mathrm{you} {}^{\prime\prime}.\!</math> For brevity, these four signs may be abbreviated to the set <math>\{ \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \, \}.\!</math> The abstract consideration of how <math>\mathrm{A}\!</math> and <math>\mathrm{B}\!</math> use this set of signs to refer to themselves and each other leads to the contemplation of a pair of 3-adic relations, the sign relations <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B},\!</math> that reflect the differential use of these signs by <math>\mathrm{A}\!</math> and <math>\mathrm{B},\!</math> respectively. Each of the sign relations, <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B},\!</math> consists of eight triples of the form <math>(x, y, z),\!</math> where the ''object'' <math>x\!</math> is an element of the ''object domain'' <math>O = \{ \mathrm{A}, \mathrm{B} \},\!</math> where the ''sign'' <math>y\!</math> is an element of the ''sign domain'' <math>S\!,</math> where the ''interpretant sign'' <math>z\!</math> is an element of the interpretant domain <math>I,\!</math> and where it happens in this case that <math>S = I = \{ \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \, \}.\!</math> In general, it is convenient to refer to the union <math>S \cup I\!</math> as the ''syntactic domain'', but in this case <math>S ~=~ I ~=~ S \cup I.\!</math> The set-up so far is summarized as follows: {| align="center" cellpadding="8" width="90%" | <math>\begin{array}{ccc} L_\mathrm{A}, L_\mathrm{B} & \subseteq & O \times S \times I \\ \\ O & = & \{ \mathrm{A}, \mathrm{B} \} \\ \\ S & = & \{ \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \, \} \\ \\ I & = & \{ \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \, \} \\ \\ \end{array}</math> |} The relation <math>L_\mathrm{A}\!</math> is the set of eight triples enumerated here: {| align="center" cellpadding="8" width="90%" | <math>\begin{array}{cccccc} \{ & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}), & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}), & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}), & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}), & \\ & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}), & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}), & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}), & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}) & \}. \end{array}</math> |} The triples in <math>L_\mathrm{A}\!</math> represent the way that interpreter <math>\mathrm{A}\!</math> uses signs. For example, the listing of the triple <math>(\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime})\!</math> in <math>L_\mathrm{A}\!</math> represents the fact that <math>\mathrm{A}\!</math> uses <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> to mean the same thing that <math>\mathrm{A}\!</math> uses <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> to mean, namely, <math>\mathrm{B}.\!</math> The relation <math>L_\mathrm{B}\!</math> is the set of eight triples enumerated here: {| align="center" cellpadding="8" width="90%" | <math>\begin{array}{cccccc} \{ & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}), & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}), & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}), & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}), & \\ & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}), & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}), & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}), & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}) & \}. \end{array}</math> |} The triples in <math>L_\mathrm{B}\!</math> represent the way that interpreter <math>\mathrm{B}\!</math> uses signs. For example, the listing of the triple <math>(\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime})\!</math> in <math>L_\mathrm{B}\!</math> represents the fact that <math>\mathrm{B}\!</math> uses <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> to mean the same thing that <math>\mathrm{B}\!</math> uses <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> to mean, namely, <math>\mathrm{B}.\!</math> The triples that make up the relations <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> are conveniently arranged in the form of ''relational data tables'', as follows: <br> {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%" |+ style="height:30px" | <math>L_\mathrm{A} ~=~ \text{Sign Relation of Interpreter A}\!</math> |- style="height:40px; background:ghostwhite" | style="width:33%" | <math>\text{Object}\!</math> | style="width:33%" | <math>\text{Sign}\!</math> | style="width:33%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} <br> {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%" |+ style="height:30px" | <math>L_\mathrm{B} ~=~ \text{Sign Relation of Interpreter B}\!</math> |- style="height:40px; background:ghostwhite" | style="width:33%" | <math>\text{Object}\!</math> | style="width:33%" | <math>\text{Sign}\!</math> | style="width:33%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- |<math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} <br> ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Triadic_relation Triadic Relation @ InterSciWiki] * [http://mywikibiz.com/Triadic_relation Triadic Relation @ MyWikiBiz] {{col-break}} * [http://beta.wikiversity.org/wiki/Triadic_relation Triadic Relation @ Wikiversity Beta] * [http://ref.subwiki.org/wiki/Triadic_relation Triadic Relation @ Subject Wikis] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Triadic_relation Triadic Relation], [http://mywikibiz.com/ MyWikiBiz] * [http://mathweb.org/wiki/Triadic_relation Triadic Relation], [http://mathweb.org/ MathWeb Wiki] {{col-break}} * [http://planetmath.org/TriadicRelation Triadic Relation], [http://planetmath.org/ PlanetMath] * [http://beta.wikiversity.org/wiki/Triadic_relation Triadic Relation], [http://beta.wikiversity.org/ Wikiversity Beta] {{col-break}} * [http://getwiki.net/-Triadic_Relation Triadic Relation], [http://getwiki.net/ GetWiki] * [http://en.wikipedia.org/w/index.php?title=Triadic_relation&oldid=108548758 Triadic Relation], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Artificial Intelligence]] [[Category:Boolean Functions]] [[Category:Charles Sanders Peirce]] [[Category:Cognitive Sciences]] [[Category:Computer Science]] [[Category:Formal Sciences]] [[Category:Hermeneutics]] [[Category:Information Systems]] [[Category:Information Theory]] [[Category:Inquiry]] [[Category:Intelligence Amplification]] [[Category:Knowledge Representation]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Philosophy]] [[Category:Pragmatics]] [[Category:Semantics]] [[Category:Semiotics]] [[Category:Syntax]] bd9cf42025014611daa34c4474265dd651674a44 0 337 653 559 2013-10-26T14:54:11Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. In logic and mathematics, '''relation reduction''' and '''relational reducibility''' have to do with the extent to which a given [[relation (mathematics)|relation]] is determined by a set of other relations, called the ''relation dataset''. The relation under examination is called the ''reductandum''. The relation dataset typically consists of a specified relation over sets of relations, called the ''reducer'', the ''method of reduction'', or the ''relational step'', plus a set of other relations, called the ''reduciens'' or the ''relational base'', each of which is properly simpler in a specified way than the relation under examination. A question of relation reduction or relational reducibility is sometimes posed as a question of '''relation reconstruction''' or '''relational reconstructibility''', since a useful way of stating the question is to ask whether the reductandum can be reconstructed from the reduciens. A relation that is not uniquely determined by a particular relation dataset is said to be ''irreducible'' in just that respect. A relation that is not uniquely determined by any relation dataset in a particular class of relation datasets is said to be ''irreducible'' in respect of that class. ==Discussion== The main thing that keeps the general problem of relational reducibility from being fully well-defined is that one would have to survey all of the conceivable ways of &ldquo;getting new relations from old&rdquo; in order to say precisely what is meant by the claim that the relation <math>L\!</math> is reducible to the set of relations <math>\{ L_j : j \in J \}.\!</math> This amounts to claiming one can be given a set of ''properly simpler'' relations <math>L_j\!</math> for values <math>j\!</math> in a given index set <math>J\!</math> and that this collection of data would suffice to fix the original relation <math>L\!</math> that one is seeking to analyze, determine, specify, or synthesize. In practice, however, apposite discussion of a particular application typically settles on either one of two different notions of reducibility as capturing the pertinent issues, namely: # Reduction under composition. # Reduction under projections. As it happens, there is an interesting relationship between these two notions of reducibility, the implications of which may be taken up partly in parallel with the discussion of the basic concepts. ==Projective reducibility of relations== It is convenient to begin with the ''projective reduction'' of relations, partly because this type of reduction is simpler and more intuitive (in the visual sense), but also because a number of conceptual tools that are needed in any case arise quite naturally in the projective setting. The work of intuiting how projections operate on multidimensional relations is often facilitated by keeping in mind the following sort of geometric image: * Picture a <math>k\!</math>-adic relation <math>L\!</math> as a body that resides in a <math>k\!</math>-dimensional space <math>X.\!</math> If the domains of the relation <math>L\!</math> are <math>X_1, \ldots, X_k,\!</math> then the ''extension'' of the relation <math>L\!</math> is a subset of the cartesian product <math>X = X_1 \times \ldots \times X_k.\!</math> In this setting the interval <math>K = [1, k] = \{ 1, \ldots, k \}\!</math> is called the ''index set'' of the ''indexed family'' of sets <math>X_1, \ldots, X_k.\!</math> For any subset <math>F\!</math> of the index set <math>K,\!</math> there is the corresponding subfamily of sets, <math>\{ X_j : j \in F \},\!</math> and there is the corresponding cartesian product over this subfamily, notated and defined as <math>\textstyle X_F = \prod_{j \in F} X_j.\!</math> For any point <math>x\!</math> in <math>X,\!</math> the ''projection'' of <math>x\!</math> on the subspace <math>X_F\!</math> is notated as <math>\mathrm{proj}_F (x).\!</math> More generally, for any relation <math>L \subseteq X,\!</math> the projection of <math>L\!</math> on the subspace <math>X_F\!</math> is written as <math>\mathrm{proj}_F (L)\!</math> or still more simply as <math>\mathrm{proj}_F L.\!</math> The question of ''projective reduction'' for <math>k\!</math>-adic relations can be stated with moderate generality in the following way: * Given a set of <math>k\!</math>-place relations in the same space <math>X\!</math> and a set of projections from <math>X\!</math> to the associated subspaces, do the projections afford sufficient data to tell the different relations apart? ==Projective reducibility of triadic relations== : ''Main article : [[Triadic relation]]'' By way of illustrating the different sorts of things that can occur in considering the projective reducibility of relations, it is convenient to reuse the four examples of 3-adic relations that are discussed in the main article on that subject. ===Examples of projectively irreducible relations=== The 3-adic relations <math>L_0\!</math> and <math>L_1\!</math> are shown in the next two Tables: <br> {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%" |+ style="height:30px" | <math>L_0 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 0 \}\!</math> |- style="height:40px; background:ghostwhite" | <math>X\!</math> || <math>Y\!</math> || <math>Z\!</math> |- | <math>0\!</math> || <math>0\!</math> || <math>0\!</math> |- | <math>0\!</math> || <math>1\!</math> || <math>1\!</math> |- | <math>1\!</math> || <math>0\!</math> || <math>1\!</math> |- | <math>1\!</math> || <math>1\!</math> || <math>0\!</math> |} <br> {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%" |+ style="height:30px" | <math>L_1 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 1 \}\!</math> |- style="height:40px; background:ghostwhite" | <math>X\!</math> || <math>Y\!</math> || <math>Z\!</math> |- | <math>0\!</math> || <math>0\!</math> || <math>1\!</math> |- | <math>0\!</math> || <math>1\!</math> || <math>0\!</math> |- | <math>1\!</math> || <math>0\!</math> || <math>0\!</math> |- | <math>1\!</math> || <math>1\!</math> || <math>1\!</math> |} <br> A ''2-adic projection'' of a 3-adic relation <math>L\!</math> is the 2-adic relation that results from deleting one column of the table for <math>L\!</math> and then deleting all but one row of any resulting rows that happen to be identical in content. In other words, the multiplicity of any repeated row is ignored. In the case of the above two relations, <math>{L_0, L_1 ~\subseteq~ X \times Y \times Z ~\cong~ \mathbb{B}^3},\!</math> the 2-adic projections are indexed by the columns or domains that remain, as shown in the following Tables. <br> {| align="center" style="width:90%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\mathrm{proj}_{XY} L_0\!</math> |- style="height:40px; background:ghostwhite" | <math>X\!</math> || <math>Y\!</math> |- | <math>0\!</math> || <math>0\!</math> |- | <math>0\!</math> || <math>1\!</math> |- | <math>1\!</math> || <math>0\!</math> |- | <math>1\!</math> || <math>1\!</math> |} | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\mathrm{proj}_{XZ} L_0\!</math> |- style="height:40px; background:ghostwhite" | <math>X\!</math> || <math>Z\!</math> |- | <math>0\!</math> || <math>0\!</math> |- | <math>0\!</math> || <math>1\!</math> |- | <math>1\!</math> || <math>1\!</math> |- | <math>1\!</math> || <math>0\!</math> |} | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\mathrm{proj}_{YZ} L_0\!</math> |- style="height:40px; background:ghostwhite" | <math>Y\!</math> || <math>Z\!</math> |- | <math>0\!</math> || <math>0\!</math> |- | <math>1\!</math> || <math>1\!</math> |- | <math>0\!</math> || <math>1\!</math> |- | <math>1\!</math> || <math>0\!</math> |} |} <br> {| align="center" style="width:90%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\mathrm{proj}_{XY} L_1\!</math> |- style="height:40px; background:ghostwhite" | <math>X\!</math> || <math>Y\!</math> |- | <math>0\!</math> || <math>0\!</math> |- | <math>0\!</math> || <math>1\!</math> |- | <math>1\!</math> || <math>0\!</math> |- | <math>1\!</math> || <math>1\!</math> |} | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\mathrm{proj}_{XZ} L_1\!</math> |- style="height:40px; background:ghostwhite" | <math>X\!</math> || <math>Z\!</math> |- | <math>0\!</math> || <math>1\!</math> |- | <math>0\!</math> || <math>0\!</math> |- | <math>1\!</math> || <math>0\!</math> |- | <math>1\!</math> || <math>1\!</math> |} | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\mathrm{proj}_{YZ} L_1\!</math> |- style="height:40px; background:ghostwhite" | <math>Y\!</math> || <math>Z\!</math> |- | <math>0\!</math> || <math>1\!</math> |- | <math>1\!</math> || <math>0\!</math> |- | <math>0\!</math> || <math>0\!</math> |- | <math>1\!</math> || <math>1\!</math> |} |} <br> It is clear on inspection that the following three equations hold: {| align="center" cellpadding="8" style="text-align:center; width:90%" | <math>\mathrm{proj}_{XY}(L_0) ~=~ \mathrm{proj}_{XY}(L_1)~\!</math> | <math>\mathrm{proj}_{XZ}(L_0) ~=~ \mathrm{proj}_{XZ}(L_1)~\!</math> | <math>\mathrm{proj}_{YZ}(L_0) ~=~ \mathrm{proj}_{YZ}(L_1)~\!</math> |} These equations say that <math>L_0\!</math> and <math>L_1\!</math> cannot be distinguished from each other solely on the basis of their 2-adic projection data. In such a case, each relation is said to be ''irreducible with respect to 2-adic projections''. Since reducibility with respect to 2-adic projections is the only interesting case where it concerns the reduction of 3-adic relations, it is customary to say more simply of such a relation that it is ''projectively irreducible'', the 2-adic basis being understood. It is immediate from the definition that projectively irreducible relations always arise in non-trivial multiplets of mutually indiscernible relations. ===Examples of projectively reducible relations=== The 3-adic relations <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> are shown in the next two Tables: <br> {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%" |+ style="height:30px" | <math>L_\mathrm{A} ~=~ \text{Sign Relation of Interpreter A}\!</math> |- style="height:40px; background:ghostwhite" | style="width:33%" | <math>\text{Object}\!</math> | style="width:33%" | <math>\text{Sign}\!</math> | style="width:33%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} <br> {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%" |+ style="height:30px" | <math>L_\mathrm{B} ~=~ \text{Sign Relation of Interpreter B}\!</math> |- style="height:40px; background:ghostwhite" | style="width:33%" | <math>\text{Object}\!</math> | style="width:33%" | <math>\text{Sign}\!</math> | style="width:33%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- |<math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} <br> In the case of the two sign relations, <math>L_\mathrm{A}, L_\mathrm{B} ~\subseteq~ X \times Y \times Z ~\cong~ O \times S \times I,\!</math> the 2-adic projections are indexed by the columns or domains that remain, as shown in the following Tables. <br> {| align="center" style="width:90%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\mathrm{proj}_{XY}(L_\mathrm{A})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Object}\!</math> | width="50%" | <math>\text{Sign}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\mathrm{proj}_{XZ}(L_\mathrm{A})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Object}\!</math> | width="50%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\mathrm{proj}_{YZ}(L_\mathrm{A})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Sign}\!</math> | width="50%" | <math>\text{Interpretant}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} |} <br> {| align="center" style="width:90%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\mathrm{proj}_{XY}(L_\mathrm{B})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Object}\!</math> | width="50%" | <math>\text{Sign}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\mathrm{proj}_{XZ}(L_\mathrm{B})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Object}\!</math> | width="50%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\mathrm{proj}_{YZ}(L_\mathrm{B})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Sign}\!</math> | width="50%" | <math>\text{Interpretant}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} |} <br> It is clear on inspection that the following three inequalities hold: {| align="center" cellpadding="8" style="text-align:left; width:90%" | <math>\mathrm{proj}_{XY}(L_\mathrm{A}) ~\ne~ \mathrm{proj}_{XY}(L_\mathrm{B})\!</math> | <math>\mathrm{proj}_{XZ}(L_\mathrm{A}) ~\ne~ \mathrm{proj}_{XZ}(L_\mathrm{B})\!</math> | <math>\mathrm{proj}_{YZ}(L_\mathrm{A}) ~\ne~ \mathrm{proj}_{YZ}(L_\mathrm{B})\!</math> |} These inequalities say that <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> can be distinguished from each other solely on the basis of their 2-adic projection data. But this is not enough to say that either one of them is projectively reducible to their 2-adic projection data. To say that a 3-adic relation is projectively reducible in that respect, one has to show that it can be distinguished from ''every'' other 3-adic relation on the basis of the 2-adic projection data alone. In other words, to show that a 3-adic relation <math>L\!</math> on <math>O \times S \times I\!</math> is ''reducible'' or ''reconstructible'' in the 2-adic projective sense, it is necessary to show that no distinct <math>L'\!</math> on <math>O \times S \times I\!</math> exists such that <math>L\!</math> and <math>L'\!</math> have the same set of projections. Proving this takes a much more comprehensive or exhaustive investigation of the space of possible relations on <math>O \times S \times I\!</math> than looking merely at one or two relations at a time. '''Fact.''' As it happens, each of the relations <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> is uniquely determined by its 2-adic projections. This can be seen by following the proof that is given below. Before tackling the proof, however, it will speed things along to recall a few ideas and notations from other articles. * If <math>L\!</math> is a relation over a set of domains that includes the domains <math>U\!</math> and <math>V,\!</math> then the abbreviated notation <math>L_{UV}\!</math> can be used for the projection <math>\mathrm{proj}_{UV}(L).\!</math> * The operation of reversing a projection asks what elements of a bigger space project onto given elements of a smaller space. The set of elements that project onto <math>x\!</math> under a given projection <math>f\!</math> is called the ''fiber'' of <math>x\!</math> under <math>f\!</math> and is written <math>f^{-1}(x)\!</math> or <math>f^{-1}x.\!</math> * If <math>X\!</math> is a finite set, the ''cardinality'' of <math>X,\!</math> written <math>\mathrm{card}(X)\!</math> or <math>|X|,\!</math> means the number of elements in <math>X.\!</math> '''Proof.''' Let <math>L\!</math> be either one of the relations <math>L_\mathrm{A}\!</math> or <math>L_\mathrm{B}.\!</math> Consider any coordinate position <math>(s, i)\!</math> in the <math>SI\!</math>-plane <math>S \times I.\!</math> If <math>(s, i)\!</math> is not in <math>L_{SI}\!</math> then there can be no element <math>(o, s, i)\!</math> in <math>L,\!</math> therefore we may restrict our attention to positions <math>(s, i)\!</math> in <math>L_{SI},\!</math> knowing that there exist at least <math>|L_{SI}| = 8\!</math> elements in <math>L,\!</math> and seeking only to determine what objects <math>o\!</math> exist such that <math>(o, s, i)\!</math> is an element in the ''fiber'' of <math>(s, i).\!</math> In other words, for what <math>o\!</math> in <math>O\!</math> is <math>(o, s, i)\!</math> in the fiber <math>\mathrm{proj}_{SI}^{-1}(s, i)?\!</math> Now, the circumstance that <math>L_{OS}\!</math> has exactly one element <math>(o, s)\!</math> for each coordinate <math>s\!</math> in <math>S\!</math> and that <math>L_{OI}\!</math> has exactly one element <math>(o, i)\!</math> for each coordinate <math>i\!</math> in <math>I,\!</math> plus the &ldquo;coincidence&rdquo; of it being the same <math>o\!</math> at any one choice for <math>(s, i),\!</math> tells us that <math>L\!</math> has just the one element <math>(o, s, i)\!</math> over each point of <math>S \times I.\!</math> All together, this proves that both <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> are reducible in an informative sense to 3-tuples of 2-adic relations, that is, they are ''projectively 2-adically reducible''. ===Summary=== The ''projective analysis'' of 3-adic relations, illustrated by means of concrete examples, has been pursued just far enough at this point to state this clearly demonstrated result: * Some 3-adic relations are, and other 3-adic relations are not, reducible to, or reconstructible from, their 2-adic projection data. In short, some 3-adic relations are projectively reducible and some 3-adic relations are projectively irreducible. ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Relation_reduction Relation Reduction @ InterSciWiki] * [http://mywikibiz.com/Relation_reduction Relation Reduction @ MyWikiBiz] {{col-break}} * [http://beta.wikiversity.org/wiki/Relation_reduction Relation Reduction @ Wikiversity Beta] * [http://ref.subwiki.org/wiki/Relation_reduction Relation Reduction @ Subject Wikis] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Relation_reduction Relation Reduction], [http://mywikibiz.com/ MyWikiBiz] * [http://mathweb.org/wiki/Relation_reduction Relation Reduction], [http://mathweb.org/wiki/ MathWeb Wiki] {{col-break}} * [http://p2pfoundation.net/Relation_Reduction Relation Reduction], [http://p2pfoundation.net/ P2P Foundation] * [http://semanticweb.org/wiki/Relation_reduction Relation Reduction], [http://semanticweb.org/ Semantic Web] {{col-break}} * [http://planetmath.org/RelationReduction Relation Reduction], [http://planetmath.org/ PlanetMath] * [http://en.wikipedia.org/w/index.php?title=Relation_reduction&oldid=39828834 Relation Reduction], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Charles Sanders Peirce]] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Database Theory]] [[Category:Discrete Mathematics]] [[Category:Formal Sciences]] [[Category:Inquiry]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Relation Theory]] [[Category:Semiotics]] [[Category:Set Theory]] e64df5b8c32f4a4e7fac863baa3e1216c551a128 0 314 654 535 2013-10-27T15:40:09Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''sign relation''' is the basic construct in the theory of signs, also known as [[semeiotic]] or [[semiotics]], as developed by [[Charles Sanders Peirce]]. ==Anthesis== {| align="center" cellpadding="6" width="90%" | <p>Thus, if a sunflower, in turning towards the sun, becomes by that very act fully capable, without further condition, of reproducing a sunflower which turns in precisely corresponding ways toward the sun, and of doing so with the same reproductive power, the sunflower would become a Representamen of the sun. (C.S. Peirce, &ldquo;Syllabus&rdquo; (''c''.&nbsp;1902), ''Collected Papers'', CP&nbsp;2.274).</p> |} In his picturesque illustration of a sign relation, along with his tracing of a corresponding sign process, or ''[[semiosis]]'', Peirce uses the technical term ''representamen'' for his concept of a sign, but the shorter word is precise enough, so long as one recognizes that its meaning in a particular theory of signs is given by a specific definition of what it means to be a sign. ==Definition== One of Peirce's clearest and most complete definitions of a sign is one that he gives, not incidentally, in the context of defining "[[logic]]", and so it is informative to view it in that setting. {| align="center" cellpadding="6" width="90%" | <p>Logic will here be defined as ''formal semiotic''. A definition of a sign will be given which no more refers to human thought than does the definition of a line as the place which a particle occupies, part by part, during a lapse of time. Namely, a sign is something, ''A'', which brings something, ''B'', its ''interpretant'' sign determined or created by it, into the same sort of correspondence with something, ''C'', its ''object'', as that in which itself stands to ''C''. It is from this definition, together with a definition of "formal", that I deduce mathematically the principles of logic. I also make a historical review of all the definitions and conceptions of logic, and show, not merely that my definition is no novelty, but that my non-psychological conception of logic has ''virtually'' been quite generally held, though not generally recognized. (C.S. Peirce, NEM&nbsp;4, 20&ndash;21).</p> |} In the general discussion of diverse theories of signs, the question frequently arises whether signhood is an absolute, essential, indelible, or ''ontological'' property of a thing, or whether it is a relational, interpretive, and mutable role that a thing can be said to have only within a particular context of relationships. Peirce's definition of a ''sign'' defines it in relation to its ''object'' and its ''interpretant sign'', and thus it defines signhood in ''[[logic of relatives|relative terms]]'', by means of a predicate with three places. In this definition, signhood is a role in a [[triadic relation]], a role that a thing bears or plays in a given context of relationships &mdash; it is not as an ''absolute'', ''non-relative'' property of a thing-in-itself, one that it possesses independently of all relationships to other things. Some of the terms that Peirce uses in his definition of a sign may need to be elaborated for the contemporary reader. :* '''Correspondence.''' From the way that Peirce uses this term throughout his work, it is clear that he means what he elsewhere calls a "triple correspondence", and thus this is just another way of referring to the whole triadic sign relation itself. In particular, his use of this term should not be taken to imply a dyadic corresponence, like the kinds of &ldquo;mirror image&rdquo; correspondence between realities and representations that are bandied about in contemporary controversies about &ldquo;correspondence theories of truth&rdquo;. :* '''Determination.''' Peirce's concept of determination is broader in several directions than the sense of the word that refers to strictly deterministic causal-temporal processes. First, and especially in this context, he is invoking a more general concept of determination, what is called a ''formal'' or ''informational'' determination, as in saying "two points determine a line", rather than the more special cases of causal and temporal determinisms. Second, he characteristically allows for what is called ''determination in measure'', that is, an order of determinism that admits a full spectrum of more and less determined relationships. :* '''Non-psychological.''' Peirce's &ldquo;non-psychological conception of logic&rdquo; must be distinguished from any variety of ''anti-psychologism''. He was quite interested in matters of psychology and had much of import to say about them. But logic and psychology operate on different planes of study even when they have occasion to view the same data, as logic is a ''[[normative science]]'' where psychology is a ''[[descriptive science]]'', and so they have very different aims, methods, and rationales. ==Signs and inquiry== : ''Main article : [[Inquiry]]'' There is a close relationship between the pragmatic theory of signs and the pragmatic theory of [[inquiry]]. In fact, the correspondence between the two studies exhibits so many congruences and parallels that it is often best to treat them as integral parts of one and the same subject. In a very real sense, inquiry is the process by which sign relations come to be established and continue to evolve. In other words, inquiry, "thinking" in its best sense, "is a term denoting the various ways in which things acquire significance" ([[John Dewey]]). Thus, there is an active and intricate form of cooperation that needs to be appreciated and maintained between these converging modes of investigation. Its proper character is best understood by realizing that the theory of inquiry is adapted to study the developmental aspects of sign relations, a subject which the theory of signs is specialized to treat from structural and comparative points of view. ==Examples of sign relations== Because the examples to follow have been artificially constructed to be as simple as possible, their detailed elaboration can run the risk of trivializing the whole theory of sign relations. Despite their simplicity, however, these examples have subtleties of their own, and their careful treatment will serve to illustrate many important issues in the general theory of signs. Imagine a discussion between two people, Ann and Bob, and attend only to that aspect of their interpretive practice that involves the use of the following nouns and pronouns: &ldquo;Ann&rdquo;, &ldquo;Bob&rdquo;, &ldquo;I&rdquo;, &ldquo;you&rdquo;. :* The ''object domain'' of this discussion fragment is the set of two people <math>\{ \text{Ann}, \text{Bob} \}.\!</math> :* The ''syntactic domain'' or the ''sign system'' of their discussion is limited to the set of four signs <math>\{ {}^{\backprime\backprime} \text{Ann} {}^{\prime\prime}, {}^{\backprime\backprime} \text{Bob} {}^{\prime\prime}, {}^{\backprime\backprime} \text{I} {}^{\prime\prime}, {}^{\backprime\backprime} \text{you} {}^{\prime\prime} \}.\!</math> In their discussion, Ann and Bob are not only the passive objects of nominative and accusative references but also the active interpreters of the language that they use. The ''system of interpretation'' (SOI) associated with each language user can be represented in the form of an individual [[three-place relation]] called the ''sign relation'' of that interpreter. Understood in terms of its ''set-theoretic extension'', a sign relation <math>L\!</math> is a ''subset'' of a ''cartesian product'' <math>O \times S \times I.\!</math> Here, <math>O, S, I\!</math> are three sets that are known as the ''object domain'', the ''sign domain'', and the ''interpretant domain'', respectively, of the sign relation <math>L \subseteq O \times S \times I.\!</math> Broadly speaking, the three domains of a sign relation can be any sets whatsoever, but the kinds of sign relations typically contemplated in formal settings are usually constrained to having <math>I \subseteq S\!</math>. In this case interpretants are just a special variety of signs and this makes it convenient to lump signs and interpretants together into a single class called the ''syntactic domain''. In the forthcoming examples <math>S\!</math> and <math>I\!</math> are identical as sets, so the same elements manifest themselves in two different roles of the sign relations in question. When it is necessary to refer to the whole set of objects and signs in the union of the domains <math>O, S, I\!</math> for a given sign relation <math>L,\!</math> one may refer to this set as the ''World'' of <math>L\!</math> and write <math>W = W_L = O \cup S \cup I.\!</math> To facilitate an interest in the abstract structures of sign relations, and to keep the notations as brief as possible as the examples become more complicated, it serves to introduce the following general notations: {| align="center" cellspacing="6" width="90%" | <math>\begin{array}{ccl} O & = & \text{Object Domain} \\[6pt] S & = & \text{Sign Domain} \\[6pt] I & = & \text{Interpretant Domain} \end{array}</math> |} Introducing a few abbreviations for use in considering the present Example, we have the following data: {| align="center" cellspacing="6" width="90%" | <math>\begin{array}{cclcl} O & = & \{ \text{Ann}, \text{Bob} \} & = & \{ \text{A}, \text{B} \} \\[6pt] S & = & \{ {}^{\backprime\backprime} \text{Ann} {}^{\prime\prime}, {}^{\backprime\backprime} \text{Bob} {}^{\prime\prime}, {}^{\backprime\backprime} \text{I} {}^{\prime\prime}, {}^{\backprime\backprime} \text{you} {}^{\prime\prime} \} & = & \{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \} \\[6pt] I & = & \{ {}^{\backprime\backprime} \text{Ann} {}^{\prime\prime}, {}^{\backprime\backprime} \text{Bob} {}^{\prime\prime}, {}^{\backprime\backprime} \text{I} {}^{\prime\prime}, {}^{\backprime\backprime} \text{you} {}^{\prime\prime} \} & = & \{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \} \end{array}</math> |} In the present example, <math>S = I = \text{Syntactic Domain}\!</math>. The next two Tables give the sign relations associated with the interpreters <math>\mathrm{A}\!</math> and <math>\mathrm{B},\!</math> respectively, putting them in the form of ''relational databases''. Thus, the rows of each Table list the ordered triples of the form <math>(o, s, i)\!</math> that make up the corresponding sign relations, <math>L_\mathrm{A}, L_\mathrm{B} \subseteq O \times S \times I.\!</math> <br> {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%" |+ style="height:30px" | <math>L_\mathrm{A} ~=~ \text{Sign Relation of Interpreter A}\!</math> |- style="height:40px; background:ghostwhite" | style="width:33%" | <math>\text{Object}\!</math> | style="width:33%" | <math>\text{Sign}\!</math> | style="width:33%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} <br> {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%" |+ style="height:30px" | <math>L_\mathrm{B} ~=~ \text{Sign Relation of Interpreter B}\!</math> |- style="height:40px; background:ghostwhite" | style="width:33%" | <math>\text{Object}\!</math> | style="width:33%" | <math>\text{Sign}\!</math> | style="width:33%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- |<math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} <br> These Tables codify a rudimentary level of interpretive practice for the agents <math>\mathrm{A}\!</math> and <math>\mathrm{B}\!</math> and provide a basis for formalizing the initial semantics that is appropriate to their common syntactic domain. Each row of a Table names an object and two co-referent signs, making up an ordered triple of the form <math>(o, s, i)\!</math> called an ''elementary relation'', that is, one element of the relation's set-theoretic extension. Already in this elementary context, there are several different meanings that might attach to the project of a ''formal semiotics'', or a formal theory of meaning for signs. In the process of discussing these alternatives, it is useful to introduce a few terms that are occasionally used in the philosophy of language to point out the needed distinctions. ==Dyadic aspects of sign relations== For an arbitrary triadic relation <math>L \subseteq O \times S \times I,\!</math> whether it is a sign relation or not, there are six dyadic relations that can be obtained by ''projecting'' <math>L\!</math> on one of the planes of the <math>OSI\!</math>-space <math>O \times S \times I.\!</math> The six dyadic projections of a triadic relation <math>L\!</math> are defined and notated as follows: <br> {| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%" | <math>\begin{matrix} L_{OS} & = & \mathrm{proj}_{OS}(L) & = & \{ (o, s) \in O \times S ~:~ (o, s, i) \in L ~\text{for some}~ i \in I \} \\[6pt] L_{SO} & = & \mathrm{proj}_{SO}(L) & = & \{ (s, o) \in S \times O ~:~ (o, s, i) \in L ~\text{for some}~ i \in I \} \\[6pt] L_{IS} & = & \mathrm{proj}_{IS}(L) & = & \{ (i, s) \in I \times S ~:~ (o, s, i) \in L ~\text{for some}~ o \in O \} \\[6pt] L_{SI} & = & \mathrm{proj}_{SI}(L) & = & \{ (s, i) \in S \times I ~:~ (o, s, i) \in L ~\text{for some}~ o \in O \} \\[6pt] L_{OI} & = & \mathrm{proj}_{OI}(L) & = & \{ (o, i) \in O \times I ~:~ (o, s, i) \in L ~\text{for some}~ s \in S \} \\[6pt] L_{IO} & = & \mathrm{proj}_{IO}(L) & = & \{ (i, o) \in I \times O ~:~ (o, s, i) \in L ~\text{for some}~ s \in S \} \end{matrix}</math> |} <br> By way of unpacking the set-theoretic notation, here is what the first definition says in ordinary language. {| align="center" cellpadding="6" width="90%" | <p>The dyadic relation that results from the projection of <math>L\!</math> on the <math>OS\!</math>-plane <math>O \times S\!</math> is written briefly as <math>L_{OS}\!</math> or written more fully as <math>\mathrm{proj}_{OS}(L),\!</math> and it is defined as the set of all ordered pairs <math>(o, s)\!</math> in the cartesian product <math>O \times S\!</math> for which there exists an ordered triple <math>(o, s, i)\!</math> in <math>L\!</math> for some interpretant <math>i\!</math> in the interpretant domain <math>I.\!</math></p> |} In the case where <math>L\!</math> is a sign relation, which it becomes by satisfying one of the definitions of a sign relation, some of the dyadic aspects of <math>L\!</math> can be recognized as formalizing aspects of sign meaning that have received their share of attention from students of signs over the centuries, and thus they can be associated with traditional concepts and terminology. Of course, traditions may vary as to the precise formation and usage of such concepts and terms. Other aspects of meaning have not received their fair share of attention, and thus remain anonymous on the contemporary scene of sign studies. ===Denotation=== One aspect of a sign's complete meaning is concerned with the reference that a sign has to its objects, which objects are collectively known as the ''denotation'' of the sign. In the pragmatic theory of sign relations, denotative references fall within the projection of the sign relation on the plane that is spanned by its object domain and its sign domain. The dyadic relation that makes up the ''denotative'', ''referential'', or ''semantic'' aspect or component of a sign relation <math>L\!</math> is notated as <math>\mathrm{Den}(L).\!</math> Information about the denotative aspect of meaning is obtained from <math>L\!</math> by taking its projection on the object-sign plane, in other words, on the 2-dimensional space that is generated by the object domain <math>O\!</math> and the sign domain <math>S.\!</math> This component of a sign relation <math>L\!</math> can be written in any of the forms, <math>\mathrm{proj}_{OS} L,\!</math> &nbsp; <math>L_{OS},\!</math> &nbsp; <math>\mathrm{proj}_{12} L,\!</math> &nbsp; <math>L_{12},\!</math> and it is defined as follows: {| align="center" cellpadding="6" width="90%" | <math>\begin{matrix} \mathrm{Den}(L) & = & \mathrm{proj}_{OS} L & = & \{ (o, s) \in O \times S ~:~ (o, s, i) \in L ~\text{for some}~ i \in I \}. \end{matrix}</math> |} Looking to the denotative aspects of <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B},\!</math> various rows of the Tables specify, for example, that <math>\mathrm{A}\!</math> uses <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> to denote <math>\mathrm{A}\!</math> and <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> to denote <math>\mathrm{B},\!</math> whereas <math>\mathrm{B}\!</math> uses <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> to denote <math>\mathrm{B}\!</math> and <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> to denote <math>\mathrm{A}.\!</math> All of these denotative references are summed up in the projections on the <math>OS\!</math>-plane, as shown in the following Tables: <br> {| align="center" style="width:90%" | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\mathrm{proj}_{OS}(L_\mathrm{A})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Object}\!</math> | width="50%" | <math>\text{Sign}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\mathrm{proj}_{OS}(L_\mathrm{B})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Object}\!</math> | width="50%" | <math>\text{Sign}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} |} <br> ===Connotation=== Another aspect of meaning concerns the connection that a sign has to its interpretants within a given sign relation. As before, this type of connection can be vacuous, singular, or plural in its collection of terminal points, and it can be formalized as the dyadic relation that is obtained as a planar projection of the triadic sign relation in question. The connection that a sign makes to an interpretant is here referred to as its ''connotation''. In the full theory of sign relations, this aspect of meaning includes the links that a sign has to affects, concepts, ideas, impressions, intentions, and the whole realm of an agent's mental states and allied activities, broadly encompassing intellectual associations, emotional impressions, motivational impulses, and real conduct. Taken at the full, in the natural setting of semiotic phenomena, this complex system of references is unlikely ever to find itself mapped in much detail, much less completely formalized, but the tangible warp of its accumulated mass is commonly alluded to as the connotative import of language. Formally speaking, however, the connotative aspect of meaning presents no additional difficulty. For a given sign relation <math>L,\!</math> the dyadic relation that constitutes the ''connotative aspect'' or ''connotative component'' of <math>L\!</math> is notated as <math>\mathrm{Con}(L).\!</math> The connotative aspect of a sign relation <math>L\!</math> is given by its projection on the plane of signs and interpretants, and is therefore defined as follows: {| align="center" cellpadding="6" width="90%" | <math>\begin{matrix} \mathrm{Con}(L) & = & \mathrm{proj}_{SI} L & = & \{ (s, i) \in S \times I ~:~ (o, s, i) \in L ~\text{for some}~ o \in O \}. \end{matrix}</math> |} All of these connotative references are summed up in the projections on the <math>SI\!</math>-plane, as shown in the following Tables: <br> {| align="center" style="width:90%" | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\mathrm{proj}_{SI}(L_\mathrm{A})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Sign}\!</math> | width="50%" | <math>\text{Interpretant}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\mathrm{proj}_{SI}(L_\mathrm{B})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Sign}\!</math> | width="50%" | <math>\text{Interpretant}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} |} <br> ===Ennotation=== The aspect of a sign's meaning that arises from the dyadic relation of its objects to its interpretants has no standard name. If an interpretant is considered to be a sign in its own right, then its independent reference to an object can be taken as belonging to another moment of denotation, but this neglects the mediational character of the whole transaction in which this occurs. Denotation and connotation have to do with dyadic relations in which the sign plays an active role, but here we have to consider a dyadic relation between objects and interpretants that is mediated by the sign from an off-stage position, as it were. As a relation between objects and interpretants that is mediated by a sign, this aspect of meaning may be referred to as the ''ennotation'' of a sign, and the dyadic relation that constitutes the ''ennotative aspect'' of a sign relation <math>L\!</math> may be notated as <math>\mathrm{Enn}(L).\!</math> The ennotational component of meaning for a sign relation <math>L\!</math> is captured by its projection on the plane of the object and interpretant domains, and it is thus defined as follows: {| align="center" cellpadding="6" width="90%" | <math>\begin{matrix} \mathrm{Enn}(L) & = & \mathrm{proj}_{OI} L & = & \{ (o, i) \in O \times I ~:~ (o, s, i) \in L ~\text{for some}~ s \in S \}. \end{matrix}</math> |} As it happens, the sign relations <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> are fully symmetric with respect to exchanging signs and interpretants, so all the data of <math>\mathrm{proj}_{OS} L_\mathrm{A}\!</math> is echoed unchanged in <math>\mathrm{proj}_{OI} L_\mathrm{A}\!</math> and all the data of <math>\mathrm{proj}_{OS} L_\mathrm{B}\!</math> is echoed unchanged in <math>\mathrm{proj}_{OI} L_\mathrm{B}.\!</math> <br> {| align="center" style="width:90%" | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\mathrm{proj}_{OI}(L_\mathrm{A})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Object}\!</math> | width="50%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\mathrm{proj}_{OI}(L_\mathrm{B})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Object}\!</math> | width="50%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} |} <br> ==Semiotic equivalence relations== (Text in Progress) ==Six ways of looking at a sign relation== In the context of 3-adic relations in general, Peirce provides the following illustration of the six ''converses'' of a 3-adic relation, that is, the six differently ordered ways of stating what is logically the same 3-adic relation: : So in a triadic fact, say, the example <br> {| align="center" cellspacing="8" style="width:72%" | align="center" | ''A'' gives ''B'' to ''C'' |} : we make no distinction in the ordinary logic of relations between the ''[[subject (grammar)|subject]] [[nominative]]'', the ''[[direct object]]'', and the ''[[indirect object]]''. We say that the proposition has three ''logical subjects''. We regard it as a mere affair of English grammar that there are six ways of expressing this: <br> {| align="center" cellspacing="8" style="width:72%" | style="width:36%" | ''A'' gives ''B'' to ''C'' | style="width:36%" | ''A'' benefits ''C'' with ''B'' |- | ''B'' enriches ''C'' at expense of ''A'' | ''C'' receives ''B'' from ''A'' |- | ''C'' thanks ''A'' for ''B'' | ''B'' leaves ''A'' for ''C'' |} : These six sentences express one and the same indivisible phenomenon. (C.S. Peirce, "The Categories Defended", MS 308 (1903), EP 2, 170-171). ===IOS=== (Text in Progress) ===ISO=== (Text in Progress) ===OIS=== <blockquote> Words spoken are symbols or signs (σύμβολα) of affections or impressions (παθήματα) of the soul (ψυχή); written words are the signs of words spoken. As writing, so also is speech not the same for all races of men. But the mental affections themselves, of which these words are primarily signs (σημεια), are the same for the whole of mankind, as are also the objects (πράγματα) of which those affections are representations or likenesses, images, copies (ομοιώματα). ([[Aristotle]], ''[[De Interpretatione]]'', 1.16^a4). </blockquote> ===OSI=== (Text in Progress) ===SIO=== <blockquote> Logic will here be defined as ''formal semiotic''. A definition of a sign will be given which no more refers to human thought than does the definition of a line as the place which a particle occupies, part by part, during a lapse of time. Namely, a sign is something, ''A'', which brings something, ''B'', its ''interpretant'' sign determined or created by it, into the same sort of correspondence with something, ''C'', its ''object'', as that in which itself stands to ''C''. It is from this definition, together with a definition of "formal", that I deduce mathematically the principles of logic. I also make a historical review of all the definitions and conceptions of logic, and show, not merely that my definition is no novelty, but that my non-psychological conception of logic has ''virtually'' been quite generally held, though not generally recognized. (C.S. Peirce, "Application to the [[Carnegie Institution]]", L75 (1902), NEM 4, 20-21). </blockquote> ===SOI=== <blockquote> A ''Sign'' is anything which is related to a Second thing, its ''Object'', in respect to a Quality, in such a way as to bring a Third thing, its ''Interpretant'', into relation to the same Object, and that in such a way as to bring a Fourth into relation to that Object in the same form, ''ad infinitum''. (CP 2.92; quoted in Fisch 1986: 274) </blockquote> ==References== ==Bibliography== ===Primary sources=== * [[Charles Sanders Peirce (Bibliography)]] ===Secondary sources=== * Awbrey, Jon, and Awbrey, Susan (1995), "Interpretation as Action : The Risk of Inquiry", ''Inquiry : Critical Thinking Across the Disciplines'' 15, 40–52. [http://www.chss.montclair.edu/inquiry/fall95/awbrey.html Online]. * Deledalle, Gérard (2000), ''C.S. Peirce's Philosophy of Signs'', Indiana University Press, Bloomington, IN. * Eisele, Carolyn (1979), in ''Studies in the Scientific and Mathematical Philosophy of C.S. Peirce'', [[Richard Milton Martin]] (ed.), Mouton, The Hague. * Esposito, Joseph (1980), ''Evolutionary Metaphysics: The Development of Peirce's Theory of Categories'', Ohio University Press (?). * Fisch, Max (1986), ''Peirce, Semeiotic, and Pragmatism'', Indiana University Press, Bloomington, IN. * Houser, N., Roberts, D.D., and Van Evra, J. (eds.)(1997), ''Studies in the Logic of C.S. Peirce'', Indiana University Press, Bloomington, IN. * Liszka, J.J. (1996), ''A General Introduction to the Semeiotic of C.S. Peirce'', Indiana University Press, Bloomington, IN. * Misak, C. (ed.)(2004), ''Cambridge Companion to C.S. Peirce'', Cambridge University Press. * Moore, E., and Robin, R. (1964), ''Studies in the Philosophy of C.S. Peirce, Second Series'', University of Massachusetts Press, Amherst, MA. * Murphey, M. (1961), ''The Development of Peirce's Thought''. Reprinted, Hackett, Indianapolis, IN, 1993. * [[Walker Percy]] (2000), pp. 271-291 in ''Signposts in a Strange Land'', P. Samway (ed.), Saint Martin's Press. ==Resources== * [http://www.helsinki.fi/science/commens/dictionary.html The Commens Dictionary of Peirce's Terms] ** [http://www.helsinki.fi/science/commens/terms/semeiotic.html Entry for ''Semeiotic''] ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Sign_relation Sign Relation @ InterSciWiki] * [http://mywikibiz.com/Sign_relation Sign Relation @ MyWikiBiz] {{col-break}} * [http://beta.wikiversity.org/wiki/Sign_relation Sign Relation @ Wikiversity Beta] * [http://ref.subwiki.org/wiki/Sign_relation Sign Relation @ Subject Wikis] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Sign_relation Sign Relation], [http://mywikibiz.com/ MyWikiBiz] * [http://mathweb.org/wiki/Sign_relation Sign Relation], [http://mathweb.org/ MathWeb Wiki] {{col-break}} * [http://planetmath.org/SignRelation Sign Relation], [http://planetmath.org/ PlanetMath] * [http://beta.wikiversity.org/wiki/Sign_relation Sign Relation], [http://beta.wikiversity.org/ Wikiversity Beta] {{col-break}} * [http://getwiki.net/-Sign_Relation Sign Relation], [http://getwiki.net/ GetWiki] * [http://en.wikipedia.org/w/index.php?title=Sign_relation&oldid=161631069 Sign Relation], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Artificial Intelligence]] [[Category:Charles Sanders Peirce]] [[Category:Cognitive Sciences]] [[Category:Computer Science]] [[Category:Formal Sciences]] [[Category:Hermeneutics]] [[Category:Information Systems]] [[Category:Information Theory]] [[Category:Inquiry]] [[Category:Intelligence Amplification]] [[Category:Knowledge Representation]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Philosophy]] [[Category:Pragmatics]] [[Category:Semantics]] [[Category:Semiotics]] [[Category:Syntax]] e05e98e773105830665e32b3cf945e6379143313 655 654 2013-10-27T16:16:01Z Jon Awbrey 3 update links wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''sign relation''' is the basic construct in the theory of signs, also known as [[semeiotic]] or [[semiotics]], as developed by [[Charles Sanders Peirce]]. ==Anthesis== {| align="center" cellpadding="6" width="90%" | <p>Thus, if a sunflower, in turning towards the sun, becomes by that very act fully capable, without further condition, of reproducing a sunflower which turns in precisely corresponding ways toward the sun, and of doing so with the same reproductive power, the sunflower would become a Representamen of the sun. (C.S. Peirce, &ldquo;Syllabus&rdquo; (''c''.&nbsp;1902), ''Collected Papers'', CP&nbsp;2.274).</p> |} In his picturesque illustration of a sign relation, along with his tracing of a corresponding sign process, or ''[[semiosis]]'', Peirce uses the technical term ''representamen'' for his concept of a sign, but the shorter word is precise enough, so long as one recognizes that its meaning in a particular theory of signs is given by a specific definition of what it means to be a sign. ==Definition== One of Peirce's clearest and most complete definitions of a sign is one that he gives, not incidentally, in the context of defining "[[logic]]", and so it is informative to view it in that setting. {| align="center" cellpadding="6" width="90%" | <p>Logic will here be defined as ''formal semiotic''. A definition of a sign will be given which no more refers to human thought than does the definition of a line as the place which a particle occupies, part by part, during a lapse of time. Namely, a sign is something, ''A'', which brings something, ''B'', its ''interpretant'' sign determined or created by it, into the same sort of correspondence with something, ''C'', its ''object'', as that in which itself stands to ''C''. It is from this definition, together with a definition of "formal", that I deduce mathematically the principles of logic. I also make a historical review of all the definitions and conceptions of logic, and show, not merely that my definition is no novelty, but that my non-psychological conception of logic has ''virtually'' been quite generally held, though not generally recognized. (C.S. Peirce, NEM&nbsp;4, 20&ndash;21).</p> |} In the general discussion of diverse theories of signs, the question frequently arises whether signhood is an absolute, essential, indelible, or ''ontological'' property of a thing, or whether it is a relational, interpretive, and mutable role that a thing can be said to have only within a particular context of relationships. Peirce's definition of a ''sign'' defines it in relation to its ''object'' and its ''interpretant sign'', and thus it defines signhood in ''[[logic of relatives|relative terms]]'', by means of a predicate with three places. In this definition, signhood is a role in a [[triadic relation]], a role that a thing bears or plays in a given context of relationships &mdash; it is not as an ''absolute'', ''non-relative'' property of a thing-in-itself, one that it possesses independently of all relationships to other things. Some of the terms that Peirce uses in his definition of a sign may need to be elaborated for the contemporary reader. :* '''Correspondence.''' From the way that Peirce uses this term throughout his work, it is clear that he means what he elsewhere calls a "triple correspondence", and thus this is just another way of referring to the whole triadic sign relation itself. In particular, his use of this term should not be taken to imply a dyadic corresponence, like the kinds of &ldquo;mirror image&rdquo; correspondence between realities and representations that are bandied about in contemporary controversies about &ldquo;correspondence theories of truth&rdquo;. :* '''Determination.''' Peirce's concept of determination is broader in several directions than the sense of the word that refers to strictly deterministic causal-temporal processes. First, and especially in this context, he is invoking a more general concept of determination, what is called a ''formal'' or ''informational'' determination, as in saying "two points determine a line", rather than the more special cases of causal and temporal determinisms. Second, he characteristically allows for what is called ''determination in measure'', that is, an order of determinism that admits a full spectrum of more and less determined relationships. :* '''Non-psychological.''' Peirce's &ldquo;non-psychological conception of logic&rdquo; must be distinguished from any variety of ''anti-psychologism''. He was quite interested in matters of psychology and had much of import to say about them. But logic and psychology operate on different planes of study even when they have occasion to view the same data, as logic is a ''[[normative science]]'' where psychology is a ''[[descriptive science]]'', and so they have very different aims, methods, and rationales. ==Signs and inquiry== : ''Main article : [[Inquiry]]'' There is a close relationship between the pragmatic theory of signs and the pragmatic theory of [[inquiry]]. In fact, the correspondence between the two studies exhibits so many congruences and parallels that it is often best to treat them as integral parts of one and the same subject. In a very real sense, inquiry is the process by which sign relations come to be established and continue to evolve. In other words, inquiry, "thinking" in its best sense, "is a term denoting the various ways in which things acquire significance" ([[John Dewey]]). Thus, there is an active and intricate form of cooperation that needs to be appreciated and maintained between these converging modes of investigation. Its proper character is best understood by realizing that the theory of inquiry is adapted to study the developmental aspects of sign relations, a subject which the theory of signs is specialized to treat from structural and comparative points of view. ==Examples of sign relations== Because the examples to follow have been artificially constructed to be as simple as possible, their detailed elaboration can run the risk of trivializing the whole theory of sign relations. Despite their simplicity, however, these examples have subtleties of their own, and their careful treatment will serve to illustrate many important issues in the general theory of signs. Imagine a discussion between two people, Ann and Bob, and attend only to that aspect of their interpretive practice that involves the use of the following nouns and pronouns: &ldquo;Ann&rdquo;, &ldquo;Bob&rdquo;, &ldquo;I&rdquo;, &ldquo;you&rdquo;. :* The ''object domain'' of this discussion fragment is the set of two people <math>\{ \text{Ann}, \text{Bob} \}.\!</math> :* The ''syntactic domain'' or the ''sign system'' of their discussion is limited to the set of four signs <math>\{ {}^{\backprime\backprime} \text{Ann} {}^{\prime\prime}, {}^{\backprime\backprime} \text{Bob} {}^{\prime\prime}, {}^{\backprime\backprime} \text{I} {}^{\prime\prime}, {}^{\backprime\backprime} \text{you} {}^{\prime\prime} \}.\!</math> In their discussion, Ann and Bob are not only the passive objects of nominative and accusative references but also the active interpreters of the language that they use. The ''system of interpretation'' (SOI) associated with each language user can be represented in the form of an individual [[three-place relation]] called the ''sign relation'' of that interpreter. Understood in terms of its ''set-theoretic extension'', a sign relation <math>L\!</math> is a ''subset'' of a ''cartesian product'' <math>O \times S \times I.\!</math> Here, <math>O, S, I\!</math> are three sets that are known as the ''object domain'', the ''sign domain'', and the ''interpretant domain'', respectively, of the sign relation <math>L \subseteq O \times S \times I.\!</math> Broadly speaking, the three domains of a sign relation can be any sets whatsoever, but the kinds of sign relations typically contemplated in formal settings are usually constrained to having <math>I \subseteq S\!</math>. In this case interpretants are just a special variety of signs and this makes it convenient to lump signs and interpretants together into a single class called the ''syntactic domain''. In the forthcoming examples <math>S\!</math> and <math>I\!</math> are identical as sets, so the same elements manifest themselves in two different roles of the sign relations in question. When it is necessary to refer to the whole set of objects and signs in the union of the domains <math>O, S, I\!</math> for a given sign relation <math>L,\!</math> one may refer to this set as the ''World'' of <math>L\!</math> and write <math>W = W_L = O \cup S \cup I.\!</math> To facilitate an interest in the abstract structures of sign relations, and to keep the notations as brief as possible as the examples become more complicated, it serves to introduce the following general notations: {| align="center" cellspacing="6" width="90%" | <math>\begin{array}{ccl} O & = & \text{Object Domain} \\[6pt] S & = & \text{Sign Domain} \\[6pt] I & = & \text{Interpretant Domain} \end{array}</math> |} Introducing a few abbreviations for use in considering the present Example, we have the following data: {| align="center" cellspacing="6" width="90%" | <math>\begin{array}{cclcl} O & = & \{ \text{Ann}, \text{Bob} \} & = & \{ \text{A}, \text{B} \} \\[6pt] S & = & \{ {}^{\backprime\backprime} \text{Ann} {}^{\prime\prime}, {}^{\backprime\backprime} \text{Bob} {}^{\prime\prime}, {}^{\backprime\backprime} \text{I} {}^{\prime\prime}, {}^{\backprime\backprime} \text{you} {}^{\prime\prime} \} & = & \{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \} \\[6pt] I & = & \{ {}^{\backprime\backprime} \text{Ann} {}^{\prime\prime}, {}^{\backprime\backprime} \text{Bob} {}^{\prime\prime}, {}^{\backprime\backprime} \text{I} {}^{\prime\prime}, {}^{\backprime\backprime} \text{you} {}^{\prime\prime} \} & = & \{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \} \end{array}</math> |} In the present example, <math>S = I = \text{Syntactic Domain}\!</math>. The next two Tables give the sign relations associated with the interpreters <math>\mathrm{A}\!</math> and <math>\mathrm{B},\!</math> respectively, putting them in the form of ''relational databases''. Thus, the rows of each Table list the ordered triples of the form <math>(o, s, i)\!</math> that make up the corresponding sign relations, <math>L_\mathrm{A}, L_\mathrm{B} \subseteq O \times S \times I.\!</math> <br> {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%" |+ style="height:30px" | <math>L_\mathrm{A} ~=~ \text{Sign Relation of Interpreter A}\!</math> |- style="height:40px; background:ghostwhite" | style="width:33%" | <math>\text{Object}\!</math> | style="width:33%" | <math>\text{Sign}\!</math> | style="width:33%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} <br> {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%" |+ style="height:30px" | <math>L_\mathrm{B} ~=~ \text{Sign Relation of Interpreter B}\!</math> |- style="height:40px; background:ghostwhite" | style="width:33%" | <math>\text{Object}\!</math> | style="width:33%" | <math>\text{Sign}\!</math> | style="width:33%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- |<math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} <br> These Tables codify a rudimentary level of interpretive practice for the agents <math>\mathrm{A}\!</math> and <math>\mathrm{B}\!</math> and provide a basis for formalizing the initial semantics that is appropriate to their common syntactic domain. Each row of a Table names an object and two co-referent signs, making up an ordered triple of the form <math>(o, s, i)\!</math> called an ''elementary relation'', that is, one element of the relation's set-theoretic extension. Already in this elementary context, there are several different meanings that might attach to the project of a ''formal semiotics'', or a formal theory of meaning for signs. In the process of discussing these alternatives, it is useful to introduce a few terms that are occasionally used in the philosophy of language to point out the needed distinctions. ==Dyadic aspects of sign relations== For an arbitrary triadic relation <math>L \subseteq O \times S \times I,\!</math> whether it is a sign relation or not, there are six dyadic relations that can be obtained by ''projecting'' <math>L\!</math> on one of the planes of the <math>OSI\!</math>-space <math>O \times S \times I.\!</math> The six dyadic projections of a triadic relation <math>L\!</math> are defined and notated as follows: <br> {| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%" | <math>\begin{matrix} L_{OS} & = & \mathrm{proj}_{OS}(L) & = & \{ (o, s) \in O \times S ~:~ (o, s, i) \in L ~\text{for some}~ i \in I \} \\[6pt] L_{SO} & = & \mathrm{proj}_{SO}(L) & = & \{ (s, o) \in S \times O ~:~ (o, s, i) \in L ~\text{for some}~ i \in I \} \\[6pt] L_{IS} & = & \mathrm{proj}_{IS}(L) & = & \{ (i, s) \in I \times S ~:~ (o, s, i) \in L ~\text{for some}~ o \in O \} \\[6pt] L_{SI} & = & \mathrm{proj}_{SI}(L) & = & \{ (s, i) \in S \times I ~:~ (o, s, i) \in L ~\text{for some}~ o \in O \} \\[6pt] L_{OI} & = & \mathrm{proj}_{OI}(L) & = & \{ (o, i) \in O \times I ~:~ (o, s, i) \in L ~\text{for some}~ s \in S \} \\[6pt] L_{IO} & = & \mathrm{proj}_{IO}(L) & = & \{ (i, o) \in I \times O ~:~ (o, s, i) \in L ~\text{for some}~ s \in S \} \end{matrix}</math> |} <br> By way of unpacking the set-theoretic notation, here is what the first definition says in ordinary language. {| align="center" cellpadding="6" width="90%" | <p>The dyadic relation that results from the projection of <math>L\!</math> on the <math>OS\!</math>-plane <math>O \times S\!</math> is written briefly as <math>L_{OS}\!</math> or written more fully as <math>\mathrm{proj}_{OS}(L),\!</math> and it is defined as the set of all ordered pairs <math>(o, s)\!</math> in the cartesian product <math>O \times S\!</math> for which there exists an ordered triple <math>(o, s, i)\!</math> in <math>L\!</math> for some interpretant <math>i\!</math> in the interpretant domain <math>I.\!</math></p> |} In the case where <math>L\!</math> is a sign relation, which it becomes by satisfying one of the definitions of a sign relation, some of the dyadic aspects of <math>L\!</math> can be recognized as formalizing aspects of sign meaning that have received their share of attention from students of signs over the centuries, and thus they can be associated with traditional concepts and terminology. Of course, traditions may vary as to the precise formation and usage of such concepts and terms. Other aspects of meaning have not received their fair share of attention, and thus remain anonymous on the contemporary scene of sign studies. ===Denotation=== One aspect of a sign's complete meaning is concerned with the reference that a sign has to its objects, which objects are collectively known as the ''denotation'' of the sign. In the pragmatic theory of sign relations, denotative references fall within the projection of the sign relation on the plane that is spanned by its object domain and its sign domain. The dyadic relation that makes up the ''denotative'', ''referential'', or ''semantic'' aspect or component of a sign relation <math>L\!</math> is notated as <math>\mathrm{Den}(L).\!</math> Information about the denotative aspect of meaning is obtained from <math>L\!</math> by taking its projection on the object-sign plane, in other words, on the 2-dimensional space that is generated by the object domain <math>O\!</math> and the sign domain <math>S.\!</math> This component of a sign relation <math>L\!</math> can be written in any of the forms, <math>\mathrm{proj}_{OS} L,\!</math> &nbsp; <math>L_{OS},\!</math> &nbsp; <math>\mathrm{proj}_{12} L,\!</math> &nbsp; <math>L_{12},\!</math> and it is defined as follows: {| align="center" cellpadding="6" width="90%" | <math>\begin{matrix} \mathrm{Den}(L) & = & \mathrm{proj}_{OS} L & = & \{ (o, s) \in O \times S ~:~ (o, s, i) \in L ~\text{for some}~ i \in I \}. \end{matrix}</math> |} Looking to the denotative aspects of <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B},\!</math> various rows of the Tables specify, for example, that <math>\mathrm{A}\!</math> uses <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> to denote <math>\mathrm{A}\!</math> and <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> to denote <math>\mathrm{B},\!</math> whereas <math>\mathrm{B}\!</math> uses <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> to denote <math>\mathrm{B}\!</math> and <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> to denote <math>\mathrm{A}.\!</math> All of these denotative references are summed up in the projections on the <math>OS\!</math>-plane, as shown in the following Tables: <br> {| align="center" style="width:90%" | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\mathrm{proj}_{OS}(L_\mathrm{A})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Object}\!</math> | width="50%" | <math>\text{Sign}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\mathrm{proj}_{OS}(L_\mathrm{B})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Object}\!</math> | width="50%" | <math>\text{Sign}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} |} <br> ===Connotation=== Another aspect of meaning concerns the connection that a sign has to its interpretants within a given sign relation. As before, this type of connection can be vacuous, singular, or plural in its collection of terminal points, and it can be formalized as the dyadic relation that is obtained as a planar projection of the triadic sign relation in question. The connection that a sign makes to an interpretant is here referred to as its ''connotation''. In the full theory of sign relations, this aspect of meaning includes the links that a sign has to affects, concepts, ideas, impressions, intentions, and the whole realm of an agent's mental states and allied activities, broadly encompassing intellectual associations, emotional impressions, motivational impulses, and real conduct. Taken at the full, in the natural setting of semiotic phenomena, this complex system of references is unlikely ever to find itself mapped in much detail, much less completely formalized, but the tangible warp of its accumulated mass is commonly alluded to as the connotative import of language. Formally speaking, however, the connotative aspect of meaning presents no additional difficulty. For a given sign relation <math>L,\!</math> the dyadic relation that constitutes the ''connotative aspect'' or ''connotative component'' of <math>L\!</math> is notated as <math>\mathrm{Con}(L).\!</math> The connotative aspect of a sign relation <math>L\!</math> is given by its projection on the plane of signs and interpretants, and is therefore defined as follows: {| align="center" cellpadding="6" width="90%" | <math>\begin{matrix} \mathrm{Con}(L) & = & \mathrm{proj}_{SI} L & = & \{ (s, i) \in S \times I ~:~ (o, s, i) \in L ~\text{for some}~ o \in O \}. \end{matrix}</math> |} All of these connotative references are summed up in the projections on the <math>SI\!</math>-plane, as shown in the following Tables: <br> {| align="center" style="width:90%" | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\mathrm{proj}_{SI}(L_\mathrm{A})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Sign}\!</math> | width="50%" | <math>\text{Interpretant}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\mathrm{proj}_{SI}(L_\mathrm{B})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Sign}\!</math> | width="50%" | <math>\text{Interpretant}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} |} <br> ===Ennotation=== The aspect of a sign's meaning that arises from the dyadic relation of its objects to its interpretants has no standard name. If an interpretant is considered to be a sign in its own right, then its independent reference to an object can be taken as belonging to another moment of denotation, but this neglects the mediational character of the whole transaction in which this occurs. Denotation and connotation have to do with dyadic relations in which the sign plays an active role, but here we have to consider a dyadic relation between objects and interpretants that is mediated by the sign from an off-stage position, as it were. As a relation between objects and interpretants that is mediated by a sign, this aspect of meaning may be referred to as the ''ennotation'' of a sign, and the dyadic relation that constitutes the ''ennotative aspect'' of a sign relation <math>L\!</math> may be notated as <math>\mathrm{Enn}(L).\!</math> The ennotational component of meaning for a sign relation <math>L\!</math> is captured by its projection on the plane of the object and interpretant domains, and it is thus defined as follows: {| align="center" cellpadding="6" width="90%" | <math>\begin{matrix} \mathrm{Enn}(L) & = & \mathrm{proj}_{OI} L & = & \{ (o, i) \in O \times I ~:~ (o, s, i) \in L ~\text{for some}~ s \in S \}. \end{matrix}</math> |} As it happens, the sign relations <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> are fully symmetric with respect to exchanging signs and interpretants, so all the data of <math>\mathrm{proj}_{OS} L_\mathrm{A}\!</math> is echoed unchanged in <math>\mathrm{proj}_{OI} L_\mathrm{A}\!</math> and all the data of <math>\mathrm{proj}_{OS} L_\mathrm{B}\!</math> is echoed unchanged in <math>\mathrm{proj}_{OI} L_\mathrm{B}.\!</math> <br> {| align="center" style="width:90%" | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\mathrm{proj}_{OI}(L_\mathrm{A})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Object}\!</math> | width="50%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\mathrm{proj}_{OI}(L_\mathrm{B})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Object}\!</math> | width="50%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} |} <br> ==Semiotic equivalence relations== (Text in Progress) ==Six ways of looking at a sign relation== In the context of 3-adic relations in general, Peirce provides the following illustration of the six ''converses'' of a 3-adic relation, that is, the six differently ordered ways of stating what is logically the same 3-adic relation: : So in a triadic fact, say, the example <br> {| align="center" cellspacing="8" style="width:72%" | align="center" | ''A'' gives ''B'' to ''C'' |} : we make no distinction in the ordinary logic of relations between the ''subject nominative'', the ''direct object'', and the ''indirect object''. We say that the proposition has three ''logical subjects''. We regard it as a mere affair of English grammar that there are six ways of expressing this: <br> {| align="center" cellspacing="8" style="width:72%" | style="width:36%" | ''A'' gives ''B'' to ''C'' | style="width:36%" | ''A'' benefits ''C'' with ''B'' |- | ''B'' enriches ''C'' at expense of ''A'' | ''C'' receives ''B'' from ''A'' |- | ''C'' thanks ''A'' for ''B'' | ''B'' leaves ''A'' for ''C'' |} : These six sentences express one and the same indivisible phenomenon. (C.S. Peirce, &ldquo;The Categories Defended&rdquo;, MS&nbsp;308 (1903), EP&nbsp;2, 170&ndash;171). ===IOS=== (Text in Progress) ===ISO=== (Text in Progress) ===OIS=== {| align="center" cellpadding="6" width="90%" | <p>Words spoken are symbols or signs (σύμβολα) of affections or impressions (παθήματα) of the soul (ψυχή); written words are the signs of words spoken. As writing, so also is speech not the same for all races of men. But the mental affections themselves, of which these words are primarily signs (σημεια), are the same for the whole of mankind, as are also the objects (πράγματα) of which those affections are representations or likenesses, images, copies (ομοιώματα). (Aristotle, ''De Interpretatione'', 1.16^a4).</p> |} ===OSI=== (Text in Progress) ===SIO=== {| align="center" cellpadding="6" width="90%" | <p>Logic will here be defined as ''formal semiotic''. A definition of a sign will be given which no more refers to human thought than does the definition of a line as the place which a particle occupies, part by part, during a lapse of time. Namely, a sign is something, ''A'', which brings something, ''B'', its ''interpretant'' sign determined or created by it, into the same sort of correspondence with something, ''C'', its ''object'', as that in which itself stands to ''C''. It is from this definition, together with a definition of &ldquo;formal&rdquo;, that I deduce mathematically the principles of logic. I also make a historical review of all the definitions and conceptions of logic, and show, not merely that my definition is no novelty, but that my non-psychological conception of logic has ''virtually'' been quite generally held, though not generally recognized. (C.S. Peirce, &ldquo;Application to the Carnegie Institution&rdquo;, L75 (1902), NEM 4, 20&ndash;21).</p> |} ===SOI=== {| align="center" cellpadding="6" width="90%" | <p>A ''Sign'' is anything which is related to a Second thing, its ''Object'', in respect to a Quality, in such a way as to bring a Third thing, its ''Interpretant'', into relation to the same Object, and that in such a way as to bring a Fourth into relation to that Object in the same form, ''ad infinitum''. (CP&nbsp;2.92, quoted in Fisch 1986, p.&nbsp;274)</p> |} ==References== ==Bibliography== ===Primary sources=== * [[Charles Sanders Peirce (Bibliography)]] ===Secondary sources=== * Awbrey, J.L., and Awbrey, S.M. (1995), &ldquo;Interpretation as Action : The Risk of Inquiry&rdquo;, ''Inquiry : Critical Thinking Across the Disciplines'' 15(1), pp. 40&ndash;52. [http://web.archive.org/web/19970626071826/http://chss.montclair.edu/inquiry/fall95/awbrey.html Archive]. [http://independent.academia.edu/JonAwbrey/Papers/1302117/Interpretation_as_Action_The_Risk_of_Inquiry Online]. * Deledalle, Gérard (2000), ''C.S. Peirce's Philosophy of Signs'', Indiana University Press, Bloomington, IN. * Eisele, Carolyn (1979), in ''Studies in the Scientific and Mathematical Philosophy of C.S. Peirce'', Richard Milton Martin (ed.), Mouton, The Hague. * Esposito, Joseph (1980), ''Evolutionary Metaphysics : The Development of Peirce's Theory of Categories'', Ohio University Press (?). * Fisch, Max (1986), ''Peirce, Semeiotic, and Pragmatism'', Indiana University Press, Bloomington, IN. * Houser, N., Roberts, D.D., and Van Evra, J. (eds.)(1997), ''Studies in the Logic of C.S. Peirce'', Indiana University Press, Bloomington, IN. * Liszka, J.J. (1996), ''A General Introduction to the Semeiotic of C.S. Peirce'', Indiana University Press, Bloomington, IN. * Misak, C. (ed.)(2004), ''Cambridge Companion to C.S. Peirce'', Cambridge University Press. * Moore, E., and Robin, R. (1964), ''Studies in the Philosophy of C.S. Peirce, Second Series'', University of Massachusetts Press, Amherst, MA. * Murphey, M. (1961), ''The Development of Peirce's Thought''. Reprinted, Hackett, Indianapolis, IN, 1993. * Percy, Walker (2000), pp. 271&ndash;291 in ''Signposts in a Strange Land'', P. Samway (ed.), Saint Martin's Press. ==Resources== * [http://www.helsinki.fi/science/commens/dictionary.html The Commens Dictionary of Peirce's Terms] ** [http://www.helsinki.fi/science/commens/terms/semeiotic.html Entry for ''Semeiotic''] ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Sign_relation Sign Relation @ InterSciWiki] * [http://mywikibiz.com/Sign_relation Sign Relation @ MyWikiBiz] {{col-break}} * [http://beta.wikiversity.org/wiki/Sign_relation Sign Relation @ Wikiversity Beta] * [http://ref.subwiki.org/wiki/Sign_relation Sign Relation @ Subject Wikis] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Sign_relation Sign Relation], [http://mywikibiz.com/ MyWikiBiz] * [http://mathweb.org/wiki/Sign_relation Sign Relation], [http://mathweb.org/ MathWeb Wiki] {{col-break}} * [http://planetmath.org/SignRelation Sign Relation], [http://planetmath.org/ PlanetMath] * [http://beta.wikiversity.org/wiki/Sign_relation Sign Relation], [http://beta.wikiversity.org/ Wikiversity Beta] {{col-break}} * [http://getwiki.net/-Sign_Relation Sign Relation], [http://getwiki.net/ GetWiki] * [http://en.wikipedia.org/w/index.php?title=Sign_relation&oldid=161631069 Sign Relation], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Artificial Intelligence]] [[Category:Charles Sanders Peirce]] [[Category:Cognitive Sciences]] [[Category:Computer Science]] [[Category:Formal Sciences]] [[Category:Hermeneutics]] [[Category:Information Systems]] [[Category:Information Theory]] [[Category:Inquiry]] [[Category:Intelligence Amplification]] [[Category:Knowledge Representation]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Philosophy]] [[Category:Pragmatics]] [[Category:Semantics]] [[Category:Semiotics]] [[Category:Syntax]] 82d5d4bdc8ab8fc740ca68cb4b7e696cd3794daf 658 655 2013-10-29T20:04:20Z Jon Awbrey 3 small fixes wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''sign relation''' is the basic construct in the theory of signs, also known as [[semeiotic]] or [[semiotics]], as developed by Charles Sanders Peirce. ==Anthesis== {| align="center" cellpadding="6" width="90%" | <p>Thus, if a sunflower, in turning towards the sun, becomes by that very act fully capable, without further condition, of reproducing a sunflower which turns in precisely corresponding ways toward the sun, and of doing so with the same reproductive power, the sunflower would become a Representamen of the sun. (C.S. Peirce, &ldquo;Syllabus&rdquo; (''c''.&nbsp;1902), ''Collected Papers'', CP&nbsp;2.274).</p> |} In his picturesque illustration of a sign relation, along with his tracing of a corresponding sign process, or ''[[semiosis]]'', Peirce uses the technical term ''representamen'' for his concept of a sign, but the shorter word is precise enough, so long as one recognizes that its meaning in a particular theory of signs is given by a specific definition of what it means to be a sign. ==Definition== One of Peirce's clearest and most complete definitions of a sign is one that he gives in the context of providing a definition for ''logic'', and so it is informative to view it in that setting. {| align="center" cellpadding="6" width="90%" | <p>Logic will here be defined as ''formal semiotic''. A definition of a sign will be given which no more refers to human thought than does the definition of a line as the place which a particle occupies, part by part, during a lapse of time. Namely, a sign is something, ''A'', which brings something, ''B'', its ''interpretant'' sign determined or created by it, into the same sort of correspondence with something, ''C'', its ''object'', as that in which itself stands to ''C''. It is from this definition, together with a definition of &ldquo;formal&rdquo;, that I deduce mathematically the principles of logic. I also make a historical review of all the definitions and conceptions of logic, and show, not merely that my definition is no novelty, but that my non-psychological conception of logic has ''virtually'' been quite generally held, though not generally recognized. (C.S. Peirce, NEM&nbsp;4, 20&ndash;21).</p> |} In the general discussion of diverse theories of signs, the question frequently arises whether signhood is an absolute, essential, indelible, or ''ontological'' property of a thing, or whether it is a relational, interpretive, and mutable role that a thing can be said to have only within a particular context of relationships. Peirce's definition of a ''sign'' defines it in relation to its ''object'' and its ''interpretant sign'', and thus it defines signhood in ''[[logic of relatives|relative terms]]'', by means of a predicate with three places. In this definition, signhood is a role in a [[triadic relation]], a role that a thing bears or plays in a given context of relationships &mdash; it is not as an ''absolute'', ''non-relative'' property of a thing-in-itself, one that it possesses independently of all relationships to other things. Some of the terms that Peirce uses in his definition of a sign may need to be elaborated for the contemporary reader. :* '''Correspondence.''' From the way that Peirce uses this term throughout his work, it is clear that he means what he elsewhere calls a "triple correspondence", and thus this is just another way of referring to the whole triadic sign relation itself. In particular, his use of this term should not be taken to imply a dyadic correspondence, like the kinds of &ldquo;mirror image&rdquo; correspondence between realities and representations that are bandied about in contemporary controversies about &ldquo;correspondence theories of truth&rdquo;. :* '''Determination.''' Peirce's concept of determination is broader in several directions than the sense of the word that refers to strictly deterministic causal-temporal processes. First, and especially in this context, he is invoking a more general concept of determination, what is called a ''formal'' or ''informational'' determination, as in saying "two points determine a line", rather than the more special cases of causal and temporal determinisms. Second, he characteristically allows for what is called ''determination in measure'', that is, an order of determinism that admits a full spectrum of more and less determined relationships. :* '''Non-psychological.''' Peirce's &ldquo;non-psychological conception of logic&rdquo; must be distinguished from any variety of ''anti-psychologism''. He was quite interested in matters of psychology and had much of import to say about them. But logic and psychology operate on different planes of study even when they have occasion to view the same data, as logic is a ''[[normative science]]'' where psychology is a ''[[descriptive science]]'', and so they have very different aims, methods, and rationales. ==Signs and inquiry== : ''Main article : [[Inquiry]]'' There is a close relationship between the pragmatic theory of signs and the pragmatic theory of [[inquiry]]. In fact, the correspondence between the two studies exhibits so many congruences and parallels that it is often best to treat them as integral parts of one and the same subject. In a very real sense, inquiry is the process by which sign relations come to be established and continue to evolve. In other words, inquiry, &ldquo;thinking&rdquo; in its best sense, &ldquo;is a term denoting the various ways in which things acquire significance&rdquo; (John Dewey). Thus, there is an active and intricate form of cooperation that needs to be appreciated and maintained between these converging modes of investigation. Its proper character is best understood by realizing that the theory of inquiry is adapted to study the developmental aspects of sign relations, a subject which the theory of signs is specialized to treat from structural and comparative points of view. ==Examples of sign relations== Because the examples to follow have been artificially constructed to be as simple as possible, their detailed elaboration can run the risk of trivializing the whole theory of sign relations. Despite their simplicity, however, these examples have subtleties of their own, and their careful treatment will serve to illustrate many important issues in the general theory of signs. Imagine a discussion between two people, Ann and Bob, and attend only to that aspect of their interpretive practice that involves the use of the following nouns and pronouns: &ldquo;Ann&rdquo;, &ldquo;Bob&rdquo;, &ldquo;I&rdquo;, &ldquo;you&rdquo;. :* The ''object domain'' of this discussion fragment is the set of two people <math>\{ \text{Ann}, \text{Bob} \}.\!</math> :* The ''syntactic domain'' or the ''sign system'' of their discussion is limited to the set of four signs <math>\{ {}^{\backprime\backprime} \text{Ann} {}^{\prime\prime}, {}^{\backprime\backprime} \text{Bob} {}^{\prime\prime}, {}^{\backprime\backprime} \text{I} {}^{\prime\prime}, {}^{\backprime\backprime} \text{you} {}^{\prime\prime} \}.\!</math> In their discussion, Ann and Bob are not only the passive objects of nominative and accusative references but also the active interpreters of the language that they use. The ''system of interpretation'' (SOI) associated with each language user can be represented in the form of an individual [[three-place relation]] called the ''sign relation'' of that interpreter. Understood in terms of its ''set-theoretic extension'', a sign relation <math>L\!</math> is a ''subset'' of a ''cartesian product'' <math>O \times S \times I.\!</math> Here, <math>O, S, I\!</math> are three sets that are known as the ''object domain'', the ''sign domain'', and the ''interpretant domain'', respectively, of the sign relation <math>L \subseteq O \times S \times I.\!</math> Broadly speaking, the three domains of a sign relation can be any sets whatsoever, but the kinds of sign relations typically contemplated in formal settings are usually constrained to having <math>I \subseteq S\!</math>. In this case interpretants are just a special variety of signs and this makes it convenient to lump signs and interpretants together into a single class called the ''syntactic domain''. In the forthcoming examples <math>S\!</math> and <math>I\!</math> are identical as sets, so the same elements manifest themselves in two different roles of the sign relations in question. When it is necessary to refer to the whole set of objects and signs in the union of the domains <math>O, S, I\!</math> for a given sign relation <math>L,\!</math> one may refer to this set as the ''World'' of <math>L\!</math> and write <math>W = W_L = O \cup S \cup I.\!</math> To facilitate an interest in the abstract structures of sign relations, and to keep the notations as brief as possible as the examples become more complicated, it serves to introduce the following general notations: {| align="center" cellspacing="6" width="90%" | <math>\begin{array}{ccl} O & = & \text{Object Domain} \\[6pt] S & = & \text{Sign Domain} \\[6pt] I & = & \text{Interpretant Domain} \end{array}</math> |} Introducing a few abbreviations for use in considering the present Example, we have the following data: {| align="center" cellspacing="6" width="90%" | <math>\begin{array}{cclcl} O & = & \{ \text{Ann}, \text{Bob} \} & = & \{ \text{A}, \text{B} \} \\[6pt] S & = & \{ {}^{\backprime\backprime} \text{Ann} {}^{\prime\prime}, {}^{\backprime\backprime} \text{Bob} {}^{\prime\prime}, {}^{\backprime\backprime} \text{I} {}^{\prime\prime}, {}^{\backprime\backprime} \text{you} {}^{\prime\prime} \} & = & \{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \} \\[6pt] I & = & \{ {}^{\backprime\backprime} \text{Ann} {}^{\prime\prime}, {}^{\backprime\backprime} \text{Bob} {}^{\prime\prime}, {}^{\backprime\backprime} \text{I} {}^{\prime\prime}, {}^{\backprime\backprime} \text{you} {}^{\prime\prime} \} & = & \{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \} \end{array}</math> |} In the present example, <math>S = I = \text{Syntactic Domain}\!</math>. The next two Tables give the sign relations associated with the interpreters <math>\mathrm{A}\!</math> and <math>\mathrm{B},\!</math> respectively, putting them in the form of ''relational databases''. Thus, the rows of each Table list the ordered triples of the form <math>(o, s, i)\!</math> that make up the corresponding sign relations, <math>L_\mathrm{A}, L_\mathrm{B} \subseteq O \times S \times I.\!</math> <br> {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%" |+ style="height:30px" | <math>L_\mathrm{A} ~=~ \text{Sign Relation of Interpreter A}\!</math> |- style="height:40px; background:ghostwhite" | style="width:33%" | <math>\text{Object}\!</math> | style="width:33%" | <math>\text{Sign}\!</math> | style="width:33%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} <br> {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%" |+ style="height:30px" | <math>L_\mathrm{B} ~=~ \text{Sign Relation of Interpreter B}\!</math> |- style="height:40px; background:ghostwhite" | style="width:33%" | <math>\text{Object}\!</math> | style="width:33%" | <math>\text{Sign}\!</math> | style="width:33%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- |<math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} <br> These Tables codify a rudimentary level of interpretive practice for the agents <math>\mathrm{A}\!</math> and <math>\mathrm{B}\!</math> and provide a basis for formalizing the initial semantics that is appropriate to their common syntactic domain. Each row of a Table names an object and two co-referent signs, making up an ordered triple of the form <math>(o, s, i)\!</math> called an ''elementary relation'', that is, one element of the relation's set-theoretic extension. Already in this elementary context, there are several different meanings that might attach to the project of a ''formal semiotics'', or a formal theory of meaning for signs. In the process of discussing these alternatives, it is useful to introduce a few terms that are occasionally used in the philosophy of language to point out the needed distinctions. ==Dyadic aspects of sign relations== For an arbitrary triadic relation <math>L \subseteq O \times S \times I,\!</math> whether it is a sign relation or not, there are six dyadic relations that can be obtained by ''projecting'' <math>L\!</math> on one of the planes of the <math>OSI\!</math>-space <math>O \times S \times I.\!</math> The six dyadic projections of a triadic relation <math>L\!</math> are defined and notated as follows: <br> {| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%" | <math>\begin{matrix} L_{OS} & = & \mathrm{proj}_{OS}(L) & = & \{ (o, s) \in O \times S ~:~ (o, s, i) \in L ~\text{for some}~ i \in I \} \\[6pt] L_{SO} & = & \mathrm{proj}_{SO}(L) & = & \{ (s, o) \in S \times O ~:~ (o, s, i) \in L ~\text{for some}~ i \in I \} \\[6pt] L_{IS} & = & \mathrm{proj}_{IS}(L) & = & \{ (i, s) \in I \times S ~:~ (o, s, i) \in L ~\text{for some}~ o \in O \} \\[6pt] L_{SI} & = & \mathrm{proj}_{SI}(L) & = & \{ (s, i) \in S \times I ~:~ (o, s, i) \in L ~\text{for some}~ o \in O \} \\[6pt] L_{OI} & = & \mathrm{proj}_{OI}(L) & = & \{ (o, i) \in O \times I ~:~ (o, s, i) \in L ~\text{for some}~ s \in S \} \\[6pt] L_{IO} & = & \mathrm{proj}_{IO}(L) & = & \{ (i, o) \in I \times O ~:~ (o, s, i) \in L ~\text{for some}~ s \in S \} \end{matrix}</math> |} <br> By way of unpacking the set-theoretic notation, here is what the first definition says in ordinary language. {| align="center" cellpadding="6" width="90%" | <p>The dyadic relation that results from the projection of <math>L\!</math> on the <math>OS\!</math>-plane <math>O \times S\!</math> is written briefly as <math>L_{OS}\!</math> or written more fully as <math>\mathrm{proj}_{OS}(L),\!</math> and it is defined as the set of all ordered pairs <math>(o, s)\!</math> in the cartesian product <math>O \times S\!</math> for which there exists an ordered triple <math>(o, s, i)\!</math> in <math>L\!</math> for some interpretant <math>i\!</math> in the interpretant domain <math>I.\!</math></p> |} In the case where <math>L\!</math> is a sign relation, which it becomes by satisfying one of the definitions of a sign relation, some of the dyadic aspects of <math>L\!</math> can be recognized as formalizing aspects of sign meaning that have received their share of attention from students of signs over the centuries, and thus they can be associated with traditional concepts and terminology. Of course, traditions may vary as to the precise formation and usage of such concepts and terms. Other aspects of meaning have not received their fair share of attention, and thus remain anonymous on the contemporary scene of sign studies. ===Denotation=== One aspect of a sign's complete meaning is concerned with the reference that a sign has to its objects, which objects are collectively known as the ''denotation'' of the sign. In the pragmatic theory of sign relations, denotative references fall within the projection of the sign relation on the plane that is spanned by its object domain and its sign domain. The dyadic relation that makes up the ''denotative'', ''referential'', or ''semantic'' aspect or component of a sign relation <math>L\!</math> is notated as <math>\mathrm{Den}(L).\!</math> Information about the denotative aspect of meaning is obtained from <math>L\!</math> by taking its projection on the object-sign plane, in other words, on the 2-dimensional space that is generated by the object domain <math>O\!</math> and the sign domain <math>S.\!</math> This component of a sign relation <math>L\!</math> can be written in any of the forms, <math>\mathrm{proj}_{OS} L,\!</math> &nbsp; <math>L_{OS},\!</math> &nbsp; <math>\mathrm{proj}_{12} L,\!</math> &nbsp; <math>L_{12},\!</math> and it is defined as follows: {| align="center" cellpadding="6" width="90%" | <math>\begin{matrix} \mathrm{Den}(L) & = & \mathrm{proj}_{OS} L & = & \{ (o, s) \in O \times S ~:~ (o, s, i) \in L ~\text{for some}~ i \in I \}. \end{matrix}</math> |} Looking to the denotative aspects of <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B},\!</math> various rows of the Tables specify, for example, that <math>\mathrm{A}\!</math> uses <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> to denote <math>\mathrm{A}\!</math> and <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> to denote <math>\mathrm{B},\!</math> whereas <math>\mathrm{B}\!</math> uses <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> to denote <math>\mathrm{B}\!</math> and <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> to denote <math>\mathrm{A}.\!</math> All of these denotative references are summed up in the projections on the <math>OS\!</math>-plane, as shown in the following Tables: <br> {| align="center" style="width:90%" | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\mathrm{proj}_{OS}(L_\mathrm{A})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Object}\!</math> | width="50%" | <math>\text{Sign}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\mathrm{proj}_{OS}(L_\mathrm{B})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Object}\!</math> | width="50%" | <math>\text{Sign}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} |} <br> ===Connotation=== Another aspect of meaning concerns the connection that a sign has to its interpretants within a given sign relation. As before, this type of connection can be vacuous, singular, or plural in its collection of terminal points, and it can be formalized as the dyadic relation that is obtained as a planar projection of the triadic sign relation in question. The connection that a sign makes to an interpretant is here referred to as its ''connotation''. In the full theory of sign relations, this aspect of meaning includes the links that a sign has to affects, concepts, ideas, impressions, intentions, and the whole realm of an agent's mental states and allied activities, broadly encompassing intellectual associations, emotional impressions, motivational impulses, and real conduct. Taken at the full, in the natural setting of semiotic phenomena, this complex system of references is unlikely ever to find itself mapped in much detail, much less completely formalized, but the tangible warp of its accumulated mass is commonly alluded to as the connotative import of language. Formally speaking, however, the connotative aspect of meaning presents no additional difficulty. For a given sign relation <math>L,\!</math> the dyadic relation that constitutes the ''connotative aspect'' or ''connotative component'' of <math>L\!</math> is notated as <math>\mathrm{Con}(L).\!</math> The connotative aspect of a sign relation <math>L\!</math> is given by its projection on the plane of signs and interpretants, and is therefore defined as follows: {| align="center" cellpadding="6" width="90%" | <math>\begin{matrix} \mathrm{Con}(L) & = & \mathrm{proj}_{SI} L & = & \{ (s, i) \in S \times I ~:~ (o, s, i) \in L ~\text{for some}~ o \in O \}. \end{matrix}</math> |} All of these connotative references are summed up in the projections on the <math>SI\!</math>-plane, as shown in the following Tables: <br> {| align="center" style="width:90%" | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\mathrm{proj}_{SI}(L_\mathrm{A})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Sign}\!</math> | width="50%" | <math>\text{Interpretant}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\mathrm{proj}_{SI}(L_\mathrm{B})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Sign}\!</math> | width="50%" | <math>\text{Interpretant}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} |} <br> ===Ennotation=== The aspect of a sign's meaning that arises from the dyadic relation of its objects to its interpretants has no standard name. If an interpretant is considered to be a sign in its own right, then its independent reference to an object can be taken as belonging to another moment of denotation, but this neglects the mediational character of the whole transaction in which this occurs. Denotation and connotation have to do with dyadic relations in which the sign plays an active role, but here we have to consider a dyadic relation between objects and interpretants that is mediated by the sign from an off-stage position, as it were. As a relation between objects and interpretants that is mediated by a sign, this aspect of meaning may be referred to as the ''ennotation'' of a sign, and the dyadic relation that constitutes the ''ennotative aspect'' of a sign relation <math>L\!</math> may be notated as <math>\mathrm{Enn}(L).\!</math> The ennotational component of meaning for a sign relation <math>L\!</math> is captured by its projection on the plane of the object and interpretant domains, and it is thus defined as follows: {| align="center" cellpadding="6" width="90%" | <math>\begin{matrix} \mathrm{Enn}(L) & = & \mathrm{proj}_{OI} L & = & \{ (o, i) \in O \times I ~:~ (o, s, i) \in L ~\text{for some}~ s \in S \}. \end{matrix}</math> |} As it happens, the sign relations <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> are fully symmetric with respect to exchanging signs and interpretants, so all the data of <math>\mathrm{proj}_{OS} L_\mathrm{A}\!</math> is echoed unchanged in <math>\mathrm{proj}_{OI} L_\mathrm{A}\!</math> and all the data of <math>\mathrm{proj}_{OS} L_\mathrm{B}\!</math> is echoed unchanged in <math>\mathrm{proj}_{OI} L_\mathrm{B}.\!</math> <br> {| align="center" style="width:90%" | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\mathrm{proj}_{OI}(L_\mathrm{A})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Object}\!</math> | width="50%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\mathrm{proj}_{OI}(L_\mathrm{B})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Object}\!</math> | width="50%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} |} <br> ==Semiotic equivalence relations== <font size="3">&#9758;</font> [http://planetmath.org/SemioticEquivalenceRelation PlanetMath &bull; Semiotic Equivalence Relation] ==Six ways of looking at a sign relation== In the context of 3-adic relations in general, Peirce provides the following illustration of the six ''converses'' of a 3-adic relation, that is, the six differently ordered ways of stating what is logically the same 3-adic relation: : So in a triadic fact, say, the example <br> {| align="center" cellspacing="8" style="width:72%" | align="center" | ''A'' gives ''B'' to ''C'' |} : we make no distinction in the ordinary logic of relations between the ''subject nominative'', the ''direct object'', and the ''indirect object''. We say that the proposition has three ''logical subjects''. We regard it as a mere affair of English grammar that there are six ways of expressing this: <br> {| align="center" cellspacing="8" style="width:72%" | style="width:36%" | ''A'' gives ''B'' to ''C'' | style="width:36%" | ''A'' benefits ''C'' with ''B'' |- | ''B'' enriches ''C'' at expense of ''A'' | ''C'' receives ''B'' from ''A'' |- | ''C'' thanks ''A'' for ''B'' | ''B'' leaves ''A'' for ''C'' |} : These six sentences express one and the same indivisible phenomenon. (C.S. Peirce, &ldquo;The Categories Defended&rdquo;, MS&nbsp;308 (1903), EP&nbsp;2, 170&ndash;171). ===IOS=== (Text in Progress) ===ISO=== (Text in Progress) ===OIS=== {| align="center" cellpadding="6" width="90%" | <p>Words spoken are symbols or signs (σύμβολα) of affections or impressions (παθήματα) of the soul (ψυχή); written words are the signs of words spoken. As writing, so also is speech not the same for all races of men. But the mental affections themselves, of which these words are primarily signs (σημεια), are the same for the whole of mankind, as are also the objects (πράγματα) of which those affections are representations or likenesses, images, copies (ομοιώματα). (Aristotle, ''De Interpretatione'', 1.16^a4).</p> |} ===OSI=== (Text in Progress) ===SIO=== {| align="center" cellpadding="6" width="90%" | <p>Logic will here be defined as ''formal semiotic''. A definition of a sign will be given which no more refers to human thought than does the definition of a line as the place which a particle occupies, part by part, during a lapse of time. Namely, a sign is something, ''A'', which brings something, ''B'', its ''interpretant'' sign determined or created by it, into the same sort of correspondence with something, ''C'', its ''object'', as that in which itself stands to ''C''. It is from this definition, together with a definition of &ldquo;formal&rdquo;, that I deduce mathematically the principles of logic. I also make a historical review of all the definitions and conceptions of logic, and show, not merely that my definition is no novelty, but that my non-psychological conception of logic has ''virtually'' been quite generally held, though not generally recognized. (C.S. Peirce, &ldquo;Application to the Carnegie Institution&rdquo;, L75 (1902), NEM 4, 20&ndash;21).</p> |} ===SOI=== {| align="center" cellpadding="6" width="90%" | <p>A ''Sign'' is anything which is related to a Second thing, its ''Object'', in respect to a Quality, in such a way as to bring a Third thing, its ''Interpretant'', into relation to the same Object, and that in such a way as to bring a Fourth into relation to that Object in the same form, ''ad infinitum''. (CP&nbsp;2.92, quoted in Fisch 1986, p.&nbsp;274)</p> |} ==References== ==Bibliography== ===Primary sources=== * [[Charles Sanders Peirce (Bibliography)]] ===Secondary sources=== * Awbrey, J.L., and Awbrey, S.M. (1995), &ldquo;Interpretation as Action : The Risk of Inquiry&rdquo;, ''Inquiry : Critical Thinking Across the Disciplines'' 15(1), pp. 40&ndash;52. [http://web.archive.org/web/19970626071826/http://chss.montclair.edu/inquiry/fall95/awbrey.html Archive]. [http://independent.academia.edu/JonAwbrey/Papers/1302117/Interpretation_as_Action_The_Risk_of_Inquiry Online]. * Deledalle, Gérard (2000), ''C.S. Peirce's Philosophy of Signs'', Indiana University Press, Bloomington, IN. * Eisele, Carolyn (1979), in ''Studies in the Scientific and Mathematical Philosophy of C.S. Peirce'', Richard Milton Martin (ed.), Mouton, The Hague. * Esposito, Joseph (1980), ''Evolutionary Metaphysics : The Development of Peirce's Theory of Categories'', Ohio University Press (?). * Fisch, Max (1986), ''Peirce, Semeiotic, and Pragmatism'', Indiana University Press, Bloomington, IN. * Houser, N., Roberts, D.D., and Van Evra, J. (eds.)(1997), ''Studies in the Logic of C.S. Peirce'', Indiana University Press, Bloomington, IN. * Liszka, J.J. (1996), ''A General Introduction to the Semeiotic of C.S. Peirce'', Indiana University Press, Bloomington, IN. * Misak, C. (ed.)(2004), ''Cambridge Companion to C.S. Peirce'', Cambridge University Press. * Moore, E., and Robin, R. (1964), ''Studies in the Philosophy of C.S. Peirce, Second Series'', University of Massachusetts Press, Amherst, MA. * Murphey, M. (1961), ''The Development of Peirce's Thought''. Reprinted, Hackett, Indianapolis, IN, 1993. * Percy, Walker (2000), pp. 271&ndash;291 in ''Signposts in a Strange Land'', P. Samway (ed.), Saint Martin's Press. ==Resources== * [http://www.helsinki.fi/science/commens/dictionary.html The Commens Dictionary of Peirce's Terms] ** [http://www.helsinki.fi/science/commens/terms/semeiotic.html Entry for ''Semeiotic''] ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Sign_relation Sign Relation @ InterSciWiki] * [http://mywikibiz.com/Sign_relation Sign Relation @ MyWikiBiz] {{col-break}} * [http://beta.wikiversity.org/wiki/Sign_relation Sign Relation @ Wikiversity Beta] * [http://ref.subwiki.org/wiki/Sign_relation Sign Relation @ Subject Wikis] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Sign_relation Sign Relation], [http://mywikibiz.com/ MyWikiBiz] * [http://mathweb.org/wiki/Sign_relation Sign Relation], [http://mathweb.org/ MathWeb Wiki] {{col-break}} * [http://planetmath.org/SignRelation Sign Relation], [http://planetmath.org/ PlanetMath] * [http://beta.wikiversity.org/wiki/Sign_relation Sign Relation], [http://beta.wikiversity.org/ Wikiversity Beta] {{col-break}} * [http://getwiki.net/-Sign_Relation Sign Relation], [http://getwiki.net/ GetWiki] * [http://en.wikipedia.org/w/index.php?title=Sign_relation&oldid=161631069 Sign Relation], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Artificial Intelligence]] [[Category:Charles Sanders Peirce]] [[Category:Cognitive Sciences]] [[Category:Computer Science]] [[Category:Formal Sciences]] [[Category:Hermeneutics]] [[Category:Information Systems]] [[Category:Information Theory]] [[Category:Inquiry]] [[Category:Intelligence Amplification]] [[Category:Knowledge Representation]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Philosophy]] [[Category:Pragmatics]] [[Category:Semantics]] [[Category:Semiotics]] [[Category:Syntax]] 5f0c6aad348d5eeb0d6a7c7f1814854b4f441b28 660 658 2013-11-01T16:20:26Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''sign relation''' is the basic construct in the theory of signs, also known as [[semeiotic]] or [[semiotics]], as developed by Charles Sanders Peirce. ==Anthesis== {| align="center" cellpadding="6" width="90%" | <p>Thus, if a sunflower, in turning towards the sun, becomes by that very act fully capable, without further condition, of reproducing a sunflower which turns in precisely corresponding ways toward the sun, and of doing so with the same reproductive power, the sunflower would become a Representamen of the sun. (C.S. Peirce, &ldquo;Syllabus&rdquo; (''c''.&nbsp;1902), ''Collected Papers'', CP&nbsp;2.274).</p> |} In his picturesque illustration of a sign relation, along with his tracing of a corresponding sign process, or ''[[semiosis]]'', Peirce uses the technical term ''representamen'' for his concept of a sign, but the shorter word is precise enough, so long as one recognizes that its meaning in a particular theory of signs is given by a specific definition of what it means to be a sign. ==Definition== One of Peirce's clearest and most complete definitions of a sign is one that he gives in the context of providing a definition for ''logic'', and so it is informative to view it in that setting. {| align="center" cellpadding="6" width="90%" | <p>Logic will here be defined as ''formal semiotic''. A definition of a sign will be given which no more refers to human thought than does the definition of a line as the place which a particle occupies, part by part, during a lapse of time. Namely, a sign is something, ''A'', which brings something, ''B'', its ''interpretant'' sign determined or created by it, into the same sort of correspondence with something, ''C'', its ''object'', as that in which itself stands to ''C''. It is from this definition, together with a definition of &ldquo;formal&rdquo;, that I deduce mathematically the principles of logic. I also make a historical review of all the definitions and conceptions of logic, and show, not merely that my definition is no novelty, but that my non-psychological conception of logic has ''virtually'' been quite generally held, though not generally recognized. (C.S. Peirce, NEM&nbsp;4, 20&ndash;21).</p> |} In the general discussion of diverse theories of signs, the question frequently arises whether signhood is an absolute, essential, indelible, or ''ontological'' property of a thing, or whether it is a relational, interpretive, and mutable role that a thing can be said to have only within a particular context of relationships. Peirce's definition of a ''sign'' defines it in relation to its ''object'' and its ''interpretant sign'', and thus it defines signhood in ''[[logic of relatives|relative terms]]'', by means of a predicate with three places. In this definition, signhood is a role in a [[triadic relation]], a role that a thing bears or plays in a given context of relationships &mdash; it is not as an ''absolute'', ''non-relative'' property of a thing-in-itself, one that it possesses independently of all relationships to other things. Some of the terms that Peirce uses in his definition of a sign may need to be elaborated for the contemporary reader. :* '''Correspondence.''' From the way that Peirce uses this term throughout his work, it is clear that he means what he elsewhere calls a &ldquo;triple correspondence&rdquo;, and thus this is just another way of referring to the whole triadic sign relation itself. In particular, his use of this term should not be taken to imply a dyadic correspondence, like the kinds of &ldquo;mirror image&rdquo; correspondence between realities and representations that are bandied about in contemporary controversies about &ldquo;correspondence theories of truth&rdquo;. :* '''Determination.''' Peirce's concept of determination is broader in several directions than the sense of the word that refers to strictly deterministic causal-temporal processes. First, and especially in this context, he is invoking a more general concept of determination, what is called a ''formal'' or ''informational'' determination, as in saying &ldquo;two points determine a line&rdquo;, rather than the more special cases of causal and temporal determinisms. Second, he characteristically allows for what is called ''determination in measure'', that is, an order of determinism that admits a full spectrum of more and less determined relationships. :* '''Non-psychological.''' Peirce's &ldquo;non-psychological conception of logic&rdquo; must be distinguished from any variety of ''anti-psychologism''. He was quite interested in matters of psychology and had much of import to say about them. But logic and psychology operate on different planes of study even when they have occasion to view the same data, as logic is a ''[[normative science]]'' where psychology is a ''[[descriptive science]]'', and so they have very different aims, methods, and rationales. ==Signs and inquiry== : ''Main article : [[Inquiry]]'' There is a close relationship between the pragmatic theory of signs and the pragmatic theory of [[inquiry]]. In fact, the correspondence between the two studies exhibits so many congruences and parallels that it is often best to treat them as integral parts of one and the same subject. In a very real sense, inquiry is the process by which sign relations come to be established and continue to evolve. In other words, inquiry, &ldquo;thinking&rdquo; in its best sense, &ldquo;is a term denoting the various ways in which things acquire significance&rdquo; (John Dewey). Thus, there is an active and intricate form of cooperation that needs to be appreciated and maintained between these converging modes of investigation. Its proper character is best understood by realizing that the theory of inquiry is adapted to study the developmental aspects of sign relations, a subject which the theory of signs is specialized to treat from structural and comparative points of view. ==Examples of sign relations== Because the examples to follow have been artificially constructed to be as simple as possible, their detailed elaboration can run the risk of trivializing the whole theory of sign relations. Despite their simplicity, however, these examples have subtleties of their own, and their careful treatment will serve to illustrate many important issues in the general theory of signs. Imagine a discussion between two people, Ann and Bob, and attend only to that aspect of their interpretive practice that involves the use of the following nouns and pronouns: &ldquo;Ann&rdquo;, &ldquo;Bob&rdquo;, &ldquo;I&rdquo;, &ldquo;you&rdquo;. :* The ''object domain'' of this discussion fragment is the set of two people <math>\{ \text{Ann}, \text{Bob} \}.\!</math> :* The ''syntactic domain'' or the ''sign system'' of their discussion is limited to the set of four signs <math>\{ {}^{\backprime\backprime} \text{Ann} {}^{\prime\prime}, {}^{\backprime\backprime} \text{Bob} {}^{\prime\prime}, {}^{\backprime\backprime} \text{I} {}^{\prime\prime}, {}^{\backprime\backprime} \text{you} {}^{\prime\prime} \}.\!</math> In their discussion, Ann and Bob are not only the passive objects of nominative and accusative references but also the active interpreters of the language that they use. The ''system of interpretation'' (SOI) associated with each language user can be represented in the form of an individual [[three-place relation]] called the ''sign relation'' of that interpreter. Understood in terms of its ''set-theoretic extension'', a sign relation <math>L\!</math> is a ''subset'' of a ''cartesian product'' <math>O \times S \times I.\!</math> Here, <math>O, S, I\!</math> are three sets that are known as the ''object domain'', the ''sign domain'', and the ''interpretant domain'', respectively, of the sign relation <math>L \subseteq O \times S \times I.\!</math> Broadly speaking, the three domains of a sign relation can be any sets whatsoever, but the kinds of sign relations typically contemplated in formal settings are usually constrained to having <math>I \subseteq S\!</math>. In this case interpretants are just a special variety of signs and this makes it convenient to lump signs and interpretants together into a single class called the ''syntactic domain''. In the forthcoming examples <math>S\!</math> and <math>I\!</math> are identical as sets, so the same elements manifest themselves in two different roles of the sign relations in question. When it is necessary to refer to the whole set of objects and signs in the union of the domains <math>O, S, I\!</math> for a given sign relation <math>L,\!</math> one may refer to this set as the ''World'' of <math>L\!</math> and write <math>W = W_L = O \cup S \cup I.\!</math> To facilitate an interest in the abstract structures of sign relations, and to keep the notations as brief as possible as the examples become more complicated, it serves to introduce the following general notations: {| align="center" cellspacing="6" width="90%" | <math>\begin{array}{ccl} O & = & \text{Object Domain} \\[6pt] S & = & \text{Sign Domain} \\[6pt] I & = & \text{Interpretant Domain} \end{array}</math> |} Introducing a few abbreviations for use in considering the present Example, we have the following data: {| align="center" cellspacing="6" width="90%" | <math>\begin{array}{cclcl} O & = & \{ \text{Ann}, \text{Bob} \} & = & \{ \text{A}, \text{B} \} \\[6pt] S & = & \{ {}^{\backprime\backprime} \text{Ann} {}^{\prime\prime}, {}^{\backprime\backprime} \text{Bob} {}^{\prime\prime}, {}^{\backprime\backprime} \text{I} {}^{\prime\prime}, {}^{\backprime\backprime} \text{you} {}^{\prime\prime} \} & = & \{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \} \\[6pt] I & = & \{ {}^{\backprime\backprime} \text{Ann} {}^{\prime\prime}, {}^{\backprime\backprime} \text{Bob} {}^{\prime\prime}, {}^{\backprime\backprime} \text{I} {}^{\prime\prime}, {}^{\backprime\backprime} \text{you} {}^{\prime\prime} \} & = & \{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \} \end{array}</math> |} In the present example, <math>S = I = \text{Syntactic Domain}\!</math>. The next two Tables give the sign relations associated with the interpreters <math>\mathrm{A}\!</math> and <math>\mathrm{B},\!</math> respectively, putting them in the form of ''relational databases''. Thus, the rows of each Table list the ordered triples of the form <math>(o, s, i)\!</math> that make up the corresponding sign relations, <math>L_\mathrm{A}, L_\mathrm{B} \subseteq O \times S \times I.\!</math> <br> {| align="center" style="width:92%" | width="50%" | {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 1a.} ~~ {L_\mathrm{A}} = \text{Sign Relation of Interpreter A}\!</math> |- style="height:40px; background:ghostwhite" | style="width:33%" | <math>\text{Object}\!</math> | style="width:33%" | <math>\text{Sign}\!</math> | style="width:33%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} | width="50%" | {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 1b.} ~~ {L_\mathrm{B}} = \text{Sign Relation of Interpreter B}\!</math> |- style="height:40px; background:ghostwhite" | style="width:33%" | <math>\text{Object}\!</math> | style="width:33%" | <math>\text{Sign}\!</math> | style="width:33%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- |<math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} |} <br> These Tables codify a rudimentary level of interpretive practice for the agents <math>\mathrm{A}\!</math> and <math>\mathrm{B}\!</math> and provide a basis for formalizing the initial semantics that is appropriate to their common syntactic domain. Each row of a Table names an object and two co-referent signs, making up an ordered triple of the form <math>(o, s, i)\!</math> called an ''elementary relation'', that is, one element of the relation's set-theoretic extension. Already in this elementary context, there are several different meanings that might attach to the project of a ''formal semiotics'', or a formal theory of meaning for signs. In the process of discussing these alternatives, it is useful to introduce a few terms that are occasionally used in the philosophy of language to point out the needed distinctions. ==Dyadic aspects of sign relations== For an arbitrary triadic relation <math>L \subseteq O \times S \times I,\!</math> whether it is a sign relation or not, there are six dyadic relations that can be obtained by ''projecting'' <math>L\!</math> on one of the planes of the <math>OSI\!</math>-space <math>O \times S \times I.\!</math> The six dyadic projections of a triadic relation <math>L\!</math> are defined and notated as follows: <br> {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%" |+ style="height:30px" | <math>\text{Table 2.} ~~ \text{Dyadic Projections of Triadic Relations}\!</math> | <math>\begin{matrix} L_{OS} & = & \mathrm{proj}_{OS}(L) & = & \{ (o, s) \in O \times S ~:~ (o, s, i) \in L ~\text{for some}~ i \in I \} \\[6pt] L_{SO} & = & \mathrm{proj}_{SO}(L) & = & \{ (s, o) \in S \times O ~:~ (o, s, i) \in L ~\text{for some}~ i \in I \} \\[6pt] L_{IS} & = & \mathrm{proj}_{IS}(L) & = & \{ (i, s) \in I \times S ~:~ (o, s, i) \in L ~\text{for some}~ o \in O \} \\[6pt] L_{SI} & = & \mathrm{proj}_{SI}(L) & = & \{ (s, i) \in S \times I ~:~ (o, s, i) \in L ~\text{for some}~ o \in O \} \\[6pt] L_{OI} & = & \mathrm{proj}_{OI}(L) & = & \{ (o, i) \in O \times I ~:~ (o, s, i) \in L ~\text{for some}~ s \in S \} \\[6pt] L_{IO} & = & \mathrm{proj}_{IO}(L) & = & \{ (i, o) \in I \times O ~:~ (o, s, i) \in L ~\text{for some}~ s \in S \} \end{matrix}</math> |} <br> By way of unpacking the set-theoretic notation, here is what the first definition says in ordinary language. {| align="center" cellpadding="6" width="90%" | <p>The dyadic relation that results from the projection of <math>L\!</math> on the <math>OS\!</math>-plane <math>O \times S\!</math> is written briefly as <math>L_{OS}\!</math> or written more fully as <math>\mathrm{proj}_{OS}(L),\!</math> and it is defined as the set of all ordered pairs <math>(o, s)\!</math> in the cartesian product <math>O \times S\!</math> for which there exists an ordered triple <math>(o, s, i)\!</math> in <math>L\!</math> for some interpretant <math>i\!</math> in the interpretant domain <math>I.\!</math></p> |} In the case where <math>L\!</math> is a sign relation, which it becomes by satisfying one of the definitions of a sign relation, some of the dyadic aspects of <math>L\!</math> can be recognized as formalizing aspects of sign meaning that have received their share of attention from students of signs over the centuries, and thus they can be associated with traditional concepts and terminology. Of course, traditions may vary as to the precise formation and usage of such concepts and terms. Other aspects of meaning have not received their fair share of attention, and thus remain anonymous on the contemporary scene of sign studies. ===Denotation=== One aspect of a sign's complete meaning is concerned with the reference that a sign has to its objects, which objects are collectively known as the ''denotation'' of the sign. In the pragmatic theory of sign relations, denotative references fall within the projection of the sign relation on the plane that is spanned by its object domain and its sign domain. The dyadic relation that makes up the ''denotative'', ''referential'', or ''semantic'' aspect or component of a sign relation <math>L\!</math> is notated as <math>\mathrm{Den}(L).\!</math> Information about the denotative aspect of meaning is obtained from <math>L\!</math> by taking its projection on the object-sign plane, in other words, on the 2-dimensional space that is generated by the object domain <math>O\!</math> and the sign domain <math>S.\!</math> This component of a sign relation <math>L\!</math> can be written in any of the forms, <math>\mathrm{proj}_{OS} L,\!</math> &nbsp; <math>L_{OS},\!</math> &nbsp; <math>\mathrm{proj}_{12} L,\!</math> &nbsp; <math>L_{12},\!</math> and it is defined as follows: {| align="center" cellpadding="6" width="90%" | <math>\begin{matrix} \mathrm{Den}(L) & = & \mathrm{proj}_{OS} L & = & \{ (o, s) \in O \times S ~:~ (o, s, i) \in L ~\text{for some}~ i \in I \}. \end{matrix}</math> |} Looking to the denotative aspects of <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B},\!</math> various rows of the Tables specify, for example, that <math>\mathrm{A}\!</math> uses <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> to denote <math>\mathrm{A}\!</math> and <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> to denote <math>\mathrm{B},\!</math> whereas <math>\mathrm{B}\!</math> uses <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> to denote <math>\mathrm{B}\!</math> and <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> to denote <math>\mathrm{A}.\!</math> All of these denotative references are summed up in the projections on the <math>OS\!</math>-plane, as shown in the following Tables: <br> {| align="center" style="width:90%" | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 3a.} ~~ \mathrm{Den}(L_\mathrm{A}) = \mathrm{proj}_{OS}(L_\mathrm{A})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Object}\!</math> | width="50%" | <math>\text{Sign}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 3b.} ~~ \mathrm{Den}(L_\mathrm{B}) = \mathrm{proj}_{OS}(L_\mathrm{B})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Object}\!</math> | width="50%" | <math>\text{Sign}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} |} <br> ===Connotation=== Another aspect of meaning concerns the connection that a sign has to its interpretants within a given sign relation. As before, this type of connection can be vacuous, singular, or plural in its collection of terminal points, and it can be formalized as the dyadic relation that is obtained as a planar projection of the triadic sign relation in question. The connection that a sign makes to an interpretant is here referred to as its ''connotation''. In the full theory of sign relations, this aspect of meaning includes the links that a sign has to affects, concepts, ideas, impressions, intentions, and the whole realm of an agent's mental states and allied activities, broadly encompassing intellectual associations, emotional impressions, motivational impulses, and real conduct. Taken at the full, in the natural setting of semiotic phenomena, this complex system of references is unlikely ever to find itself mapped in much detail, much less completely formalized, but the tangible warp of its accumulated mass is commonly alluded to as the connotative import of language. Formally speaking, however, the connotative aspect of meaning presents no additional difficulty. For a given sign relation <math>L,\!</math> the dyadic relation that constitutes the ''connotative aspect'' or ''connotative component'' of <math>L\!</math> is notated as <math>\mathrm{Con}(L).\!</math> The connotative aspect of a sign relation <math>L\!</math> is given by its projection on the plane of signs and interpretants, and is therefore defined as follows: {| align="center" cellpadding="6" width="90%" | <math>\begin{matrix} \mathrm{Con}(L) & = & \mathrm{proj}_{SI} L & = & \{ (s, i) \in S \times I ~:~ (o, s, i) \in L ~\text{for some}~ o \in O \}. \end{matrix}</math> |} All of these connotative references are summed up in the projections on the <math>SI\!</math>-plane, as shown in the following Tables: <br> {| align="center" style="width:90%" | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 4a.} ~~ \mathrm{Con}(L_\mathrm{A}) = \mathrm{proj}_{SI}(L_\mathrm{A})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Sign}\!</math> | width="50%" | <math>\text{Interpretant}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 4b.} ~~ \mathrm{Con}(L_\mathrm{B}) = \mathrm{proj}_{SI}(L_\mathrm{B})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Sign}\!</math> | width="50%" | <math>\text{Interpretant}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} |} <br> ===Ennotation=== The aspect of a sign's meaning that arises from the dyadic relation of its objects to its interpretants has no standard name. If an interpretant is considered to be a sign in its own right, then its independent reference to an object can be taken as belonging to another moment of denotation, but this neglects the mediational character of the whole transaction in which this occurs. Denotation and connotation have to do with dyadic relations in which the sign plays an active role, but here we have to consider a dyadic relation between objects and interpretants that is mediated by the sign from an off-stage position, as it were. As a relation between objects and interpretants that is mediated by a sign, this aspect of meaning may be referred to as the ''ennotation'' of a sign, and the dyadic relation that constitutes the ''ennotative aspect'' of a sign relation <math>L\!</math> may be notated as <math>\mathrm{Enn}(L).\!</math> The ennotational component of meaning for a sign relation <math>L\!</math> is captured by its projection on the plane of the object and interpretant domains, and it is thus defined as follows: {| align="center" cellpadding="6" width="90%" | <math>\begin{matrix} \mathrm{Enn}(L) & = & \mathrm{proj}_{OI} L & = & \{ (o, i) \in O \times I ~:~ (o, s, i) \in L ~\text{for some}~ s \in S \}. \end{matrix}</math> |} As it happens, the sign relations <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> are fully symmetric with respect to exchanging signs and interpretants, so all the data of <math>\mathrm{proj}_{OS} L_\mathrm{A}\!</math> is echoed unchanged in <math>\mathrm{proj}_{OI} L_\mathrm{A}\!</math> and all the data of <math>\mathrm{proj}_{OS} L_\mathrm{B}\!</math> is echoed unchanged in <math>\mathrm{proj}_{OI} L_\mathrm{B}.\!</math> <br> {| align="center" style="width:90%" | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 5a.} ~~ \mathrm{Enn}(L_\mathrm{A}) = \mathrm{proj}_{OI}(L_\mathrm{A})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Object}\!</math> | width="50%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 5b.} ~~ \mathrm{Enn}(L_\mathrm{B}) = \mathrm{proj}_{OI}(L_\mathrm{B})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Object}\!</math> | width="50%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} |} <br> ==Semiotic equivalence relations== A ''semiotic equivalence relation'' (SER) is a special type of equivalence relation that arises in the analysis of sign relations. As a general rule, any equivalence relation is closely associated with a family of equivalence classes that partition the underlying set of elements, frequently called the ''domain'' or ''space'' of the relation. In the case of a SER, the equivalence classes are called ''semiotic equivalence classes'' (SECs) and the partition is called a ''semiotic partition'' (SEP). The sign relations <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> have many interesting properties that are not possessed by sign relations in general. Some of these properties have to do with the relation between signs and their interpretant signs, as reflected in the projections of <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> on the <math>SI\!</math>-plane, notated as <math>\mathrm{proj}_{SI} L_\mathrm{A}\!</math> and <math>\mathrm{proj}_{SI} L_\mathrm{B},\!</math> respectively. The 2-adic relations on <math>S \times I\!</math> induced by these projections are also referred to as the ''connotative components'' of the corresponding sign relations, notated as <math>\mathrm{Con}(L_\mathrm{A})\!</math> and <math>\mathrm{Con}(L_\mathrm{B}),\!</math> respectively. Tables&nbsp;6a and 6b show the corresponding connotative components. <br> {| align="center" style="width:90%" | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 6a.} ~~ \mathrm{Con}(L_\mathrm{A}) = \mathrm{proj}_{SI}(L_\mathrm{A})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>S\!</math> | width="50%" | <math>I\!</math> |- | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}</math> | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}</math> |- | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}</math> | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}</math> |} | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 6b.} ~~ \mathrm{Con}(L_\mathrm{B}) = \mathrm{proj}_{SI}(L_\mathrm{B})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>S\!</math> | width="50%" | <math>I\!</math> |- | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}</math> | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}</math> |- | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}</math> | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}</math> |} |} <br> One nice property possessed by the sign relations <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> is that their connotative components <math>\mathrm{Con}(L_\mathrm{A})\!</math> and <math>\mathrm{Con}(L_\mathrm{B})\!</math> form a pair of equivalence relations on their common syntactic domain <math>S = I.\!</math> It is convenient to refer to such a structure as a ''semiotic equivalence relation'' (SER) since it equates signs that mean the same thing to some interpreter. Each of the SERs, <math>\mathrm{Con}(L_\mathrm{A}), \mathrm{Con}(L_\mathrm{B}) \subseteq S \times I \cong S \times S\!</math> partitions the whole collection of signs into ''semiotic equivalence classes'' (SECs). This makes for a strong form of representation in that the structure of the interpreters' common object domain <math>\{ \mathrm{A}, \mathrm{B} \}\!</math> is reflected or reconstructed, part for part, in the structure of each of their ''semiotic partitions'' (SEPs) of the syntactic domain <math>\{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}.\!</math> But it needs to be observed that the semiotic partitions for interpreters <math>\mathrm{A}\!</math> and <math>\mathrm{B}\!</math> are not the same, indeed, they are orthogonal to each other. This makes it difficult to interpret either one of the partitions or equivalence relations on the syntactic domain as corresponding to any sort of objective structure or invariant reality, independent of the individual interpreter's point of view. Information about the contrasting patterns of semiotic equivalence induced by the interpreters <math>\mathrm{A}\!</math> and <math>\mathrm{B}\!</math> is summarized in Tables&nbsp;7a and 7b. The form of these Tables should be sufficient to explain what is meant by saying that the SEPs for <math>\mathrm{A}\!</math> and <math>\mathrm{B}\!</math> are orthogonal to each other. <br> {| align="center" style="width:92%" | width="50%" | {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:92%" |+ style="height:30px" | <math>\text{Table 7a.} ~~ \text{Semiotic Partition for Interpreter A}\!</math> | {| align="center" border="0" cellpadding="10" cellspacing="0" style="text-align:center; width:100%" | width="50%" | <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime}\!</math> | width="50%" | <math>{}^{\backprime\backprime} \text{i} {}^{\prime\prime}\!</math> |} |- | {| align="center" border="0" cellpadding="10" cellspacing="0" style="text-align:center; width:100%" | width="50%" | <math>{}^{\backprime\backprime} \text{u} {}^{\prime\prime}\!</math> | width="50%" | <math>{}^{\backprime\backprime} \text{B} {}^{\prime\prime}\!</math> |} |} | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:92%" |+ style="height:30px" | <math>\text{Table 7b.} ~~ \text{Semiotic Partition for Interpreter B}\!</math> | {| align="center" border="0" cellpadding="10" cellspacing="0" style="text-align:center; width:50%" | <math>\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\[20pt] {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}</math> |} | {| align="center" border="0" cellpadding="10" cellspacing="0" style="text-align:center; width:50%" | <math>\begin{matrix} {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\[20pt] {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \end{matrix}</math> |} |} |} <br> A few items of notation are useful in discussing equivalence relations in general and semiotic equivalence relations in particular. As a general consideration, if <math>E\!</math> is an equivalence relation on a set <math>X,\!</math> then every element <math>x\!</math> of <math>X\!</math> belongs to a unique equivalence class under <math>E\!</math> called ''the equivalence class of <math>x\!</math> under <math>E\!</math>''. Convention provides the ''square bracket notation'' for denoting this equivalence class, either in the subscripted form <math>[x]_E\!</math> or in the simpler form <math>[x]\!</math> when the subscript <math>E\!</math> is understood. A statement that the elements <math>x\!</math> and <math>y\!</math> are equivalent under <math>E\!</math> is called an ''equation'' or an ''equivalence'' and may be expressed in any of the following ways: {| align="center" style="text-align:center; width:100%" | <math>\begin{array}{clc} (x, y) & \in & E \\[4pt] x & \in & [y]_E \\[4pt] y & \in & [x]_E \\[4pt] [x]_E & = & [y]_E \\[4pt] x & =_E & y \end{array}</math> |} Thus we have the following definitions: {| align="center" style="text-align:center; width:100%" | <math>\begin{array}{ccc} [x]_E & = & \{ y \in X : (x, y) \in E \} \\[6pt] x =_E y & \Leftrightarrow & (x, y) \in E \end{array}</math> |} In the application to sign relations it is useful to extend the square bracket notation in the following ways. If <math>L\!</math> is a sign relation whose connotative component or syntactic projection <math>L_{SI}\!</math> is an equivalence relation on <math>S = I,\!</math> let <math>[s]_L\!</math> be the equivalence class of <math>s\!</math> under <math>L_{SI}.\!</math> That is to say, <math>[s]_L = [s]_{L_{SI}}.\!</math> A statement that the signs <math>x\!</math> and <math>y\!</math> are equivalent under a semiotic equivalence relation <math>L_{SI}\!</math> is called a ''semiotic equation'' (SEQ) and may be written in either of the following equivalent forms: {| align="center" style="text-align:center; width:100%" | <math>\begin{array}{clc} [x]_L & = & [y]_L \\[6pt] x & =_L & y \end{array}</math> |} In many situations there is one further adaptation of the square bracket notation for semiotic equivalence classes that can be useful. Namely, when there is known to exist a particular triple <math>(o, s, i)\!</math> in a sign relation <math>L,\!</math> it is permissible to let <math>[o]_L\!</math> be defined as <math>[s]_L.\!</math> These modifications are designed to make the notation for semiotic equivalence classes harmonize as well as possible with the frequent use of similar devices for the denotations of signs and expressions. The semiotic equivalence relation for interpreter <math>\mathrm{A}\!</math> yields the following semiotic equations: {| align="center" style="text-align:center; width:100%" | <math>\begin{matrix} [ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} ]_{L_\mathrm{A}} & = & [ {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} ]_{L_\mathrm{A}} \\[6pt] [ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} ]_{L_\mathrm{A}} & = & [ {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} ]_{L_\mathrm{A}} \end{matrix}</math> |} or {| align="center" style="text-align:center; width:100%" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} & =_{L_\mathrm{A}} & {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} & =_{L_\mathrm{A}} & {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \end{matrix}</math> |} Thus it induces the semiotic partition: {| align="center" cellpadding="12" style="text-align:center; width:100%" | <math>\{ \{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \}, \{ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \} \}\!</math> |} The semiotic equivalence relation for interpreter <math>\mathrm{B}\!</math> yields the following semiotic equations: {| align="center" style="text-align:center; width:100%" | <math>\begin{matrix} [ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} ]_{L_\mathrm{B}} & = & [ {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} ]_{L_\mathrm{B}} \\[6pt] [ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} ]_{L_\mathrm{B}} & = & [ {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} ]_{L_\mathrm{B}} \end{matrix}</math> |} or {| align="center" style="text-align:center; width:100%" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} & =_{L_\mathrm{B}} & {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} & =_{L_\mathrm{B}} & {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \end{matrix}</math> |} Thus it induces the semiotic partition: {| align="center" cellpadding="12" style="text-align:center; width:100%" | <math>\{ \{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}, \{ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \} \}\!</math> |} ==Six ways of looking at a sign relation== In the context of 3-adic relations in general, Peirce provides the following illustration of the six ''converses'' of a 3-adic relation, that is, the six differently ordered ways of stating what is logically the same 3-adic relation: : So in a triadic fact, say, the example <br> {| align="center" cellspacing="8" style="width:72%" | align="center" | ''A'' gives ''B'' to ''C'' |} : we make no distinction in the ordinary logic of relations between the ''subject nominative'', the ''direct object'', and the ''indirect object''. We say that the proposition has three ''logical subjects''. We regard it as a mere affair of English grammar that there are six ways of expressing this: <br> {| align="center" cellspacing="8" style="width:72%" | style="width:36%" | ''A'' gives ''B'' to ''C'' | style="width:36%" | ''A'' benefits ''C'' with ''B'' |- | ''B'' enriches ''C'' at expense of ''A'' | ''C'' receives ''B'' from ''A'' |- | ''C'' thanks ''A'' for ''B'' | ''B'' leaves ''A'' for ''C'' |} : These six sentences express one and the same indivisible phenomenon. (C.S. Peirce, &ldquo;The Categories Defended&rdquo;, MS&nbsp;308 (1903), EP&nbsp;2, 170&ndash;171). ===IOS=== (Text in Progress) ===ISO=== (Text in Progress) ===OIS=== {| align="center" cellpadding="6" width="90%" | <p>Words spoken are symbols or signs (σύμβολα) of affections or impressions (παθήματα) of the soul (ψυχή); written words are the signs of words spoken. As writing, so also is speech not the same for all races of men. But the mental affections themselves, of which these words are primarily signs (σημεια), are the same for the whole of mankind, as are also the objects (πράγματα) of which those affections are representations or likenesses, images, copies (ομοιώματα). (Aristotle, ''De Interpretatione'', 1.16^a4).</p> |} ===OSI=== (Text in Progress) ===SIO=== {| align="center" cellpadding="6" width="90%" | <p>Logic will here be defined as ''formal semiotic''. A definition of a sign will be given which no more refers to human thought than does the definition of a line as the place which a particle occupies, part by part, during a lapse of time. Namely, a sign is something, ''A'', which brings something, ''B'', its ''interpretant'' sign determined or created by it, into the same sort of correspondence with something, ''C'', its ''object'', as that in which itself stands to ''C''. It is from this definition, together with a definition of &ldquo;formal&rdquo;, that I deduce mathematically the principles of logic. I also make a historical review of all the definitions and conceptions of logic, and show, not merely that my definition is no novelty, but that my non-psychological conception of logic has ''virtually'' been quite generally held, though not generally recognized. (C.S. Peirce, &ldquo;Application to the Carnegie Institution&rdquo;, L75 (1902), NEM 4, 20&ndash;21).</p> |} ===SOI=== {| align="center" cellpadding="6" width="90%" | <p>A ''Sign'' is anything which is related to a Second thing, its ''Object'', in respect to a Quality, in such a way as to bring a Third thing, its ''Interpretant'', into relation to the same Object, and that in such a way as to bring a Fourth into relation to that Object in the same form, ''ad infinitum''. (CP&nbsp;2.92, quoted in Fisch 1986, p.&nbsp;274)</p> |} ==References== ==Bibliography== ===Primary sources=== * [[Charles Sanders Peirce (Bibliography)]] ===Secondary sources=== * Awbrey, J.L., and Awbrey, S.M. (1995), &ldquo;Interpretation as Action : The Risk of Inquiry&rdquo;, ''Inquiry : Critical Thinking Across the Disciplines'' 15(1), pp. 40&ndash;52. [http://web.archive.org/web/19970626071826/http://chss.montclair.edu/inquiry/fall95/awbrey.html Archive]. [http://independent.academia.edu/JonAwbrey/Papers/1302117/Interpretation_as_Action_The_Risk_of_Inquiry Online]. * Deledalle, Gérard (2000), ''C.S. Peirce's Philosophy of Signs'', Indiana University Press, Bloomington, IN. * Eisele, Carolyn (1979), in ''Studies in the Scientific and Mathematical Philosophy of C.S. Peirce'', Richard Milton Martin (ed.), Mouton, The Hague. * Esposito, Joseph (1980), ''Evolutionary Metaphysics : The Development of Peirce's Theory of Categories'', Ohio University Press (?). * Fisch, Max (1986), ''Peirce, Semeiotic, and Pragmatism'', Indiana University Press, Bloomington, IN. * Houser, N., Roberts, D.D., and Van Evra, J. (eds.)(1997), ''Studies in the Logic of C.S. Peirce'', Indiana University Press, Bloomington, IN. * Liszka, J.J. (1996), ''A General Introduction to the Semeiotic of C.S. Peirce'', Indiana University Press, Bloomington, IN. * Misak, C. (ed.)(2004), ''Cambridge Companion to C.S. Peirce'', Cambridge University Press. * Moore, E., and Robin, R. (1964), ''Studies in the Philosophy of C.S. Peirce, Second Series'', University of Massachusetts Press, Amherst, MA. * Murphey, M. (1961), ''The Development of Peirce's Thought''. Reprinted, Hackett, Indianapolis, IN, 1993. * Percy, Walker (2000), pp. 271&ndash;291 in ''Signposts in a Strange Land'', P. Samway (ed.), Saint Martin's Press. ==Resources== * [http://www.helsinki.fi/science/commens/dictionary.html The Commens Dictionary of Peirce's Terms] ** [http://www.helsinki.fi/science/commens/terms/semeiotic.html Entry for ''Semeiotic''] ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Sign_relation Sign Relation @ InterSciWiki] * [http://mywikibiz.com/Sign_relation Sign Relation @ MyWikiBiz] {{col-break}} * [http://beta.wikiversity.org/wiki/Sign_relation Sign Relation @ Wikiversity Beta] * [http://ref.subwiki.org/wiki/Sign_relation Sign Relation @ Subject Wikis] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Sign_relation Sign Relation], [http://mywikibiz.com/ MyWikiBiz] * [http://mathweb.org/wiki/Sign_relation Sign Relation], [http://mathweb.org/ MathWeb Wiki] {{col-break}} * [http://planetmath.org/SignRelation Sign Relation], [http://planetmath.org/ PlanetMath] * [http://beta.wikiversity.org/wiki/Sign_relation Sign Relation], [http://beta.wikiversity.org/ Wikiversity Beta] {{col-break}} * [http://getwiki.net/-Sign_Relation Sign Relation], [http://getwiki.net/ GetWiki] * [http://en.wikipedia.org/w/index.php?title=Sign_relation&oldid=161631069 Sign Relation], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Artificial Intelligence]] [[Category:Charles Sanders Peirce]] [[Category:Cognitive Sciences]] [[Category:Computer Science]] [[Category:Formal Sciences]] [[Category:Hermeneutics]] [[Category:Information Systems]] [[Category:Information Theory]] [[Category:Inquiry]] [[Category:Intelligence Amplification]] [[Category:Knowledge Representation]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Philosophy]] [[Category:Pragmatics]] [[Category:Semantics]] [[Category:Semiotics]] [[Category:Syntax]] 62512cd81c52b4370695f36aaa0d842c460aa31f 661 660 2013-11-02T19:08:13Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''sign relation''' is the basic construct in the theory of signs, also known as [[semeiotic]] or [[semiotics]], as developed by Charles Sanders Peirce. ==Anthesis== {| align="center" cellpadding="6" width="90%" | <p>Thus, if a sunflower, in turning towards the sun, becomes by that very act fully capable, without further condition, of reproducing a sunflower which turns in precisely corresponding ways toward the sun, and of doing so with the same reproductive power, the sunflower would become a Representamen of the sun. (C.S. Peirce, &ldquo;Syllabus&rdquo; (''c''.&nbsp;1902), ''Collected Papers'', CP&nbsp;2.274).</p> |} In his picturesque illustration of a sign relation, along with his tracing of a corresponding sign process, or ''[[semiosis]]'', Peirce uses the technical term ''representamen'' for his concept of a sign, but the shorter word is precise enough, so long as one recognizes that its meaning in a particular theory of signs is given by a specific definition of what it means to be a sign. ==Definition== One of Peirce's clearest and most complete definitions of a sign is one that he gives in the context of providing a definition for ''logic'', and so it is informative to view it in that setting. {| align="center" cellpadding="6" width="90%" | <p>Logic will here be defined as ''formal semiotic''. A definition of a sign will be given which no more refers to human thought than does the definition of a line as the place which a particle occupies, part by part, during a lapse of time. Namely, a sign is something, ''A'', which brings something, ''B'', its ''interpretant'' sign determined or created by it, into the same sort of correspondence with something, ''C'', its ''object'', as that in which itself stands to ''C''. It is from this definition, together with a definition of &ldquo;formal&rdquo;, that I deduce mathematically the principles of logic. I also make a historical review of all the definitions and conceptions of logic, and show, not merely that my definition is no novelty, but that my non-psychological conception of logic has ''virtually'' been quite generally held, though not generally recognized. (C.S. Peirce, NEM&nbsp;4, 20&ndash;21).</p> |} In the general discussion of diverse theories of signs, the question frequently arises whether signhood is an absolute, essential, indelible, or ''ontological'' property of a thing, or whether it is a relational, interpretive, and mutable role that a thing can be said to have only within a particular context of relationships. Peirce's definition of a ''sign'' defines it in relation to its ''object'' and its ''interpretant sign'', and thus it defines signhood in ''[[logic of relatives|relative terms]]'', by means of a predicate with three places. In this definition, signhood is a role in a [[triadic relation]], a role that a thing bears or plays in a given context of relationships &mdash; it is not as an ''absolute'', ''non-relative'' property of a thing-in-itself, one that it possesses independently of all relationships to other things. Some of the terms that Peirce uses in his definition of a sign may need to be elaborated for the contemporary reader. :* '''Correspondence.''' From the way that Peirce uses this term throughout his work, it is clear that he means what he elsewhere calls a &ldquo;triple correspondence&rdquo;, and thus this is just another way of referring to the whole triadic sign relation itself. In particular, his use of this term should not be taken to imply a dyadic correspondence, like the kinds of &ldquo;mirror image&rdquo; correspondence between realities and representations that are bandied about in contemporary controversies about &ldquo;correspondence theories of truth&rdquo;. :* '''Determination.''' Peirce's concept of determination is broader in several directions than the sense of the word that refers to strictly deterministic causal-temporal processes. First, and especially in this context, he is invoking a more general concept of determination, what is called a ''formal'' or ''informational'' determination, as in saying &ldquo;two points determine a line&rdquo;, rather than the more special cases of causal and temporal determinisms. Second, he characteristically allows for what is called ''determination in measure'', that is, an order of determinism that admits a full spectrum of more and less determined relationships. :* '''Non-psychological.''' Peirce's &ldquo;non-psychological conception of logic&rdquo; must be distinguished from any variety of ''anti-psychologism''. He was quite interested in matters of psychology and had much of import to say about them. But logic and psychology operate on different planes of study even when they have occasion to view the same data, as logic is a ''[[normative science]]'' where psychology is a ''[[descriptive science]]'', and so they have very different aims, methods, and rationales. ==Signs and inquiry== : ''Main article : [[Inquiry]]'' There is a close relationship between the pragmatic theory of signs and the pragmatic theory of [[inquiry]]. In fact, the correspondence between the two studies exhibits so many congruences and parallels that it is often best to treat them as integral parts of one and the same subject. In a very real sense, inquiry is the process by which sign relations come to be established and continue to evolve. In other words, inquiry, &ldquo;thinking&rdquo; in its best sense, &ldquo;is a term denoting the various ways in which things acquire significance&rdquo; (John Dewey). Thus, there is an active and intricate form of cooperation that needs to be appreciated and maintained between these converging modes of investigation. Its proper character is best understood by realizing that the theory of inquiry is adapted to study the developmental aspects of sign relations, a subject which the theory of signs is specialized to treat from structural and comparative points of view. ==Examples of sign relations== Because the examples to follow have been artificially constructed to be as simple as possible, their detailed elaboration can run the risk of trivializing the whole theory of sign relations. Despite their simplicity, however, these examples have subtleties of their own, and their careful treatment will serve to illustrate many important issues in the general theory of signs. Imagine a discussion between two people, Ann and Bob, and attend only to that aspect of their interpretive practice that involves the use of the following nouns and pronouns: &ldquo;Ann&rdquo;, &ldquo;Bob&rdquo;, &ldquo;I&rdquo;, &ldquo;you&rdquo;. :* The ''object domain'' of this discussion fragment is the set of two people <math>\{ \text{Ann}, \text{Bob} \}.\!</math> :* The ''syntactic domain'' or the ''sign system'' of their discussion is limited to the set of four signs <math>\{ {}^{\backprime\backprime} \text{Ann} {}^{\prime\prime}, {}^{\backprime\backprime} \text{Bob} {}^{\prime\prime}, {}^{\backprime\backprime} \text{I} {}^{\prime\prime}, {}^{\backprime\backprime} \text{you} {}^{\prime\prime} \}.\!</math> In their discussion, Ann and Bob are not only the passive objects of nominative and accusative references but also the active interpreters of the language that they use. The ''system of interpretation'' (SOI) associated with each language user can be represented in the form of an individual [[three-place relation]] called the ''sign relation'' of that interpreter. Understood in terms of its ''set-theoretic extension'', a sign relation <math>L\!</math> is a ''subset'' of a ''cartesian product'' <math>O \times S \times I.\!</math> Here, <math>O, S, I\!</math> are three sets that are known as the ''object domain'', the ''sign domain'', and the ''interpretant domain'', respectively, of the sign relation <math>L \subseteq O \times S \times I.\!</math> Broadly speaking, the three domains of a sign relation can be any sets whatsoever, but the kinds of sign relations typically contemplated in formal settings are usually constrained to having <math>I \subseteq S\!</math>. In this case interpretants are just a special variety of signs and this makes it convenient to lump signs and interpretants together into a single class called the ''syntactic domain''. In the forthcoming examples <math>S\!</math> and <math>I\!</math> are identical as sets, so the same elements manifest themselves in two different roles of the sign relations in question. When it is necessary to refer to the whole set of objects and signs in the union of the domains <math>O, S, I\!</math> for a given sign relation <math>L,\!</math> one may refer to this set as the ''World'' of <math>L\!</math> and write <math>W = W_L = O \cup S \cup I.\!</math> To facilitate an interest in the abstract structures of sign relations, and to keep the notations as brief as possible as the examples become more complicated, it serves to introduce the following general notations: {| align="center" cellspacing="6" width="90%" | <math>\begin{array}{ccl} O & = & \text{Object Domain} \\[6pt] S & = & \text{Sign Domain} \\[6pt] I & = & \text{Interpretant Domain} \end{array}</math> |} Introducing a few abbreviations for use in considering the present Example, we have the following data: {| align="center" cellspacing="6" width="90%" | <math>\begin{array}{cclcl} O & = & \{ \text{Ann}, \text{Bob} \} & = & \{ \text{A}, \text{B} \} \\[6pt] S & = & \{ {}^{\backprime\backprime} \text{Ann} {}^{\prime\prime}, {}^{\backprime\backprime} \text{Bob} {}^{\prime\prime}, {}^{\backprime\backprime} \text{I} {}^{\prime\prime}, {}^{\backprime\backprime} \text{you} {}^{\prime\prime} \} & = & \{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \} \\[6pt] I & = & \{ {}^{\backprime\backprime} \text{Ann} {}^{\prime\prime}, {}^{\backprime\backprime} \text{Bob} {}^{\prime\prime}, {}^{\backprime\backprime} \text{I} {}^{\prime\prime}, {}^{\backprime\backprime} \text{you} {}^{\prime\prime} \} & = & \{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \} \end{array}</math> |} In the present example, <math>S = I = \text{Syntactic Domain}\!</math>. The next two Tables give the sign relations associated with the interpreters <math>\mathrm{A}\!</math> and <math>\mathrm{B},\!</math> respectively, putting them in the form of ''relational databases''. Thus, the rows of each Table list the ordered triples of the form <math>(o, s, i)\!</math> that make up the corresponding sign relations, <math>L_\mathrm{A}, L_\mathrm{B} \subseteq O \times S \times I.\!</math> <br> {| align="center" style="width:92%" | width="50%" | {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 1a.} ~~ {L_\mathrm{A}} = \text{Sign Relation of Interpreter A}\!</math> |- style="height:40px; background:ghostwhite" | style="width:33%" | <math>\text{Object}\!</math> | style="width:33%" | <math>\text{Sign}\!</math> | style="width:33%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} | width="50%" | {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 1b.} ~~ {L_\mathrm{B}} = \text{Sign Relation of Interpreter B}\!</math> |- style="height:40px; background:ghostwhite" | style="width:33%" | <math>\text{Object}\!</math> | style="width:33%" | <math>\text{Sign}\!</math> | style="width:33%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- |<math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} |} <br> These Tables codify a rudimentary level of interpretive practice for the agents <math>\mathrm{A}\!</math> and <math>\mathrm{B}\!</math> and provide a basis for formalizing the initial semantics that is appropriate to their common syntactic domain. Each row of a Table names an object and two co-referent signs, making up an ordered triple of the form <math>(o, s, i)\!</math> called an ''elementary relation'', that is, one element of the relation's set-theoretic extension. Already in this elementary context, there are several different meanings that might attach to the project of a ''formal semiotics'', or a formal theory of meaning for signs. In the process of discussing these alternatives, it is useful to introduce a few terms that are occasionally used in the philosophy of language to point out the needed distinctions. ==Dyadic aspects of sign relations== For an arbitrary triadic relation <math>L \subseteq O \times S \times I,\!</math> whether it is a sign relation or not, there are six dyadic relations that can be obtained by ''projecting'' <math>L\!</math> on one of the planes of the <math>OSI\!</math>-space <math>O \times S \times I.\!</math> The six dyadic projections of a triadic relation <math>L\!</math> are defined and notated as follows: <br> {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%" |+ style="height:30px" | <math>\text{Table 2.} ~~ \text{Dyadic Projections of Triadic Relations}\!</math> | <math>\begin{matrix} L_{OS} & = & \mathrm{proj}_{OS}(L) & = & \{ (o, s) \in O \times S ~:~ (o, s, i) \in L ~\text{for some}~ i \in I \} \\[6pt] L_{SO} & = & \mathrm{proj}_{SO}(L) & = & \{ (s, o) \in S \times O ~:~ (o, s, i) \in L ~\text{for some}~ i \in I \} \\[6pt] L_{IS} & = & \mathrm{proj}_{IS}(L) & = & \{ (i, s) \in I \times S ~:~ (o, s, i) \in L ~\text{for some}~ o \in O \} \\[6pt] L_{SI} & = & \mathrm{proj}_{SI}(L) & = & \{ (s, i) \in S \times I ~:~ (o, s, i) \in L ~\text{for some}~ o \in O \} \\[6pt] L_{OI} & = & \mathrm{proj}_{OI}(L) & = & \{ (o, i) \in O \times I ~:~ (o, s, i) \in L ~\text{for some}~ s \in S \} \\[6pt] L_{IO} & = & \mathrm{proj}_{IO}(L) & = & \{ (i, o) \in I \times O ~:~ (o, s, i) \in L ~\text{for some}~ s \in S \} \end{matrix}</math> |} <br> By way of unpacking the set-theoretic notation, here is what the first definition says in ordinary language. {| align="center" cellpadding="6" width="90%" | <p>The dyadic relation that results from the projection of <math>L\!</math> on the <math>OS\!</math>-plane <math>O \times S\!</math> is written briefly as <math>L_{OS}\!</math> or written more fully as <math>\mathrm{proj}_{OS}(L),\!</math> and it is defined as the set of all ordered pairs <math>(o, s)\!</math> in the cartesian product <math>O \times S\!</math> for which there exists an ordered triple <math>(o, s, i)\!</math> in <math>L\!</math> for some interpretant <math>i\!</math> in the interpretant domain <math>I.\!</math></p> |} In the case where <math>L\!</math> is a sign relation, which it becomes by satisfying one of the definitions of a sign relation, some of the dyadic aspects of <math>L\!</math> can be recognized as formalizing aspects of sign meaning that have received their share of attention from students of signs over the centuries, and thus they can be associated with traditional concepts and terminology. Of course, traditions may vary as to the precise formation and usage of such concepts and terms. Other aspects of meaning have not received their fair share of attention, and thus remain anonymous on the contemporary scene of sign studies. ===Denotation=== One aspect of a sign's complete meaning is concerned with the reference that a sign has to its objects, which objects are collectively known as the ''denotation'' of the sign. In the pragmatic theory of sign relations, denotative references fall within the projection of the sign relation on the plane that is spanned by its object domain and its sign domain. The dyadic relation that makes up the ''denotative'', ''referential'', or ''semantic'' aspect or component of a sign relation <math>L\!</math> is notated as <math>\mathrm{Den}(L).\!</math> Information about the denotative aspect of meaning is obtained from <math>L\!</math> by taking its projection on the object-sign plane, in other words, on the 2-dimensional space that is generated by the object domain <math>O\!</math> and the sign domain <math>S.\!</math> This component of a sign relation <math>L\!</math> can be written in any of the forms, <math>\mathrm{proj}_{OS} L,\!</math> &nbsp; <math>L_{OS},\!</math> &nbsp; <math>\mathrm{proj}_{12} L,\!</math> &nbsp; <math>L_{12},\!</math> and it is defined as follows: {| align="center" cellpadding="6" width="90%" | <math>\begin{matrix} \mathrm{Den}(L) & = & \mathrm{proj}_{OS} L & = & \{ (o, s) \in O \times S ~:~ (o, s, i) \in L ~\text{for some}~ i \in I \}. \end{matrix}</math> |} Looking to the denotative aspects of <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B},\!</math> various rows of the Tables specify, for example, that <math>\mathrm{A}\!</math> uses <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> to denote <math>\mathrm{A}\!</math> and <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> to denote <math>\mathrm{B},\!</math> whereas <math>\mathrm{B}\!</math> uses <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> to denote <math>\mathrm{B}\!</math> and <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> to denote <math>\mathrm{A}.\!</math> All of these denotative references are summed up in the projections on the <math>OS\!</math>-plane, as shown in the following Tables: <br> {| align="center" style="width:90%" | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 3a.} ~~ \mathrm{Den}(L_\mathrm{A}) = \mathrm{proj}_{OS}(L_\mathrm{A})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Object}\!</math> | width="50%" | <math>\text{Sign}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 3b.} ~~ \mathrm{Den}(L_\mathrm{B}) = \mathrm{proj}_{OS}(L_\mathrm{B})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Object}\!</math> | width="50%" | <math>\text{Sign}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} |} <br> ===Connotation=== Another aspect of meaning concerns the connection that a sign has to its interpretants within a given sign relation. As before, this type of connection can be vacuous, singular, or plural in its collection of terminal points, and it can be formalized as the dyadic relation that is obtained as a planar projection of the triadic sign relation in question. The connection that a sign makes to an interpretant is here referred to as its ''connotation''. In the full theory of sign relations, this aspect of meaning includes the links that a sign has to affects, concepts, ideas, impressions, intentions, and the whole realm of an agent's mental states and allied activities, broadly encompassing intellectual associations, emotional impressions, motivational impulses, and real conduct. Taken at the full, in the natural setting of semiotic phenomena, this complex system of references is unlikely ever to find itself mapped in much detail, much less completely formalized, but the tangible warp of its accumulated mass is commonly alluded to as the connotative import of language. Formally speaking, however, the connotative aspect of meaning presents no additional difficulty. For a given sign relation <math>L,\!</math> the dyadic relation that constitutes the ''connotative aspect'' or ''connotative component'' of <math>L\!</math> is notated as <math>\mathrm{Con}(L).\!</math> The connotative aspect of a sign relation <math>L\!</math> is given by its projection on the plane of signs and interpretants, and is therefore defined as follows: {| align="center" cellpadding="6" width="90%" | <math>\begin{matrix} \mathrm{Con}(L) & = & \mathrm{proj}_{SI} L & = & \{ (s, i) \in S \times I ~:~ (o, s, i) \in L ~\text{for some}~ o \in O \}. \end{matrix}</math> |} All of these connotative references are summed up in the projections on the <math>SI\!</math>-plane, as shown in the following Tables: <br> {| align="center" style="width:90%" | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 4a.} ~~ \mathrm{Con}(L_\mathrm{A}) = \mathrm{proj}_{SI}(L_\mathrm{A})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Sign}\!</math> | width="50%" | <math>\text{Interpretant}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 4b.} ~~ \mathrm{Con}(L_\mathrm{B}) = \mathrm{proj}_{SI}(L_\mathrm{B})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Sign}\!</math> | width="50%" | <math>\text{Interpretant}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} |} <br> ===Ennotation=== The aspect of a sign's meaning that arises from the dyadic relation of its objects to its interpretants has no standard name. If an interpretant is considered to be a sign in its own right, then its independent reference to an object can be taken as belonging to another moment of denotation, but this neglects the mediational character of the whole transaction in which this occurs. Denotation and connotation have to do with dyadic relations in which the sign plays an active role, but here we have to consider a dyadic relation between objects and interpretants that is mediated by the sign from an off-stage position, as it were. As a relation between objects and interpretants that is mediated by a sign, this aspect of meaning may be referred to as the ''ennotation'' of a sign, and the dyadic relation that constitutes the ''ennotative aspect'' of a sign relation <math>L\!</math> may be notated as <math>\mathrm{Enn}(L).\!</math> The ennotational component of meaning for a sign relation <math>L\!</math> is captured by its projection on the plane of the object and interpretant domains, and it is thus defined as follows: {| align="center" cellpadding="6" width="90%" | <math>\begin{matrix} \mathrm{Enn}(L) & = & \mathrm{proj}_{OI} L & = & \{ (o, i) \in O \times I ~:~ (o, s, i) \in L ~\text{for some}~ s \in S \}. \end{matrix}</math> |} As it happens, the sign relations <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> are fully symmetric with respect to exchanging signs and interpretants, so all the data of <math>\mathrm{proj}_{OS} L_\mathrm{A}\!</math> is echoed unchanged in <math>\mathrm{proj}_{OI} L_\mathrm{A}\!</math> and all the data of <math>\mathrm{proj}_{OS} L_\mathrm{B}\!</math> is echoed unchanged in <math>\mathrm{proj}_{OI} L_\mathrm{B}.\!</math> <br> {| align="center" style="width:90%" | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 5a.} ~~ \mathrm{Enn}(L_\mathrm{A}) = \mathrm{proj}_{OI}(L_\mathrm{A})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Object}\!</math> | width="50%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 5b.} ~~ \mathrm{Enn}(L_\mathrm{B}) = \mathrm{proj}_{OI}(L_\mathrm{B})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Object}\!</math> | width="50%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} |} <br> ==Semiotic equivalence relations== A ''semiotic equivalence relation'' (SER) is a special type of equivalence relation that arises in the analysis of sign relations. As a general rule, any equivalence relation is closely associated with a family of equivalence classes that partition the underlying set of elements, frequently called the ''domain'' or ''space'' of the relation. In the case of a SER, the equivalence classes are called ''semiotic equivalence classes'' (SECs) and the partition is called a ''semiotic partition'' (SEP). The sign relations <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> have many interesting properties that are not possessed by sign relations in general. Some of these properties have to do with the relation between signs and their interpretant signs, as reflected in the projections of <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> on the <math>SI\!</math>-plane, notated as <math>\mathrm{proj}_{SI} L_\mathrm{A}\!</math> and <math>\mathrm{proj}_{SI} L_\mathrm{B},\!</math> respectively. The 2-adic relations on <math>S \times I\!</math> induced by these projections are also referred to as the ''connotative components'' of the corresponding sign relations, notated as <math>\mathrm{Con}(L_\mathrm{A})\!</math> and <math>\mathrm{Con}(L_\mathrm{B}),\!</math> respectively. Tables&nbsp;6a and 6b show the corresponding connotative components. <br> {| align="center" style="width:90%" | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 6a.} ~~ \mathrm{Con}(L_\mathrm{A}) = \mathrm{proj}_{SI}(L_\mathrm{A})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>S\!</math> | width="50%" | <math>I\!</math> |- | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}</math> | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}</math> |- | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}</math> | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}</math> |} | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 6b.} ~~ \mathrm{Con}(L_\mathrm{B}) = \mathrm{proj}_{SI}(L_\mathrm{B})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>S\!</math> | width="50%" | <math>I\!</math> |- | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}</math> | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}</math> |- | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}</math> | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}</math> |} |} <br> One nice property possessed by the sign relations <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> is that their connotative components <math>\mathrm{Con}(L_\mathrm{A})\!</math> and <math>\mathrm{Con}(L_\mathrm{B})\!</math> form a pair of equivalence relations on their common syntactic domain <math>S = I.\!</math> It is convenient to refer to such a structure as a ''semiotic equivalence relation'' (SER) since it equates signs that mean the same thing to some interpreter. Each of the SERs, <math>\mathrm{Con}(L_\mathrm{A}), \mathrm{Con}(L_\mathrm{B}) \subseteq S \times I \cong S \times S\!</math> partitions the whole collection of signs into ''semiotic equivalence classes'' (SECs). This makes for a strong form of representation in that the structure of the interpreters' common object domain <math>\{ \mathrm{A}, \mathrm{B} \}\!</math> is reflected or reconstructed, part for part, in the structure of each of their ''semiotic partitions'' (SEPs) of the syntactic domain <math>\{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}.\!</math> But it needs to be observed that the semiotic partitions for interpreters <math>\mathrm{A}\!</math> and <math>\mathrm{B}\!</math> are not the same, indeed, they are orthogonal to each other. This makes it difficult to interpret either one of the partitions or equivalence relations on the syntactic domain as corresponding to any sort of objective structure or invariant reality, independent of the individual interpreter's point of view. Information about the contrasting patterns of semiotic equivalence induced by the interpreters <math>\mathrm{A}\!</math> and <math>\mathrm{B}\!</math> is summarized in Tables&nbsp;7a and 7b. The form of these Tables should be sufficient to explain what is meant by saying that the SEPs for <math>\mathrm{A}\!</math> and <math>\mathrm{B}\!</math> are orthogonal to each other. <br> {| align="center" style="width:92%" | width="50%" | {| align="center" cellpadding="20" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 7a.} ~~ \text{Semiotic Partition for Interpreter A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | style="border-top:1px solid black" | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | style="border-top:1px solid black" | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |} | width="50%" | {| align="center" cellpadding="20" cellspacing="1" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 7b.} ~~ \text{Semiotic Partition for Interpreter B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | style="border-left:1px solid black" | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | style="border-left:1px solid black" | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |} |} <br> A few items of notation are useful in discussing equivalence relations in general and semiotic equivalence relations in particular. As a general consideration, if <math>E\!</math> is an equivalence relation on a set <math>X,\!</math> then every element <math>x\!</math> of <math>X\!</math> belongs to a unique equivalence class under <math>E\!</math> called ''the equivalence class of <math>x\!</math> under <math>E\!</math>''. Convention provides the ''square bracket notation'' for denoting this equivalence class, either in the subscripted form <math>[x]_E\!</math> or in the simpler form <math>[x]\!</math> when the subscript <math>E\!</math> is understood. A statement that the elements <math>x\!</math> and <math>y\!</math> are equivalent under <math>E\!</math> is called an ''equation'' or an ''equivalence'' and may be expressed in any of the following ways: {| align="center" style="text-align:center; width:100%" | <math>\begin{array}{clc} (x, y) & \in & E \\[4pt] x & \in & [y]_E \\[4pt] y & \in & [x]_E \\[4pt] [x]_E & = & [y]_E \\[4pt] x & =_E & y \end{array}</math> |} Thus we have the following definitions: {| align="center" style="text-align:center; width:100%" | <math>\begin{array}{ccc} [x]_E & = & \{ y \in X : (x, y) \in E \} \\[6pt] x =_E y & \Leftrightarrow & (x, y) \in E \end{array}</math> |} In the application to sign relations it is useful to extend the square bracket notation in the following ways. If <math>L\!</math> is a sign relation whose connotative component or syntactic projection <math>L_{SI}\!</math> is an equivalence relation on <math>S = I,\!</math> let <math>[s]_L\!</math> be the equivalence class of <math>s\!</math> under <math>L_{SI}.\!</math> That is to say, <math>[s]_L = [s]_{L_{SI}}.\!</math> A statement that the signs <math>x\!</math> and <math>y\!</math> are equivalent under a semiotic equivalence relation <math>L_{SI}\!</math> is called a ''semiotic equation'' (SEQ) and may be written in either of the following equivalent forms: {| align="center" style="text-align:center; width:100%" | <math>\begin{array}{clc} [x]_L & = & [y]_L \\[6pt] x & =_L & y \end{array}</math> |} In many situations there is one further adaptation of the square bracket notation for semiotic equivalence classes that can be useful. Namely, when there is known to exist a particular triple <math>(o, s, i)\!</math> in a sign relation <math>L,\!</math> it is permissible to let <math>[o]_L\!</math> be defined as <math>[s]_L.\!</math> These modifications are designed to make the notation for semiotic equivalence classes harmonize as well as possible with the frequent use of similar devices for the denotations of signs and expressions. The semiotic equivalence relation for interpreter <math>\mathrm{A}\!</math> yields the following semiotic equations: {| align="center" style="text-align:center; width:100%" | <math>\begin{matrix} [ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} ]_{L_\mathrm{A}} & = & [ {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} ]_{L_\mathrm{A}} \\[6pt] [ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} ]_{L_\mathrm{A}} & = & [ {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} ]_{L_\mathrm{A}} \end{matrix}</math> |} or {| align="center" style="text-align:center; width:100%" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} & =_{L_\mathrm{A}} & {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} & =_{L_\mathrm{A}} & {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \end{matrix}</math> |} Thus it induces the semiotic partition: {| align="center" cellpadding="12" style="text-align:center; width:100%" | <math>\{ \{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \}, \{ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \} \}\!</math> |} The semiotic equivalence relation for interpreter <math>\mathrm{B}\!</math> yields the following semiotic equations: {| align="center" style="text-align:center; width:100%" | <math>\begin{matrix} [ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} ]_{L_\mathrm{B}} & = & [ {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} ]_{L_\mathrm{B}} \\[6pt] [ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} ]_{L_\mathrm{B}} & = & [ {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} ]_{L_\mathrm{B}} \end{matrix}</math> |} or {| align="center" style="text-align:center; width:100%" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} & =_{L_\mathrm{B}} & {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} & =_{L_\mathrm{B}} & {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \end{matrix}</math> |} Thus it induces the semiotic partition: {| align="center" cellpadding="12" style="text-align:center; width:100%" | <math>\{ \{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}, \{ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \} \}\!</math> |} ==Six ways of looking at a sign relation== In the context of 3-adic relations in general, Peirce provides the following illustration of the six ''converses'' of a 3-adic relation, that is, the six differently ordered ways of stating what is logically the same 3-adic relation: : So in a triadic fact, say, the example <br> {| align="center" cellspacing="8" style="width:72%" | align="center" | ''A'' gives ''B'' to ''C'' |} : we make no distinction in the ordinary logic of relations between the ''subject nominative'', the ''direct object'', and the ''indirect object''. We say that the proposition has three ''logical subjects''. We regard it as a mere affair of English grammar that there are six ways of expressing this: <br> {| align="center" cellspacing="8" style="width:72%" | style="width:36%" | ''A'' gives ''B'' to ''C'' | style="width:36%" | ''A'' benefits ''C'' with ''B'' |- | ''B'' enriches ''C'' at expense of ''A'' | ''C'' receives ''B'' from ''A'' |- | ''C'' thanks ''A'' for ''B'' | ''B'' leaves ''A'' for ''C'' |} : These six sentences express one and the same indivisible phenomenon. (C.S. Peirce, &ldquo;The Categories Defended&rdquo;, MS&nbsp;308 (1903), EP&nbsp;2, 170&ndash;171). ===IOS=== (Text in Progress) ===ISO=== (Text in Progress) ===OIS=== {| align="center" cellpadding="6" width="90%" | <p>Words spoken are symbols or signs (σύμβολα) of affections or impressions (παθήματα) of the soul (ψυχή); written words are the signs of words spoken. As writing, so also is speech not the same for all races of men. But the mental affections themselves, of which these words are primarily signs (σημεια), are the same for the whole of mankind, as are also the objects (πράγματα) of which those affections are representations or likenesses, images, copies (ομοιώματα). (Aristotle, ''De Interpretatione'', 1.16^a4).</p> |} ===OSI=== (Text in Progress) ===SIO=== {| align="center" cellpadding="6" width="90%" | <p>Logic will here be defined as ''formal semiotic''. A definition of a sign will be given which no more refers to human thought than does the definition of a line as the place which a particle occupies, part by part, during a lapse of time. Namely, a sign is something, ''A'', which brings something, ''B'', its ''interpretant'' sign determined or created by it, into the same sort of correspondence with something, ''C'', its ''object'', as that in which itself stands to ''C''. It is from this definition, together with a definition of &ldquo;formal&rdquo;, that I deduce mathematically the principles of logic. I also make a historical review of all the definitions and conceptions of logic, and show, not merely that my definition is no novelty, but that my non-psychological conception of logic has ''virtually'' been quite generally held, though not generally recognized. (C.S. Peirce, &ldquo;Application to the Carnegie Institution&rdquo;, L75 (1902), NEM 4, 20&ndash;21).</p> |} ===SOI=== {| align="center" cellpadding="6" width="90%" | <p>A ''Sign'' is anything which is related to a Second thing, its ''Object'', in respect to a Quality, in such a way as to bring a Third thing, its ''Interpretant'', into relation to the same Object, and that in such a way as to bring a Fourth into relation to that Object in the same form, ''ad infinitum''. (CP&nbsp;2.92, quoted in Fisch 1986, p.&nbsp;274)</p> |} ==References== ==Bibliography== ===Primary sources=== * [[Charles Sanders Peirce (Bibliography)]] ===Secondary sources=== * Awbrey, J.L., and Awbrey, S.M. (1995), &ldquo;Interpretation as Action : The Risk of Inquiry&rdquo;, ''Inquiry : Critical Thinking Across the Disciplines'' 15(1), pp. 40&ndash;52. [http://web.archive.org/web/19970626071826/http://chss.montclair.edu/inquiry/fall95/awbrey.html Archive]. [http://independent.academia.edu/JonAwbrey/Papers/1302117/Interpretation_as_Action_The_Risk_of_Inquiry Online]. * Deledalle, Gérard (2000), ''C.S. Peirce's Philosophy of Signs'', Indiana University Press, Bloomington, IN. * Eisele, Carolyn (1979), in ''Studies in the Scientific and Mathematical Philosophy of C.S. Peirce'', Richard Milton Martin (ed.), Mouton, The Hague. * Esposito, Joseph (1980), ''Evolutionary Metaphysics : The Development of Peirce's Theory of Categories'', Ohio University Press (?). * Fisch, Max (1986), ''Peirce, Semeiotic, and Pragmatism'', Indiana University Press, Bloomington, IN. * Houser, N., Roberts, D.D., and Van Evra, J. (eds.)(1997), ''Studies in the Logic of C.S. Peirce'', Indiana University Press, Bloomington, IN. * Liszka, J.J. (1996), ''A General Introduction to the Semeiotic of C.S. Peirce'', Indiana University Press, Bloomington, IN. * Misak, C. (ed.)(2004), ''Cambridge Companion to C.S. Peirce'', Cambridge University Press. * Moore, E., and Robin, R. (1964), ''Studies in the Philosophy of C.S. Peirce, Second Series'', University of Massachusetts Press, Amherst, MA. * Murphey, M. (1961), ''The Development of Peirce's Thought''. Reprinted, Hackett, Indianapolis, IN, 1993. * Percy, Walker (2000), pp. 271&ndash;291 in ''Signposts in a Strange Land'', P. Samway (ed.), Saint Martin's Press. ==Resources== * [http://www.helsinki.fi/science/commens/dictionary.html The Commens Dictionary of Peirce's Terms] ** [http://www.helsinki.fi/science/commens/terms/semeiotic.html Entry for ''Semeiotic''] ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Sign_relation Sign Relation @ InterSciWiki] * [http://mywikibiz.com/Sign_relation Sign Relation @ MyWikiBiz] {{col-break}} * [http://beta.wikiversity.org/wiki/Sign_relation Sign Relation @ Wikiversity Beta] * [http://ref.subwiki.org/wiki/Sign_relation Sign Relation @ Subject Wikis] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Sign_relation Sign Relation], [http://mywikibiz.com/ MyWikiBiz] * [http://mathweb.org/wiki/Sign_relation Sign Relation], [http://mathweb.org/ MathWeb Wiki] {{col-break}} * [http://planetmath.org/SignRelation Sign Relation], [http://planetmath.org/ PlanetMath] * [http://beta.wikiversity.org/wiki/Sign_relation Sign Relation], [http://beta.wikiversity.org/ Wikiversity Beta] {{col-break}} * [http://getwiki.net/-Sign_Relation Sign Relation], [http://getwiki.net/ GetWiki] * [http://en.wikipedia.org/w/index.php?title=Sign_relation&oldid=161631069 Sign Relation], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Artificial Intelligence]] [[Category:Charles Sanders Peirce]] [[Category:Cognitive Sciences]] [[Category:Computer Science]] [[Category:Formal Sciences]] [[Category:Hermeneutics]] [[Category:Information Systems]] [[Category:Information Theory]] [[Category:Inquiry]] [[Category:Intelligence Amplification]] [[Category:Knowledge Representation]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Philosophy]] [[Category:Pragmatics]] [[Category:Semantics]] [[Category:Semiotics]] [[Category:Syntax]] b6857bd96213472d92dbcb682e55c050d882ab87 662 661 2013-11-03T02:02:28Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''sign relation''' is the basic construct in the theory of signs, also known as [[semeiotic]] or [[semiotics]], as developed by Charles Sanders Peirce. ==Anthesis== {| align="center" cellpadding="6" width="90%" | <p>Thus, if a sunflower, in turning towards the sun, becomes by that very act fully capable, without further condition, of reproducing a sunflower which turns in precisely corresponding ways toward the sun, and of doing so with the same reproductive power, the sunflower would become a Representamen of the sun. (C.S. Peirce, &ldquo;Syllabus&rdquo; (''c''.&nbsp;1902), ''Collected Papers'', CP&nbsp;2.274).</p> |} In his picturesque illustration of a sign relation, along with his tracing of a corresponding sign process, or ''[[semiosis]]'', Peirce uses the technical term ''representamen'' for his concept of a sign, but the shorter word is precise enough, so long as one recognizes that its meaning in a particular theory of signs is given by a specific definition of what it means to be a sign. ==Definition== One of Peirce's clearest and most complete definitions of a sign is one that he gives in the context of providing a definition for ''logic'', and so it is informative to view it in that setting. {| align="center" cellpadding="6" width="90%" | <p>Logic will here be defined as ''formal semiotic''. A definition of a sign will be given which no more refers to human thought than does the definition of a line as the place which a particle occupies, part by part, during a lapse of time. Namely, a sign is something, ''A'', which brings something, ''B'', its ''interpretant'' sign determined or created by it, into the same sort of correspondence with something, ''C'', its ''object'', as that in which itself stands to ''C''. It is from this definition, together with a definition of &ldquo;formal&rdquo;, that I deduce mathematically the principles of logic. I also make a historical review of all the definitions and conceptions of logic, and show, not merely that my definition is no novelty, but that my non-psychological conception of logic has ''virtually'' been quite generally held, though not generally recognized. (C.S. Peirce, NEM&nbsp;4, 20&ndash;21).</p> |} In the general discussion of diverse theories of signs, the question frequently arises whether signhood is an absolute, essential, indelible, or ''ontological'' property of a thing, or whether it is a relational, interpretive, and mutable role that a thing can be said to have only within a particular context of relationships. Peirce's definition of a ''sign'' defines it in relation to its ''object'' and its ''interpretant sign'', and thus it defines signhood in ''[[logic of relatives|relative terms]]'', by means of a predicate with three places. In this definition, signhood is a role in a [[triadic relation]], a role that a thing bears or plays in a given context of relationships &mdash; it is not as an ''absolute'', ''non-relative'' property of a thing-in-itself, one that it possesses independently of all relationships to other things. Some of the terms that Peirce uses in his definition of a sign may need to be elaborated for the contemporary reader. :* '''Correspondence.''' From the way that Peirce uses this term throughout his work, it is clear that he means what he elsewhere calls a &ldquo;triple correspondence&rdquo;, and thus this is just another way of referring to the whole triadic sign relation itself. In particular, his use of this term should not be taken to imply a dyadic correspondence, like the kinds of &ldquo;mirror image&rdquo; correspondence between realities and representations that are bandied about in contemporary controversies about &ldquo;correspondence theories of truth&rdquo;. :* '''Determination.''' Peirce's concept of determination is broader in several directions than the sense of the word that refers to strictly deterministic causal-temporal processes. First, and especially in this context, he is invoking a more general concept of determination, what is called a ''formal'' or ''informational'' determination, as in saying &ldquo;two points determine a line&rdquo;, rather than the more special cases of causal and temporal determinisms. Second, he characteristically allows for what is called ''determination in measure'', that is, an order of determinism that admits a full spectrum of more and less determined relationships. :* '''Non-psychological.''' Peirce's &ldquo;non-psychological conception of logic&rdquo; must be distinguished from any variety of ''anti-psychologism''. He was quite interested in matters of psychology and had much of import to say about them. But logic and psychology operate on different planes of study even when they have occasion to view the same data, as logic is a ''[[normative science]]'' where psychology is a ''[[descriptive science]]'', and so they have very different aims, methods, and rationales. ==Signs and inquiry== : ''Main article : [[Inquiry]]'' There is a close relationship between the pragmatic theory of signs and the pragmatic theory of [[inquiry]]. In fact, the correspondence between the two studies exhibits so many congruences and parallels that it is often best to treat them as integral parts of one and the same subject. In a very real sense, inquiry is the process by which sign relations come to be established and continue to evolve. In other words, inquiry, &ldquo;thinking&rdquo; in its best sense, &ldquo;is a term denoting the various ways in which things acquire significance&rdquo; (John Dewey). Thus, there is an active and intricate form of cooperation that needs to be appreciated and maintained between these converging modes of investigation. Its proper character is best understood by realizing that the theory of inquiry is adapted to study the developmental aspects of sign relations, a subject which the theory of signs is specialized to treat from structural and comparative points of view. ==Examples of sign relations== Because the examples to follow have been artificially constructed to be as simple as possible, their detailed elaboration can run the risk of trivializing the whole theory of sign relations. Despite their simplicity, however, these examples have subtleties of their own, and their careful treatment will serve to illustrate many important issues in the general theory of signs. Imagine a discussion between two people, Ann and Bob, and attend only to that aspect of their interpretive practice that involves the use of the following nouns and pronouns: &ldquo;Ann&rdquo;, &ldquo;Bob&rdquo;, &ldquo;I&rdquo;, &ldquo;you&rdquo;. :* The ''object domain'' of this discussion fragment is the set of two people <math>\{ \text{Ann}, \text{Bob} \}.\!</math> :* The ''syntactic domain'' or the ''sign system'' of their discussion is limited to the set of four signs <math>\{ {}^{\backprime\backprime} \text{Ann} {}^{\prime\prime}, {}^{\backprime\backprime} \text{Bob} {}^{\prime\prime}, {}^{\backprime\backprime} \text{I} {}^{\prime\prime}, {}^{\backprime\backprime} \text{you} {}^{\prime\prime} \}.\!</math> In their discussion, Ann and Bob are not only the passive objects of nominative and accusative references but also the active interpreters of the language that they use. The ''system of interpretation'' (SOI) associated with each language user can be represented in the form of an individual [[three-place relation]] called the ''sign relation'' of that interpreter. Understood in terms of its ''set-theoretic extension'', a sign relation <math>L\!</math> is a ''subset'' of a ''cartesian product'' <math>O \times S \times I.\!</math> Here, <math>O, S, I\!</math> are three sets that are known as the ''object domain'', the ''sign domain'', and the ''interpretant domain'', respectively, of the sign relation <math>L \subseteq O \times S \times I.\!</math> Broadly speaking, the three domains of a sign relation can be any sets whatsoever, but the kinds of sign relations typically contemplated in formal settings are usually constrained to having <math>I \subseteq S\!</math>. In this case interpretants are just a special variety of signs and this makes it convenient to lump signs and interpretants together into a single class called the ''syntactic domain''. In the forthcoming examples <math>S\!</math> and <math>I\!</math> are identical as sets, so the same elements manifest themselves in two different roles of the sign relations in question. When it is necessary to refer to the whole set of objects and signs in the union of the domains <math>O, S, I\!</math> for a given sign relation <math>L,\!</math> one may refer to this set as the ''World'' of <math>L\!</math> and write <math>W = W_L = O \cup S \cup I.\!</math> To facilitate an interest in the abstract structures of sign relations, and to keep the notations as brief as possible as the examples become more complicated, it serves to introduce the following general notations: {| align="center" cellspacing="6" width="90%" | <math>\begin{array}{ccl} O & = & \text{Object Domain} \\[6pt] S & = & \text{Sign Domain} \\[6pt] I & = & \text{Interpretant Domain} \end{array}</math> |} Introducing a few abbreviations for use in considering the present Example, we have the following data: {| align="center" cellspacing="6" width="90%" | <math>\begin{array}{cclcl} O & = & \{ \text{Ann}, \text{Bob} \} & = & \{ \mathrm{A}, \mathrm{B} \} \\[6pt] S & = & \{ {}^{\backprime\backprime} \text{Ann} {}^{\prime\prime}, {}^{\backprime\backprime} \text{Bob} {}^{\prime\prime}, {}^{\backprime\backprime} \text{I} {}^{\prime\prime}, {}^{\backprime\backprime} \text{you} {}^{\prime\prime} \} & = & \{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \} \\[6pt] I & = & \{ {}^{\backprime\backprime} \text{Ann} {}^{\prime\prime}, {}^{\backprime\backprime} \text{Bob} {}^{\prime\prime}, {}^{\backprime\backprime} \text{I} {}^{\prime\prime}, {}^{\backprime\backprime} \text{you} {}^{\prime\prime} \} & = & \{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \} \end{array}</math> |} In the present example, <math>S = I = \text{Syntactic Domain}\!</math>. The next two Tables give the sign relations associated with the interpreters <math>\mathrm{A}\!</math> and <math>\mathrm{B},\!</math> respectively, putting them in the form of ''relational databases''. Thus, the rows of each Table list the ordered triples of the form <math>(o, s, i)\!</math> that make up the corresponding sign relations, <math>L_\mathrm{A}, L_\mathrm{B} \subseteq O \times S \times I.\!</math> <br> {| align="center" style="width:92%" | width="50%" | {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 1a.} ~~ {L_\mathrm{A}} = \text{Sign Relation of Interpreter A}\!</math> |- style="height:40px; background:ghostwhite" | style="width:33%" | <math>\text{Object}\!</math> | style="width:33%" | <math>\text{Sign}\!</math> | style="width:33%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} | width="50%" | {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 1b.} ~~ {L_\mathrm{B}} = \text{Sign Relation of Interpreter B}\!</math> |- style="height:40px; background:ghostwhite" | style="width:33%" | <math>\text{Object}\!</math> | style="width:33%" | <math>\text{Sign}\!</math> | style="width:33%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- |<math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} |} <br> These Tables codify a rudimentary level of interpretive practice for the agents <math>\mathrm{A}\!</math> and <math>\mathrm{B}\!</math> and provide a basis for formalizing the initial semantics that is appropriate to their common syntactic domain. Each row of a Table names an object and two co-referent signs, making up an ordered triple of the form <math>(o, s, i)\!</math> called an ''elementary relation'', that is, one element of the relation's set-theoretic extension. Already in this elementary context, there are several different meanings that might attach to the project of a ''formal semiotics'', or a formal theory of meaning for signs. In the process of discussing these alternatives, it is useful to introduce a few terms that are occasionally used in the philosophy of language to point out the needed distinctions. ==Dyadic aspects of sign relations== For an arbitrary triadic relation <math>L \subseteq O \times S \times I,\!</math> whether it is a sign relation or not, there are six dyadic relations that can be obtained by ''projecting'' <math>L\!</math> on one of the planes of the <math>OSI\!</math>-space <math>O \times S \times I.\!</math> The six dyadic projections of a triadic relation <math>L\!</math> are defined and notated as follows: <br> {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%" |+ style="height:30px" | <math>\text{Table 2.} ~~ \text{Dyadic Projections of Triadic Relations}\!</math> | <math>\begin{matrix} L_{OS} & = & \mathrm{proj}_{OS}(L) & = & \{ (o, s) \in O \times S ~:~ (o, s, i) \in L ~\text{for some}~ i \in I \} \\[6pt] L_{SO} & = & \mathrm{proj}_{SO}(L) & = & \{ (s, o) \in S \times O ~:~ (o, s, i) \in L ~\text{for some}~ i \in I \} \\[6pt] L_{IS} & = & \mathrm{proj}_{IS}(L) & = & \{ (i, s) \in I \times S ~:~ (o, s, i) \in L ~\text{for some}~ o \in O \} \\[6pt] L_{SI} & = & \mathrm{proj}_{SI}(L) & = & \{ (s, i) \in S \times I ~:~ (o, s, i) \in L ~\text{for some}~ o \in O \} \\[6pt] L_{OI} & = & \mathrm{proj}_{OI}(L) & = & \{ (o, i) \in O \times I ~:~ (o, s, i) \in L ~\text{for some}~ s \in S \} \\[6pt] L_{IO} & = & \mathrm{proj}_{IO}(L) & = & \{ (i, o) \in I \times O ~:~ (o, s, i) \in L ~\text{for some}~ s \in S \} \end{matrix}</math> |} <br> By way of unpacking the set-theoretic notation, here is what the first definition says in ordinary language. {| align="center" cellpadding="6" width="90%" | <p>The dyadic relation that results from the projection of <math>L\!</math> on the <math>OS\!</math>-plane <math>O \times S\!</math> is written briefly as <math>L_{OS}\!</math> or written more fully as <math>\mathrm{proj}_{OS}(L),\!</math> and it is defined as the set of all ordered pairs <math>(o, s)\!</math> in the cartesian product <math>O \times S\!</math> for which there exists an ordered triple <math>(o, s, i)\!</math> in <math>L\!</math> for some interpretant <math>i\!</math> in the interpretant domain <math>I.\!</math></p> |} In the case where <math>L\!</math> is a sign relation, which it becomes by satisfying one of the definitions of a sign relation, some of the dyadic aspects of <math>L\!</math> can be recognized as formalizing aspects of sign meaning that have received their share of attention from students of signs over the centuries, and thus they can be associated with traditional concepts and terminology. Of course, traditions may vary as to the precise formation and usage of such concepts and terms. Other aspects of meaning have not received their fair share of attention, and thus remain anonymous on the contemporary scene of sign studies. ===Denotation=== One aspect of a sign's complete meaning is concerned with the reference that a sign has to its objects, which objects are collectively known as the ''denotation'' of the sign. In the pragmatic theory of sign relations, denotative references fall within the projection of the sign relation on the plane that is spanned by its object domain and its sign domain. The dyadic relation that makes up the ''denotative'', ''referential'', or ''semantic'' aspect or component of a sign relation <math>L\!</math> is notated as <math>\mathrm{Den}(L).\!</math> Information about the denotative aspect of meaning is obtained from <math>L\!</math> by taking its projection on the object-sign plane, in other words, on the 2-dimensional space that is generated by the object domain <math>O\!</math> and the sign domain <math>S.\!</math> This component of a sign relation <math>L\!</math> can be written in any of the forms, <math>\mathrm{proj}_{OS} L,\!</math> &nbsp; <math>L_{OS},\!</math> &nbsp; <math>\mathrm{proj}_{12} L,\!</math> &nbsp; <math>L_{12},\!</math> and it is defined as follows: {| align="center" cellpadding="6" width="90%" | <math>\begin{matrix} \mathrm{Den}(L) & = & \mathrm{proj}_{OS} L & = & \{ (o, s) \in O \times S ~:~ (o, s, i) \in L ~\text{for some}~ i \in I \}. \end{matrix}</math> |} Looking to the denotative aspects of <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B},\!</math> various rows of the Tables specify, for example, that <math>\mathrm{A}\!</math> uses <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> to denote <math>\mathrm{A}\!</math> and <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> to denote <math>\mathrm{B},\!</math> whereas <math>\mathrm{B}\!</math> uses <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> to denote <math>\mathrm{B}\!</math> and <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> to denote <math>\mathrm{A}.\!</math> All of these denotative references are summed up in the projections on the <math>OS\!</math>-plane, as shown in the following Tables: <br> {| align="center" style="width:90%" | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 3a.} ~~ \mathrm{Den}(L_\mathrm{A}) = \mathrm{proj}_{OS}(L_\mathrm{A})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Object}\!</math> | width="50%" | <math>\text{Sign}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 3b.} ~~ \mathrm{Den}(L_\mathrm{B}) = \mathrm{proj}_{OS}(L_\mathrm{B})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Object}\!</math> | width="50%" | <math>\text{Sign}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} |} <br> ===Connotation=== Another aspect of meaning concerns the connection that a sign has to its interpretants within a given sign relation. As before, this type of connection can be vacuous, singular, or plural in its collection of terminal points, and it can be formalized as the dyadic relation that is obtained as a planar projection of the triadic sign relation in question. The connection that a sign makes to an interpretant is here referred to as its ''connotation''. In the full theory of sign relations, this aspect of meaning includes the links that a sign has to affects, concepts, ideas, impressions, intentions, and the whole realm of an agent's mental states and allied activities, broadly encompassing intellectual associations, emotional impressions, motivational impulses, and real conduct. Taken at the full, in the natural setting of semiotic phenomena, this complex system of references is unlikely ever to find itself mapped in much detail, much less completely formalized, but the tangible warp of its accumulated mass is commonly alluded to as the connotative import of language. Formally speaking, however, the connotative aspect of meaning presents no additional difficulty. For a given sign relation <math>L,\!</math> the dyadic relation that constitutes the ''connotative aspect'' or ''connotative component'' of <math>L\!</math> is notated as <math>\mathrm{Con}(L).\!</math> The connotative aspect of a sign relation <math>L\!</math> is given by its projection on the plane of signs and interpretants, and is therefore defined as follows: {| align="center" cellpadding="6" width="90%" | <math>\begin{matrix} \mathrm{Con}(L) & = & \mathrm{proj}_{SI} L & = & \{ (s, i) \in S \times I ~:~ (o, s, i) \in L ~\text{for some}~ o \in O \}. \end{matrix}</math> |} All of these connotative references are summed up in the projections on the <math>SI\!</math>-plane, as shown in the following Tables: <br> {| align="center" style="width:90%" | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 4a.} ~~ \mathrm{Con}(L_\mathrm{A}) = \mathrm{proj}_{SI}(L_\mathrm{A})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Sign}\!</math> | width="50%" | <math>\text{Interpretant}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 4b.} ~~ \mathrm{Con}(L_\mathrm{B}) = \mathrm{proj}_{SI}(L_\mathrm{B})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Sign}\!</math> | width="50%" | <math>\text{Interpretant}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} |} <br> ===Ennotation=== The aspect of a sign's meaning that arises from the dyadic relation of its objects to its interpretants has no standard name. If an interpretant is considered to be a sign in its own right, then its independent reference to an object can be taken as belonging to another moment of denotation, but this neglects the mediational character of the whole transaction in which this occurs. Denotation and connotation have to do with dyadic relations in which the sign plays an active role, but here we have to consider a dyadic relation between objects and interpretants that is mediated by the sign from an off-stage position, as it were. As a relation between objects and interpretants that is mediated by a sign, this aspect of meaning may be referred to as the ''ennotation'' of a sign, and the dyadic relation that constitutes the ''ennotative aspect'' of a sign relation <math>L\!</math> may be notated as <math>\mathrm{Enn}(L).\!</math> The ennotational component of meaning for a sign relation <math>L\!</math> is captured by its projection on the plane of the object and interpretant domains, and it is thus defined as follows: {| align="center" cellpadding="6" width="90%" | <math>\begin{matrix} \mathrm{Enn}(L) & = & \mathrm{proj}_{OI} L & = & \{ (o, i) \in O \times I ~:~ (o, s, i) \in L ~\text{for some}~ s \in S \}. \end{matrix}</math> |} As it happens, the sign relations <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> are fully symmetric with respect to exchanging signs and interpretants, so all the data of <math>\mathrm{proj}_{OS} L_\mathrm{A}\!</math> is echoed unchanged in <math>\mathrm{proj}_{OI} L_\mathrm{A}\!</math> and all the data of <math>\mathrm{proj}_{OS} L_\mathrm{B}\!</math> is echoed unchanged in <math>\mathrm{proj}_{OI} L_\mathrm{B}.\!</math> <br> {| align="center" style="width:90%" | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 5a.} ~~ \mathrm{Enn}(L_\mathrm{A}) = \mathrm{proj}_{OI}(L_\mathrm{A})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Object}\!</math> | width="50%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 5b.} ~~ \mathrm{Enn}(L_\mathrm{B}) = \mathrm{proj}_{OI}(L_\mathrm{B})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Object}\!</math> | width="50%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} |} <br> ==Semiotic equivalence relations== A ''semiotic equivalence relation'' (SER) is a special type of equivalence relation that arises in the analysis of sign relations. As a general rule, any equivalence relation is closely associated with a family of equivalence classes that partition the underlying set of elements, frequently called the ''domain'' or ''space'' of the relation. In the case of a SER, the equivalence classes are called ''semiotic equivalence classes'' (SECs) and the partition is called a ''semiotic partition'' (SEP). The sign relations <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> have many interesting properties that are not possessed by sign relations in general. Some of these properties have to do with the relation between signs and their interpretant signs, as reflected in the projections of <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> on the <math>SI\!</math>-plane, notated as <math>\mathrm{proj}_{SI} L_\mathrm{A}\!</math> and <math>\mathrm{proj}_{SI} L_\mathrm{B},\!</math> respectively. The 2-adic relations on <math>S \times I\!</math> induced by these projections are also referred to as the ''connotative components'' of the corresponding sign relations, notated as <math>\mathrm{Con}(L_\mathrm{A})\!</math> and <math>\mathrm{Con}(L_\mathrm{B}),\!</math> respectively. Tables&nbsp;6a and 6b show the corresponding connotative components. <br> {| align="center" style="width:90%" | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 6a.} ~~ \mathrm{Con}(L_\mathrm{A}) = \mathrm{proj}_{SI}(L_\mathrm{A})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>S\!</math> | width="50%" | <math>I\!</math> |- | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \end{matrix}</math> | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \end{matrix}</math> |- | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \end{matrix}</math> | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \end{matrix}</math> |} | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 6b.} ~~ \mathrm{Con}(L_\mathrm{B}) = \mathrm{proj}_{SI}(L_\mathrm{B})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>S\!</math> | width="50%" | <math>I\!</math> |- | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \end{matrix}</math> | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \end{matrix}</math> |- | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \end{matrix}</math> | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \end{matrix}</math> |} |} <br> One nice property possessed by the sign relations <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> is that their connotative components <math>\mathrm{Con}(L_\mathrm{A})\!</math> and <math>\mathrm{Con}(L_\mathrm{B})\!</math> form a pair of equivalence relations on their common syntactic domain <math>S = I.\!</math> It is convenient to refer to such a structure as a ''semiotic equivalence relation'' (SER) since it equates signs that mean the same thing to some interpreter. Each of the SERs, <math>\mathrm{Con}(L_\mathrm{A}), \mathrm{Con}(L_\mathrm{B}) \subseteq S \times I \cong S \times S\!</math> partitions the whole collection of signs into ''semiotic equivalence classes'' (SECs). This makes for a strong form of representation in that the structure of the interpreters' common object domain <math>\{ \mathrm{A}, \mathrm{B} \}\!</math> is reflected or reconstructed, part for part, in the structure of each of their ''semiotic partitions'' (SEPs) of the syntactic domain <math>\{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}.\!</math> But it needs to be observed that the semiotic partitions for interpreters <math>\mathrm{A}\!</math> and <math>\mathrm{B}\!</math> are not the same, indeed, they are orthogonal to each other. This makes it difficult to interpret either one of the partitions or equivalence relations on the syntactic domain as corresponding to any sort of objective structure or invariant reality, independent of the individual interpreter's point of view. Information about the contrasting patterns of semiotic equivalence induced by the interpreters <math>\mathrm{A}\!</math> and <math>\mathrm{B}\!</math> is summarized in Tables&nbsp;7a and 7b. The form of these Tables should be sufficient to explain what is meant by saying that the SEPs for <math>\mathrm{A}\!</math> and <math>\mathrm{B}\!</math> are orthogonal to each other. <br> {| align="center" style="width:92%" | width="50%" | {| align="center" cellpadding="20" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 7a.} ~~ \text{Semiotic Partition for Interpreter A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | style="border-top:1px solid black" | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | style="border-top:1px solid black" | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |} | width="50%" | {| align="center" cellpadding="20" cellspacing="1" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 7b.} ~~ \text{Semiotic Partition for Interpreter B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | style="border-left:1px solid black" | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | style="border-left:1px solid black" | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |} |} <br> A few items of notation are useful in discussing equivalence relations in general and semiotic equivalence relations in particular. As a general consideration, if <math>E\!</math> is an equivalence relation on a set <math>X,\!</math> then every element <math>x\!</math> of <math>X\!</math> belongs to a unique equivalence class under <math>E\!</math> called ''the equivalence class of <math>x\!</math> under <math>E\!</math>''. Convention provides the ''square bracket notation'' for denoting this equivalence class, either in the subscripted form <math>[x]_E\!</math> or in the simpler form <math>[x]\!</math> when the subscript <math>E\!</math> is understood. A statement that the elements <math>x\!</math> and <math>y\!</math> are equivalent under <math>E\!</math> is called an ''equation'' or an ''equivalence'' and may be expressed in any of the following ways: {| align="center" style="text-align:center; width:100%" | <math>\begin{array}{clc} (x, y) & \in & E \\[4pt] x & \in & [y]_E \\[4pt] y & \in & [x]_E \\[4pt] [x]_E & = & [y]_E \\[4pt] x & =_E & y \end{array}</math> |} Thus we have the following definitions: {| align="center" style="text-align:center; width:100%" | <math>\begin{array}{ccc} [x]_E & = & \{ y \in X : (x, y) \in E \} \\[6pt] x =_E y & \Leftrightarrow & (x, y) \in E \end{array}</math> |} In the application to sign relations it is useful to extend the square bracket notation in the following ways. If <math>L\!</math> is a sign relation whose connotative component or syntactic projection <math>L_{SI}\!</math> is an equivalence relation on <math>S = I,\!</math> let <math>[s]_L\!</math> be the equivalence class of <math>s\!</math> under <math>L_{SI}.\!</math> That is to say, <math>[s]_L = [s]_{L_{SI}}.\!</math> A statement that the signs <math>x\!</math> and <math>y\!</math> are equivalent under a semiotic equivalence relation <math>L_{SI}\!</math> is called a ''semiotic equation'' (SEQ) and may be written in either of the following equivalent forms: {| align="center" style="text-align:center; width:100%" | <math>\begin{array}{clc} [x]_L & = & [y]_L \\[6pt] x & =_L & y \end{array}</math> |} In many situations there is one further adaptation of the square bracket notation for semiotic equivalence classes that can be useful. Namely, when there is known to exist a particular triple <math>(o, s, i)\!</math> in a sign relation <math>L,\!</math> it is permissible to let <math>[o]_L\!</math> be defined as <math>[s]_L.\!</math> These modifications are designed to make the notation for semiotic equivalence classes harmonize as well as possible with the frequent use of similar devices for the denotations of signs and expressions. The semiotic equivalence relation for interpreter <math>\mathrm{A}\!</math> yields the following semiotic equations: {| align="center" style="text-align:center; width:100%" | <math>\begin{matrix} [ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} ]_{L_\mathrm{A}} & = & [ {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} ]_{L_\mathrm{A}} \\[6pt] [ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} ]_{L_\mathrm{A}} & = & [ {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} ]_{L_\mathrm{A}} \end{matrix}</math> |} or {| align="center" style="text-align:center; width:100%" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} & =_{L_\mathrm{A}} & {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} & =_{L_\mathrm{A}} & {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \end{matrix}</math> |} Thus it induces the semiotic partition: {| align="center" cellpadding="12" style="text-align:center; width:100%" | <math>\{ \{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \}, \{ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \} \}\!</math> |} The semiotic equivalence relation for interpreter <math>\mathrm{B}\!</math> yields the following semiotic equations: {| align="center" style="text-align:center; width:100%" | <math>\begin{matrix} [ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} ]_{L_\mathrm{B}} & = & [ {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} ]_{L_\mathrm{B}} \\[6pt] [ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} ]_{L_\mathrm{B}} & = & [ {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} ]_{L_\mathrm{B}} \end{matrix}</math> |} or {| align="center" style="text-align:center; width:100%" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} & =_{L_\mathrm{B}} & {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} & =_{L_\mathrm{B}} & {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \end{matrix}</math> |} Thus it induces the semiotic partition: {| align="center" cellpadding="12" style="text-align:center; width:100%" | <math>\{ \{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}, \{ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \} \}\!</math> |} ==Six ways of looking at a sign relation== In the context of 3-adic relations in general, Peirce provides the following illustration of the six ''converses'' of a 3-adic relation, that is, the six differently ordered ways of stating what is logically the same 3-adic relation: : So in a triadic fact, say, the example <br> {| align="center" cellspacing="8" style="width:72%" | align="center" | ''A'' gives ''B'' to ''C'' |} : we make no distinction in the ordinary logic of relations between the ''subject nominative'', the ''direct object'', and the ''indirect object''. We say that the proposition has three ''logical subjects''. We regard it as a mere affair of English grammar that there are six ways of expressing this: <br> {| align="center" cellspacing="8" style="width:72%" | style="width:36%" | ''A'' gives ''B'' to ''C'' | style="width:36%" | ''A'' benefits ''C'' with ''B'' |- | ''B'' enriches ''C'' at expense of ''A'' | ''C'' receives ''B'' from ''A'' |- | ''C'' thanks ''A'' for ''B'' | ''B'' leaves ''A'' for ''C'' |} : These six sentences express one and the same indivisible phenomenon. (C.S. Peirce, &ldquo;The Categories Defended&rdquo;, MS&nbsp;308 (1903), EP&nbsp;2, 170&ndash;171). ===IOS=== (Text in Progress) ===ISO=== (Text in Progress) ===OIS=== {| align="center" cellpadding="6" width="90%" | <p>Words spoken are symbols or signs (σύμβολα) of affections or impressions (παθήματα) of the soul (ψυχή); written words are the signs of words spoken. As writing, so also is speech not the same for all races of men. But the mental affections themselves, of which these words are primarily signs (σημεια), are the same for the whole of mankind, as are also the objects (πράγματα) of which those affections are representations or likenesses, images, copies (ομοιώματα). (Aristotle, ''De Interpretatione'', 1.16^a4).</p> |} ===OSI=== (Text in Progress) ===SIO=== {| align="center" cellpadding="6" width="90%" | <p>Logic will here be defined as ''formal semiotic''. A definition of a sign will be given which no more refers to human thought than does the definition of a line as the place which a particle occupies, part by part, during a lapse of time. Namely, a sign is something, ''A'', which brings something, ''B'', its ''interpretant'' sign determined or created by it, into the same sort of correspondence with something, ''C'', its ''object'', as that in which itself stands to ''C''. It is from this definition, together with a definition of &ldquo;formal&rdquo;, that I deduce mathematically the principles of logic. I also make a historical review of all the definitions and conceptions of logic, and show, not merely that my definition is no novelty, but that my non-psychological conception of logic has ''virtually'' been quite generally held, though not generally recognized. (C.S. Peirce, &ldquo;Application to the Carnegie Institution&rdquo;, L75 (1902), NEM 4, 20&ndash;21).</p> |} ===SOI=== {| align="center" cellpadding="6" width="90%" | <p>A ''Sign'' is anything which is related to a Second thing, its ''Object'', in respect to a Quality, in such a way as to bring a Third thing, its ''Interpretant'', into relation to the same Object, and that in such a way as to bring a Fourth into relation to that Object in the same form, ''ad infinitum''. (CP&nbsp;2.92, quoted in Fisch 1986, p.&nbsp;274)</p> |} ==References== ==Bibliography== ===Primary sources=== * [[Charles Sanders Peirce (Bibliography)]] ===Secondary sources=== * Awbrey, J.L., and Awbrey, S.M. (1995), &ldquo;Interpretation as Action : The Risk of Inquiry&rdquo;, ''Inquiry : Critical Thinking Across the Disciplines'' 15(1), pp. 40&ndash;52. [http://web.archive.org/web/19970626071826/http://chss.montclair.edu/inquiry/fall95/awbrey.html Archive]. [http://independent.academia.edu/JonAwbrey/Papers/1302117/Interpretation_as_Action_The_Risk_of_Inquiry Online]. * Deledalle, Gérard (2000), ''C.S. Peirce's Philosophy of Signs'', Indiana University Press, Bloomington, IN. * Eisele, Carolyn (1979), in ''Studies in the Scientific and Mathematical Philosophy of C.S. Peirce'', Richard Milton Martin (ed.), Mouton, The Hague. * Esposito, Joseph (1980), ''Evolutionary Metaphysics : The Development of Peirce's Theory of Categories'', Ohio University Press (?). * Fisch, Max (1986), ''Peirce, Semeiotic, and Pragmatism'', Indiana University Press, Bloomington, IN. * Houser, N., Roberts, D.D., and Van Evra, J. (eds.)(1997), ''Studies in the Logic of C.S. Peirce'', Indiana University Press, Bloomington, IN. * Liszka, J.J. (1996), ''A General Introduction to the Semeiotic of C.S. Peirce'', Indiana University Press, Bloomington, IN. * Misak, C. (ed.)(2004), ''Cambridge Companion to C.S. Peirce'', Cambridge University Press. * Moore, E., and Robin, R. (1964), ''Studies in the Philosophy of C.S. Peirce, Second Series'', University of Massachusetts Press, Amherst, MA. * Murphey, M. (1961), ''The Development of Peirce's Thought''. Reprinted, Hackett, Indianapolis, IN, 1993. * Percy, Walker (2000), pp. 271&ndash;291 in ''Signposts in a Strange Land'', P. Samway (ed.), Saint Martin's Press. ==Resources== * [http://www.helsinki.fi/science/commens/dictionary.html The Commens Dictionary of Peirce's Terms] ** [http://www.helsinki.fi/science/commens/terms/semeiotic.html Entry for ''Semeiotic''] ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Sign_relation Sign Relation @ InterSciWiki] * [http://mywikibiz.com/Sign_relation Sign Relation @ MyWikiBiz] {{col-break}} * [http://beta.wikiversity.org/wiki/Sign_relation Sign Relation @ Wikiversity Beta] * [http://ref.subwiki.org/wiki/Sign_relation Sign Relation @ Subject Wikis] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Sign_relation Sign Relation], [http://mywikibiz.com/ MyWikiBiz] * [http://mathweb.org/wiki/Sign_relation Sign Relation], [http://mathweb.org/ MathWeb Wiki] {{col-break}} * [http://planetmath.org/SignRelation Sign Relation], [http://planetmath.org/ PlanetMath] * [http://beta.wikiversity.org/wiki/Sign_relation Sign Relation], [http://beta.wikiversity.org/ Wikiversity Beta] {{col-break}} * [http://getwiki.net/-Sign_Relation Sign Relation], [http://getwiki.net/ GetWiki] * [http://en.wikipedia.org/w/index.php?title=Sign_relation&oldid=161631069 Sign Relation], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Artificial Intelligence]] [[Category:Charles Sanders Peirce]] [[Category:Cognitive Sciences]] [[Category:Computer Science]] [[Category:Formal Sciences]] [[Category:Hermeneutics]] [[Category:Information Systems]] [[Category:Information Theory]] [[Category:Inquiry]] [[Category:Intelligence Amplification]] [[Category:Knowledge Representation]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Philosophy]] [[Category:Pragmatics]] [[Category:Semantics]] [[Category:Semiotics]] [[Category:Syntax]] 847aec15a8c0821c12449b635b20a0541e4ecdba 663 662 2013-11-03T17:32:09Z Jon Awbrey 3 + links to NKOS wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''sign relation''' is the basic construct in the theory of signs, also known as [[semeiotic]] or [[semiotics]], as developed by Charles Sanders Peirce. ==Anthesis== {| align="center" cellpadding="6" width="90%" | <p>Thus, if a sunflower, in turning towards the sun, becomes by that very act fully capable, without further condition, of reproducing a sunflower which turns in precisely corresponding ways toward the sun, and of doing so with the same reproductive power, the sunflower would become a Representamen of the sun. (C.S. Peirce, &ldquo;Syllabus&rdquo; (''c''.&nbsp;1902), ''Collected Papers'', CP&nbsp;2.274).</p> |} In his picturesque illustration of a sign relation, along with his tracing of a corresponding sign process, or ''[[semiosis]]'', Peirce uses the technical term ''representamen'' for his concept of a sign, but the shorter word is precise enough, so long as one recognizes that its meaning in a particular theory of signs is given by a specific definition of what it means to be a sign. ==Definition== One of Peirce's clearest and most complete definitions of a sign is one that he gives in the context of providing a definition for ''logic'', and so it is informative to view it in that setting. {| align="center" cellpadding="6" width="90%" | <p>Logic will here be defined as ''formal semiotic''. A definition of a sign will be given which no more refers to human thought than does the definition of a line as the place which a particle occupies, part by part, during a lapse of time. Namely, a sign is something, ''A'', which brings something, ''B'', its ''interpretant'' sign determined or created by it, into the same sort of correspondence with something, ''C'', its ''object'', as that in which itself stands to ''C''. It is from this definition, together with a definition of &ldquo;formal&rdquo;, that I deduce mathematically the principles of logic. I also make a historical review of all the definitions and conceptions of logic, and show, not merely that my definition is no novelty, but that my non-psychological conception of logic has ''virtually'' been quite generally held, though not generally recognized. (C.S. Peirce, NEM&nbsp;4, 20&ndash;21).</p> |} In the general discussion of diverse theories of signs, the question frequently arises whether signhood is an absolute, essential, indelible, or ''ontological'' property of a thing, or whether it is a relational, interpretive, and mutable role that a thing can be said to have only within a particular context of relationships. Peirce's definition of a ''sign'' defines it in relation to its ''object'' and its ''interpretant sign'', and thus it defines signhood in ''[[logic of relatives|relative terms]]'', by means of a predicate with three places. In this definition, signhood is a role in a [[triadic relation]], a role that a thing bears or plays in a given context of relationships &mdash; it is not as an ''absolute'', ''non-relative'' property of a thing-in-itself, one that it possesses independently of all relationships to other things. Some of the terms that Peirce uses in his definition of a sign may need to be elaborated for the contemporary reader. :* '''Correspondence.''' From the way that Peirce uses this term throughout his work, it is clear that he means what he elsewhere calls a &ldquo;triple correspondence&rdquo;, and thus this is just another way of referring to the whole triadic sign relation itself. In particular, his use of this term should not be taken to imply a dyadic correspondence, like the kinds of &ldquo;mirror image&rdquo; correspondence between realities and representations that are bandied about in contemporary controversies about &ldquo;correspondence theories of truth&rdquo;. :* '''Determination.''' Peirce's concept of determination is broader in several directions than the sense of the word that refers to strictly deterministic causal-temporal processes. First, and especially in this context, he is invoking a more general concept of determination, what is called a ''formal'' or ''informational'' determination, as in saying &ldquo;two points determine a line&rdquo;, rather than the more special cases of causal and temporal determinisms. Second, he characteristically allows for what is called ''determination in measure'', that is, an order of determinism that admits a full spectrum of more and less determined relationships. :* '''Non-psychological.''' Peirce's &ldquo;non-psychological conception of logic&rdquo; must be distinguished from any variety of ''anti-psychologism''. He was quite interested in matters of psychology and had much of import to say about them. But logic and psychology operate on different planes of study even when they have occasion to view the same data, as logic is a ''[[normative science]]'' where psychology is a ''[[descriptive science]]'', and so they have very different aims, methods, and rationales. ==Signs and inquiry== : ''Main article : [[Inquiry]]'' There is a close relationship between the pragmatic theory of signs and the pragmatic theory of [[inquiry]]. In fact, the correspondence between the two studies exhibits so many congruences and parallels that it is often best to treat them as integral parts of one and the same subject. In a very real sense, inquiry is the process by which sign relations come to be established and continue to evolve. In other words, inquiry, &ldquo;thinking&rdquo; in its best sense, &ldquo;is a term denoting the various ways in which things acquire significance&rdquo; (John Dewey). Thus, there is an active and intricate form of cooperation that needs to be appreciated and maintained between these converging modes of investigation. Its proper character is best understood by realizing that the theory of inquiry is adapted to study the developmental aspects of sign relations, a subject which the theory of signs is specialized to treat from structural and comparative points of view. ==Examples of sign relations== Because the examples to follow have been artificially constructed to be as simple as possible, their detailed elaboration can run the risk of trivializing the whole theory of sign relations. Despite their simplicity, however, these examples have subtleties of their own, and their careful treatment will serve to illustrate many important issues in the general theory of signs. Imagine a discussion between two people, Ann and Bob, and attend only to that aspect of their interpretive practice that involves the use of the following nouns and pronouns: &ldquo;Ann&rdquo;, &ldquo;Bob&rdquo;, &ldquo;I&rdquo;, &ldquo;you&rdquo;. :* The ''object domain'' of this discussion fragment is the set of two people <math>\{ \text{Ann}, \text{Bob} \}.\!</math> :* The ''syntactic domain'' or the ''sign system'' of their discussion is limited to the set of four signs <math>\{ {}^{\backprime\backprime} \text{Ann} {}^{\prime\prime}, {}^{\backprime\backprime} \text{Bob} {}^{\prime\prime}, {}^{\backprime\backprime} \text{I} {}^{\prime\prime}, {}^{\backprime\backprime} \text{you} {}^{\prime\prime} \}.\!</math> In their discussion, Ann and Bob are not only the passive objects of nominative and accusative references but also the active interpreters of the language that they use. The ''system of interpretation'' (SOI) associated with each language user can be represented in the form of an individual [[three-place relation]] called the ''sign relation'' of that interpreter. Understood in terms of its ''set-theoretic extension'', a sign relation <math>L\!</math> is a ''subset'' of a ''cartesian product'' <math>O \times S \times I.\!</math> Here, <math>O, S, I\!</math> are three sets that are known as the ''object domain'', the ''sign domain'', and the ''interpretant domain'', respectively, of the sign relation <math>L \subseteq O \times S \times I.\!</math> Broadly speaking, the three domains of a sign relation can be any sets whatsoever, but the kinds of sign relations typically contemplated in formal settings are usually constrained to having <math>I \subseteq S\!</math>. In this case interpretants are just a special variety of signs and this makes it convenient to lump signs and interpretants together into a single class called the ''syntactic domain''. In the forthcoming examples <math>S\!</math> and <math>I\!</math> are identical as sets, so the same elements manifest themselves in two different roles of the sign relations in question. When it is necessary to refer to the whole set of objects and signs in the union of the domains <math>O, S, I\!</math> for a given sign relation <math>L,\!</math> one may refer to this set as the ''World'' of <math>L\!</math> and write <math>W = W_L = O \cup S \cup I.\!</math> To facilitate an interest in the abstract structures of sign relations, and to keep the notations as brief as possible as the examples become more complicated, it serves to introduce the following general notations: {| align="center" cellspacing="6" width="90%" | <math>\begin{array}{ccl} O & = & \text{Object Domain} \\[6pt] S & = & \text{Sign Domain} \\[6pt] I & = & \text{Interpretant Domain} \end{array}</math> |} Introducing a few abbreviations for use in considering the present Example, we have the following data: {| align="center" cellspacing="6" width="90%" | <math>\begin{array}{cclcl} O & = & \{ \text{Ann}, \text{Bob} \} & = & \{ \mathrm{A}, \mathrm{B} \} \\[6pt] S & = & \{ {}^{\backprime\backprime} \text{Ann} {}^{\prime\prime}, {}^{\backprime\backprime} \text{Bob} {}^{\prime\prime}, {}^{\backprime\backprime} \text{I} {}^{\prime\prime}, {}^{\backprime\backprime} \text{you} {}^{\prime\prime} \} & = & \{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \} \\[6pt] I & = & \{ {}^{\backprime\backprime} \text{Ann} {}^{\prime\prime}, {}^{\backprime\backprime} \text{Bob} {}^{\prime\prime}, {}^{\backprime\backprime} \text{I} {}^{\prime\prime}, {}^{\backprime\backprime} \text{you} {}^{\prime\prime} \} & = & \{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \} \end{array}</math> |} In the present example, <math>S = I = \text{Syntactic Domain}\!</math>. The next two Tables give the sign relations associated with the interpreters <math>\mathrm{A}\!</math> and <math>\mathrm{B},\!</math> respectively, putting them in the form of ''relational databases''. Thus, the rows of each Table list the ordered triples of the form <math>(o, s, i)\!</math> that make up the corresponding sign relations, <math>L_\mathrm{A}, L_\mathrm{B} \subseteq O \times S \times I.\!</math> <br> {| align="center" style="width:92%" | width="50%" | {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 1a.} ~~ {L_\mathrm{A}} = \text{Sign Relation of Interpreter A}\!</math> |- style="height:40px; background:ghostwhite" | style="width:33%" | <math>\text{Object}\!</math> | style="width:33%" | <math>\text{Sign}\!</math> | style="width:33%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} | width="50%" | {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 1b.} ~~ {L_\mathrm{B}} = \text{Sign Relation of Interpreter B}\!</math> |- style="height:40px; background:ghostwhite" | style="width:33%" | <math>\text{Object}\!</math> | style="width:33%" | <math>\text{Sign}\!</math> | style="width:33%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- |<math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} |} <br> These Tables codify a rudimentary level of interpretive practice for the agents <math>\mathrm{A}\!</math> and <math>\mathrm{B}\!</math> and provide a basis for formalizing the initial semantics that is appropriate to their common syntactic domain. Each row of a Table names an object and two co-referent signs, making up an ordered triple of the form <math>(o, s, i)\!</math> called an ''elementary relation'', that is, one element of the relation's set-theoretic extension. Already in this elementary context, there are several different meanings that might attach to the project of a ''formal semiotics'', or a formal theory of meaning for signs. In the process of discussing these alternatives, it is useful to introduce a few terms that are occasionally used in the philosophy of language to point out the needed distinctions. ==Dyadic aspects of sign relations== For an arbitrary triadic relation <math>L \subseteq O \times S \times I,\!</math> whether it is a sign relation or not, there are six dyadic relations that can be obtained by ''projecting'' <math>L\!</math> on one of the planes of the <math>OSI\!</math>-space <math>O \times S \times I.\!</math> The six dyadic projections of a triadic relation <math>L\!</math> are defined and notated as follows: <br> {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%" |+ style="height:30px" | <math>\text{Table 2.} ~~ \text{Dyadic Projections of Triadic Relations}\!</math> | <math>\begin{matrix} L_{OS} & = & \mathrm{proj}_{OS}(L) & = & \{ (o, s) \in O \times S ~:~ (o, s, i) \in L ~\text{for some}~ i \in I \} \\[6pt] L_{SO} & = & \mathrm{proj}_{SO}(L) & = & \{ (s, o) \in S \times O ~:~ (o, s, i) \in L ~\text{for some}~ i \in I \} \\[6pt] L_{IS} & = & \mathrm{proj}_{IS}(L) & = & \{ (i, s) \in I \times S ~:~ (o, s, i) \in L ~\text{for some}~ o \in O \} \\[6pt] L_{SI} & = & \mathrm{proj}_{SI}(L) & = & \{ (s, i) \in S \times I ~:~ (o, s, i) \in L ~\text{for some}~ o \in O \} \\[6pt] L_{OI} & = & \mathrm{proj}_{OI}(L) & = & \{ (o, i) \in O \times I ~:~ (o, s, i) \in L ~\text{for some}~ s \in S \} \\[6pt] L_{IO} & = & \mathrm{proj}_{IO}(L) & = & \{ (i, o) \in I \times O ~:~ (o, s, i) \in L ~\text{for some}~ s \in S \} \end{matrix}</math> |} <br> By way of unpacking the set-theoretic notation, here is what the first definition says in ordinary language. {| align="center" cellpadding="6" width="90%" | <p>The dyadic relation that results from the projection of <math>L\!</math> on the <math>OS\!</math>-plane <math>O \times S\!</math> is written briefly as <math>L_{OS}\!</math> or written more fully as <math>\mathrm{proj}_{OS}(L),\!</math> and it is defined as the set of all ordered pairs <math>(o, s)\!</math> in the cartesian product <math>O \times S\!</math> for which there exists an ordered triple <math>(o, s, i)\!</math> in <math>L\!</math> for some interpretant <math>i\!</math> in the interpretant domain <math>I.\!</math></p> |} In the case where <math>L\!</math> is a sign relation, which it becomes by satisfying one of the definitions of a sign relation, some of the dyadic aspects of <math>L\!</math> can be recognized as formalizing aspects of sign meaning that have received their share of attention from students of signs over the centuries, and thus they can be associated with traditional concepts and terminology. Of course, traditions may vary as to the precise formation and usage of such concepts and terms. Other aspects of meaning have not received their fair share of attention, and thus remain anonymous on the contemporary scene of sign studies. ===Denotation=== One aspect of a sign's complete meaning is concerned with the reference that a sign has to its objects, which objects are collectively known as the ''denotation'' of the sign. In the pragmatic theory of sign relations, denotative references fall within the projection of the sign relation on the plane that is spanned by its object domain and its sign domain. The dyadic relation that makes up the ''denotative'', ''referential'', or ''semantic'' aspect or component of a sign relation <math>L\!</math> is notated as <math>\mathrm{Den}(L).\!</math> Information about the denotative aspect of meaning is obtained from <math>L\!</math> by taking its projection on the object-sign plane, in other words, on the 2-dimensional space that is generated by the object domain <math>O\!</math> and the sign domain <math>S.\!</math> This component of a sign relation <math>L\!</math> can be written in any of the forms, <math>\mathrm{proj}_{OS} L,\!</math> &nbsp; <math>L_{OS},\!</math> &nbsp; <math>\mathrm{proj}_{12} L,\!</math> &nbsp; <math>L_{12},\!</math> and it is defined as follows: {| align="center" cellpadding="6" width="90%" | <math>\begin{matrix} \mathrm{Den}(L) & = & \mathrm{proj}_{OS} L & = & \{ (o, s) \in O \times S ~:~ (o, s, i) \in L ~\text{for some}~ i \in I \}. \end{matrix}</math> |} Looking to the denotative aspects of <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B},\!</math> various rows of the Tables specify, for example, that <math>\mathrm{A}\!</math> uses <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> to denote <math>\mathrm{A}\!</math> and <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> to denote <math>\mathrm{B},\!</math> whereas <math>\mathrm{B}\!</math> uses <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> to denote <math>\mathrm{B}\!</math> and <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> to denote <math>\mathrm{A}.\!</math> All of these denotative references are summed up in the projections on the <math>OS\!</math>-plane, as shown in the following Tables: <br> {| align="center" style="width:90%" | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 3a.} ~~ \mathrm{Den}(L_\mathrm{A}) = \mathrm{proj}_{OS}(L_\mathrm{A})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Object}\!</math> | width="50%" | <math>\text{Sign}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 3b.} ~~ \mathrm{Den}(L_\mathrm{B}) = \mathrm{proj}_{OS}(L_\mathrm{B})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Object}\!</math> | width="50%" | <math>\text{Sign}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} |} <br> ===Connotation=== Another aspect of meaning concerns the connection that a sign has to its interpretants within a given sign relation. As before, this type of connection can be vacuous, singular, or plural in its collection of terminal points, and it can be formalized as the dyadic relation that is obtained as a planar projection of the triadic sign relation in question. The connection that a sign makes to an interpretant is here referred to as its ''connotation''. In the full theory of sign relations, this aspect of meaning includes the links that a sign has to affects, concepts, ideas, impressions, intentions, and the whole realm of an agent's mental states and allied activities, broadly encompassing intellectual associations, emotional impressions, motivational impulses, and real conduct. Taken at the full, in the natural setting of semiotic phenomena, this complex system of references is unlikely ever to find itself mapped in much detail, much less completely formalized, but the tangible warp of its accumulated mass is commonly alluded to as the connotative import of language. Formally speaking, however, the connotative aspect of meaning presents no additional difficulty. For a given sign relation <math>L,\!</math> the dyadic relation that constitutes the ''connotative aspect'' or ''connotative component'' of <math>L\!</math> is notated as <math>\mathrm{Con}(L).\!</math> The connotative aspect of a sign relation <math>L\!</math> is given by its projection on the plane of signs and interpretants, and is therefore defined as follows: {| align="center" cellpadding="6" width="90%" | <math>\begin{matrix} \mathrm{Con}(L) & = & \mathrm{proj}_{SI} L & = & \{ (s, i) \in S \times I ~:~ (o, s, i) \in L ~\text{for some}~ o \in O \}. \end{matrix}</math> |} All of these connotative references are summed up in the projections on the <math>SI\!</math>-plane, as shown in the following Tables: <br> {| align="center" style="width:90%" | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 4a.} ~~ \mathrm{Con}(L_\mathrm{A}) = \mathrm{proj}_{SI}(L_\mathrm{A})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Sign}\!</math> | width="50%" | <math>\text{Interpretant}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 4b.} ~~ \mathrm{Con}(L_\mathrm{B}) = \mathrm{proj}_{SI}(L_\mathrm{B})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Sign}\!</math> | width="50%" | <math>\text{Interpretant}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} |} <br> ===Ennotation=== The aspect of a sign's meaning that arises from the dyadic relation of its objects to its interpretants has no standard name. If an interpretant is considered to be a sign in its own right, then its independent reference to an object can be taken as belonging to another moment of denotation, but this neglects the mediational character of the whole transaction in which this occurs. Denotation and connotation have to do with dyadic relations in which the sign plays an active role, but here we have to consider a dyadic relation between objects and interpretants that is mediated by the sign from an off-stage position, as it were. As a relation between objects and interpretants that is mediated by a sign, this aspect of meaning may be referred to as the ''ennotation'' of a sign, and the dyadic relation that constitutes the ''ennotative aspect'' of a sign relation <math>L\!</math> may be notated as <math>\mathrm{Enn}(L).\!</math> The ennotational component of meaning for a sign relation <math>L\!</math> is captured by its projection on the plane of the object and interpretant domains, and it is thus defined as follows: {| align="center" cellpadding="6" width="90%" | <math>\begin{matrix} \mathrm{Enn}(L) & = & \mathrm{proj}_{OI} L & = & \{ (o, i) \in O \times I ~:~ (o, s, i) \in L ~\text{for some}~ s \in S \}. \end{matrix}</math> |} As it happens, the sign relations <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> are fully symmetric with respect to exchanging signs and interpretants, so all the data of <math>\mathrm{proj}_{OS} L_\mathrm{A}\!</math> is echoed unchanged in <math>\mathrm{proj}_{OI} L_\mathrm{A}\!</math> and all the data of <math>\mathrm{proj}_{OS} L_\mathrm{B}\!</math> is echoed unchanged in <math>\mathrm{proj}_{OI} L_\mathrm{B}.\!</math> <br> {| align="center" style="width:90%" | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 5a.} ~~ \mathrm{Enn}(L_\mathrm{A}) = \mathrm{proj}_{OI}(L_\mathrm{A})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Object}\!</math> | width="50%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 5b.} ~~ \mathrm{Enn}(L_\mathrm{B}) = \mathrm{proj}_{OI}(L_\mathrm{B})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Object}\!</math> | width="50%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} |} <br> ==Semiotic equivalence relations== A ''semiotic equivalence relation'' (SER) is a special type of equivalence relation that arises in the analysis of sign relations. As a general rule, any equivalence relation is closely associated with a family of equivalence classes that partition the underlying set of elements, frequently called the ''domain'' or ''space'' of the relation. In the case of a SER, the equivalence classes are called ''semiotic equivalence classes'' (SECs) and the partition is called a ''semiotic partition'' (SEP). The sign relations <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> have many interesting properties that are not possessed by sign relations in general. Some of these properties have to do with the relation between signs and their interpretant signs, as reflected in the projections of <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> on the <math>SI\!</math>-plane, notated as <math>\mathrm{proj}_{SI} L_\mathrm{A}\!</math> and <math>\mathrm{proj}_{SI} L_\mathrm{B},\!</math> respectively. The 2-adic relations on <math>S \times I\!</math> induced by these projections are also referred to as the ''connotative components'' of the corresponding sign relations, notated as <math>\mathrm{Con}(L_\mathrm{A})\!</math> and <math>\mathrm{Con}(L_\mathrm{B}),\!</math> respectively. Tables&nbsp;6a and 6b show the corresponding connotative components. <br> {| align="center" style="width:90%" | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 6a.} ~~ \mathrm{Con}(L_\mathrm{A}) = \mathrm{proj}_{SI}(L_\mathrm{A})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>S\!</math> | width="50%" | <math>I\!</math> |- | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \end{matrix}</math> | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \end{matrix}</math> |- | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \end{matrix}</math> | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \end{matrix}</math> |} | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 6b.} ~~ \mathrm{Con}(L_\mathrm{B}) = \mathrm{proj}_{SI}(L_\mathrm{B})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>S\!</math> | width="50%" | <math>I\!</math> |- | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \end{matrix}</math> | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \end{matrix}</math> |- | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \end{matrix}</math> | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \end{matrix}</math> |} |} <br> One nice property possessed by the sign relations <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> is that their connotative components <math>\mathrm{Con}(L_\mathrm{A})\!</math> and <math>\mathrm{Con}(L_\mathrm{B})\!</math> form a pair of equivalence relations on their common syntactic domain <math>S = I.\!</math> It is convenient to refer to such a structure as a ''semiotic equivalence relation'' (SER) since it equates signs that mean the same thing to some interpreter. Each of the SERs, <math>\mathrm{Con}(L_\mathrm{A}), \mathrm{Con}(L_\mathrm{B}) \subseteq S \times I \cong S \times S\!</math> partitions the whole collection of signs into ''semiotic equivalence classes'' (SECs). This makes for a strong form of representation in that the structure of the interpreters' common object domain <math>\{ \mathrm{A}, \mathrm{B} \}\!</math> is reflected or reconstructed, part for part, in the structure of each of their ''semiotic partitions'' (SEPs) of the syntactic domain <math>\{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}.\!</math> But it needs to be observed that the semiotic partitions for interpreters <math>\mathrm{A}\!</math> and <math>\mathrm{B}\!</math> are not the same, indeed, they are orthogonal to each other. This makes it difficult to interpret either one of the partitions or equivalence relations on the syntactic domain as corresponding to any sort of objective structure or invariant reality, independent of the individual interpreter's point of view. Information about the contrasting patterns of semiotic equivalence induced by the interpreters <math>\mathrm{A}\!</math> and <math>\mathrm{B}\!</math> is summarized in Tables&nbsp;7a and 7b. The form of these Tables should be sufficient to explain what is meant by saying that the SEPs for <math>\mathrm{A}\!</math> and <math>\mathrm{B}\!</math> are orthogonal to each other. <br> {| align="center" style="width:92%" | width="50%" | {| align="center" cellpadding="20" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 7a.} ~~ \text{Semiotic Partition for Interpreter A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | style="border-top:1px solid black" | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | style="border-top:1px solid black" | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |} | width="50%" | {| align="center" cellpadding="20" cellspacing="1" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 7b.} ~~ \text{Semiotic Partition for Interpreter B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | style="border-left:1px solid black" | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | style="border-left:1px solid black" | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |} |} <br> A few items of notation are useful in discussing equivalence relations in general and semiotic equivalence relations in particular. As a general consideration, if <math>E\!</math> is an equivalence relation on a set <math>X,\!</math> then every element <math>x\!</math> of <math>X\!</math> belongs to a unique equivalence class under <math>E\!</math> called ''the equivalence class of <math>x\!</math> under <math>E\!</math>''. Convention provides the ''square bracket notation'' for denoting this equivalence class, either in the subscripted form <math>[x]_E\!</math> or in the simpler form <math>[x]\!</math> when the subscript <math>E\!</math> is understood. A statement that the elements <math>x\!</math> and <math>y\!</math> are equivalent under <math>E\!</math> is called an ''equation'' or an ''equivalence'' and may be expressed in any of the following ways: {| align="center" style="text-align:center; width:100%" | <math>\begin{array}{clc} (x, y) & \in & E \\[4pt] x & \in & [y]_E \\[4pt] y & \in & [x]_E \\[4pt] [x]_E & = & [y]_E \\[4pt] x & =_E & y \end{array}</math> |} Thus we have the following definitions: {| align="center" style="text-align:center; width:100%" | <math>\begin{array}{ccc} [x]_E & = & \{ y \in X : (x, y) \in E \} \\[6pt] x =_E y & \Leftrightarrow & (x, y) \in E \end{array}</math> |} In the application to sign relations it is useful to extend the square bracket notation in the following ways. If <math>L\!</math> is a sign relation whose connotative component or syntactic projection <math>L_{SI}\!</math> is an equivalence relation on <math>S = I,\!</math> let <math>[s]_L\!</math> be the equivalence class of <math>s\!</math> under <math>L_{SI}.\!</math> That is to say, <math>[s]_L = [s]_{L_{SI}}.\!</math> A statement that the signs <math>x\!</math> and <math>y\!</math> are equivalent under a semiotic equivalence relation <math>L_{SI}\!</math> is called a ''semiotic equation'' (SEQ) and may be written in either of the following equivalent forms: {| align="center" style="text-align:center; width:100%" | <math>\begin{array}{clc} [x]_L & = & [y]_L \\[6pt] x & =_L & y \end{array}</math> |} In many situations there is one further adaptation of the square bracket notation for semiotic equivalence classes that can be useful. Namely, when there is known to exist a particular triple <math>(o, s, i)\!</math> in a sign relation <math>L,\!</math> it is permissible to let <math>[o]_L\!</math> be defined as <math>[s]_L.\!</math> These modifications are designed to make the notation for semiotic equivalence classes harmonize as well as possible with the frequent use of similar devices for the denotations of signs and expressions. The semiotic equivalence relation for interpreter <math>\mathrm{A}\!</math> yields the following semiotic equations: {| align="center" style="text-align:center; width:100%" | <math>\begin{matrix} [ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} ]_{L_\mathrm{A}} & = & [ {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} ]_{L_\mathrm{A}} \\[6pt] [ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} ]_{L_\mathrm{A}} & = & [ {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} ]_{L_\mathrm{A}} \end{matrix}</math> |} or {| align="center" style="text-align:center; width:100%" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} & =_{L_\mathrm{A}} & {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} & =_{L_\mathrm{A}} & {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \end{matrix}</math> |} Thus it induces the semiotic partition: {| align="center" cellpadding="12" style="text-align:center; width:100%" | <math>\{ \{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \}, \{ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \} \}\!</math> |} The semiotic equivalence relation for interpreter <math>\mathrm{B}\!</math> yields the following semiotic equations: {| align="center" style="text-align:center; width:100%" | <math>\begin{matrix} [ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} ]_{L_\mathrm{B}} & = & [ {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} ]_{L_\mathrm{B}} \\[6pt] [ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} ]_{L_\mathrm{B}} & = & [ {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} ]_{L_\mathrm{B}} \end{matrix}</math> |} or {| align="center" style="text-align:center; width:100%" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} & =_{L_\mathrm{B}} & {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} & =_{L_\mathrm{B}} & {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \end{matrix}</math> |} Thus it induces the semiotic partition: {| align="center" cellpadding="12" style="text-align:center; width:100%" | <math>\{ \{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}, \{ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \} \}\!</math> |} ==Six ways of looking at a sign relation== In the context of 3-adic relations in general, Peirce provides the following illustration of the six ''converses'' of a 3-adic relation, that is, the six differently ordered ways of stating what is logically the same 3-adic relation: : So in a triadic fact, say, the example <br> {| align="center" cellspacing="8" style="width:72%" | align="center" | ''A'' gives ''B'' to ''C'' |} : we make no distinction in the ordinary logic of relations between the ''subject nominative'', the ''direct object'', and the ''indirect object''. We say that the proposition has three ''logical subjects''. We regard it as a mere affair of English grammar that there are six ways of expressing this: <br> {| align="center" cellspacing="8" style="width:72%" | style="width:36%" | ''A'' gives ''B'' to ''C'' | style="width:36%" | ''A'' benefits ''C'' with ''B'' |- | ''B'' enriches ''C'' at expense of ''A'' | ''C'' receives ''B'' from ''A'' |- | ''C'' thanks ''A'' for ''B'' | ''B'' leaves ''A'' for ''C'' |} : These six sentences express one and the same indivisible phenomenon. (C.S. Peirce, &ldquo;The Categories Defended&rdquo;, MS&nbsp;308 (1903), EP&nbsp;2, 170&ndash;171). ===IOS=== (Text in Progress) ===ISO=== (Text in Progress) ===OIS=== {| align="center" cellpadding="6" width="90%" | <p>Words spoken are symbols or signs (σύμβολα) of affections or impressions (παθήματα) of the soul (ψυχή); written words are the signs of words spoken. As writing, so also is speech not the same for all races of men. But the mental affections themselves, of which these words are primarily signs (σημεια), are the same for the whole of mankind, as are also the objects (πράγματα) of which those affections are representations or likenesses, images, copies (ομοιώματα). (Aristotle, ''De Interpretatione'', 1.16^a4).</p> |} ===OSI=== (Text in Progress) ===SIO=== {| align="center" cellpadding="6" width="90%" | <p>Logic will here be defined as ''formal semiotic''. A definition of a sign will be given which no more refers to human thought than does the definition of a line as the place which a particle occupies, part by part, during a lapse of time. Namely, a sign is something, ''A'', which brings something, ''B'', its ''interpretant'' sign determined or created by it, into the same sort of correspondence with something, ''C'', its ''object'', as that in which itself stands to ''C''. It is from this definition, together with a definition of &ldquo;formal&rdquo;, that I deduce mathematically the principles of logic. I also make a historical review of all the definitions and conceptions of logic, and show, not merely that my definition is no novelty, but that my non-psychological conception of logic has ''virtually'' been quite generally held, though not generally recognized. (C.S. Peirce, &ldquo;Application to the Carnegie Institution&rdquo;, L75 (1902), NEM 4, 20&ndash;21).</p> |} ===SOI=== {| align="center" cellpadding="6" width="90%" | <p>A ''Sign'' is anything which is related to a Second thing, its ''Object'', in respect to a Quality, in such a way as to bring a Third thing, its ''Interpretant'', into relation to the same Object, and that in such a way as to bring a Fourth into relation to that Object in the same form, ''ad infinitum''. (CP&nbsp;2.92, quoted in Fisch 1986, p.&nbsp;274)</p> |} ==References== ==Bibliography== ===Primary sources=== * [[Charles Sanders Peirce (Bibliography)]] ===Secondary sources=== * Awbrey, J.L., and Awbrey, S.M. (1995), &ldquo;Interpretation as Action : The Risk of Inquiry&rdquo;, ''Inquiry : Critical Thinking Across the Disciplines'' 15(1), pp. 40&ndash;52. [http://web.archive.org/web/19970626071826/http://chss.montclair.edu/inquiry/fall95/awbrey.html Archive]. [http://independent.academia.edu/JonAwbrey/Papers/1302117/Interpretation_as_Action_The_Risk_of_Inquiry Online]. * Deledalle, Gérard (2000), ''C.S. Peirce's Philosophy of Signs'', Indiana University Press, Bloomington, IN. * Eisele, Carolyn (1979), in ''Studies in the Scientific and Mathematical Philosophy of C.S. Peirce'', Richard Milton Martin (ed.), Mouton, The Hague. * Esposito, Joseph (1980), ''Evolutionary Metaphysics : The Development of Peirce's Theory of Categories'', Ohio University Press (?). * Fisch, Max (1986), ''Peirce, Semeiotic, and Pragmatism'', Indiana University Press, Bloomington, IN. * Houser, N., Roberts, D.D., and Van Evra, J. (eds.)(1997), ''Studies in the Logic of C.S. Peirce'', Indiana University Press, Bloomington, IN. * Liszka, J.J. (1996), ''A General Introduction to the Semeiotic of C.S. Peirce'', Indiana University Press, Bloomington, IN. * Misak, C. (ed.)(2004), ''Cambridge Companion to C.S. Peirce'', Cambridge University Press. * Moore, E., and Robin, R. (1964), ''Studies in the Philosophy of C.S. Peirce, Second Series'', University of Massachusetts Press, Amherst, MA. * Murphey, M. (1961), ''The Development of Peirce's Thought''. Reprinted, Hackett, Indianapolis, IN, 1993. * Percy, Walker (2000), pp. 271&ndash;291 in ''Signposts in a Strange Land'', P. Samway (ed.), Saint Martin's Press. ==Resources== * [http://www.helsinki.fi/science/commens/dictionary.html The Commens Dictionary of Peirce's Terms] ** [http://www.helsinki.fi/science/commens/terms/semeiotic.html Semeiotic, Semiotic, Semeotic, Semeiotics] * [http://forum.wolframscience.com/archive/ A New Kind Of Science &bull; Forum Archive] ** [http://forum.wolframscience.com/archive/topic/647.html Excerpts from Peirce on Sign Relations] ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Sign_relation Sign Relation @ InterSciWiki] * [http://mywikibiz.com/Sign_relation Sign Relation @ MyWikiBiz] {{col-break}} * [http://beta.wikiversity.org/wiki/Sign_relation Sign Relation @ Wikiversity Beta] * [http://ref.subwiki.org/wiki/Sign_relation Sign Relation @ Subject Wikis] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Sign_relation Sign Relation], [http://mywikibiz.com/ MyWikiBiz] * [http://mathweb.org/wiki/Sign_relation Sign Relation], [http://mathweb.org/ MathWeb Wiki] {{col-break}} * [http://planetmath.org/SignRelation Sign Relation], [http://planetmath.org/ PlanetMath] * [http://beta.wikiversity.org/wiki/Sign_relation Sign Relation], [http://beta.wikiversity.org/ Wikiversity Beta] {{col-break}} * [http://getwiki.net/-Sign_Relation Sign Relation], [http://getwiki.net/ GetWiki] * [http://en.wikipedia.org/w/index.php?title=Sign_relation&oldid=161631069 Sign Relation], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Artificial Intelligence]] [[Category:Charles Sanders Peirce]] [[Category:Cognitive Sciences]] [[Category:Computer Science]] [[Category:Formal Sciences]] [[Category:Hermeneutics]] [[Category:Information Systems]] [[Category:Information Theory]] [[Category:Inquiry]] [[Category:Intelligence Amplification]] [[Category:Knowledge Representation]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Philosophy]] [[Category:Pragmatics]] [[Category:Semantics]] [[Category:Semiotics]] [[Category:Syntax]] b6e3e4f14721db9cb48f07c8c37c9bdd3e948b6d 664 663 2013-11-04T03:02:04Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''sign relation''' is the basic construct in the theory of signs, also known as [[semeiotic]] or [[semiotics]], as developed by Charles Sanders Peirce. ==Anthesis== {| align="center" cellpadding="6" width="90%" | <p>Thus, if a sunflower, in turning towards the sun, becomes by that very act fully capable, without further condition, of reproducing a sunflower which turns in precisely corresponding ways toward the sun, and of doing so with the same reproductive power, the sunflower would become a Representamen of the sun. (C.S. Peirce, &ldquo;Syllabus&rdquo; (''c''.&nbsp;1902), ''Collected Papers'', CP&nbsp;2.274).</p> |} In his picturesque illustration of a sign relation, along with his tracing of a corresponding sign process, or ''[[semiosis]]'', Peirce uses the technical term ''representamen'' for his concept of a sign, but the shorter word is precise enough, so long as one recognizes that its meaning in a particular theory of signs is given by a specific definition of what it means to be a sign. ==Definition== One of Peirce's clearest and most complete definitions of a sign is one that he gives in the context of providing a definition for ''logic'', and so it is informative to view it in that setting. {| align="center" cellpadding="6" width="90%" | <p>Logic will here be defined as ''formal semiotic''. A definition of a sign will be given which no more refers to human thought than does the definition of a line as the place which a particle occupies, part by part, during a lapse of time. Namely, a sign is something, ''A'', which brings something, ''B'', its ''interpretant'' sign determined or created by it, into the same sort of correspondence with something, ''C'', its ''object'', as that in which itself stands to ''C''. It is from this definition, together with a definition of &ldquo;formal&rdquo;, that I deduce mathematically the principles of logic. I also make a historical review of all the definitions and conceptions of logic, and show, not merely that my definition is no novelty, but that my non-psychological conception of logic has ''virtually'' been quite generally held, though not generally recognized. (C.S. Peirce, NEM&nbsp;4, 20&ndash;21).</p> |} In the general discussion of diverse theories of signs, the question frequently arises whether signhood is an absolute, essential, indelible, or ''ontological'' property of a thing, or whether it is a relational, interpretive, and mutable role that a thing can be said to have only within a particular context of relationships. Peirce's definition of a ''sign'' defines it in relation to its ''object'' and its ''interpretant sign'', and thus it defines signhood in ''[[logic of relatives|relative terms]]'', by means of a predicate with three places. In this definition, signhood is a role in a [[triadic relation]], a role that a thing bears or plays in a given context of relationships &mdash; it is not as an ''absolute'', ''non-relative'' property of a thing-in-itself, one that it possesses independently of all relationships to other things. Some of the terms that Peirce uses in his definition of a sign may need to be elaborated for the contemporary reader. :* '''Correspondence.''' From the way that Peirce uses this term throughout his work, it is clear that he means what he elsewhere calls a &ldquo;triple correspondence&rdquo;, and thus this is just another way of referring to the whole triadic sign relation itself. In particular, his use of this term should not be taken to imply a dyadic correspondence, like the kinds of &ldquo;mirror image&rdquo; correspondence between realities and representations that are bandied about in contemporary controversies about &ldquo;correspondence theories of truth&rdquo;. :* '''Determination.''' Peirce's concept of determination is broader in several directions than the sense of the word that refers to strictly deterministic causal-temporal processes. First, and especially in this context, he is invoking a more general concept of determination, what is called a ''formal'' or ''informational'' determination, as in saying &ldquo;two points determine a line&rdquo;, rather than the more special cases of causal and temporal determinisms. Second, he characteristically allows for what is called ''determination in measure'', that is, an order of determinism that admits a full spectrum of more and less determined relationships. :* '''Non-psychological.''' Peirce's &ldquo;non-psychological conception of logic&rdquo; must be distinguished from any variety of ''anti-psychologism''. He was quite interested in matters of psychology and had much of import to say about them. But logic and psychology operate on different planes of study even when they have occasion to view the same data, as logic is a ''[[normative science]]'' where psychology is a ''[[descriptive science]]'', and so they have very different aims, methods, and rationales. ==Signs and inquiry== : ''Main article : [[Inquiry]]'' There is a close relationship between the pragmatic theory of signs and the pragmatic theory of [[inquiry]]. In fact, the correspondence between the two studies exhibits so many congruences and parallels that it is often best to treat them as integral parts of one and the same subject. In a very real sense, inquiry is the process by which sign relations come to be established and continue to evolve. In other words, inquiry, &ldquo;thinking&rdquo; in its best sense, &ldquo;is a term denoting the various ways in which things acquire significance&rdquo; (John Dewey). Thus, there is an active and intricate form of cooperation that needs to be appreciated and maintained between these converging modes of investigation. Its proper character is best understood by realizing that the theory of inquiry is adapted to study the developmental aspects of sign relations, a subject which the theory of signs is specialized to treat from structural and comparative points of view. ==Examples of sign relations== Because the examples to follow have been artificially constructed to be as simple as possible, their detailed elaboration can run the risk of trivializing the whole theory of sign relations. Despite their simplicity, however, these examples have subtleties of their own, and their careful treatment will serve to illustrate many important issues in the general theory of signs. Imagine a discussion between two people, Ann and Bob, and attend only to that aspect of their interpretive practice that involves the use of the following nouns and pronouns: &ldquo;Ann&rdquo;, &ldquo;Bob&rdquo;, &ldquo;I&rdquo;, &ldquo;you&rdquo;. :* The ''object domain'' of this discussion fragment is the set of two people <math>\{ \text{Ann}, \text{Bob} \}.\!</math> :* The ''syntactic domain'' or the ''sign system'' of their discussion is limited to the set of four signs <math>\{ {}^{\backprime\backprime} \text{Ann} {}^{\prime\prime}, {}^{\backprime\backprime} \text{Bob} {}^{\prime\prime}, {}^{\backprime\backprime} \text{I} {}^{\prime\prime}, {}^{\backprime\backprime} \text{you} {}^{\prime\prime} \}.\!</math> In their discussion, Ann and Bob are not only the passive objects of nominative and accusative references but also the active interpreters of the language that they use. The ''system of interpretation'' (SOI) associated with each language user can be represented in the form of an individual [[three-place relation]] called the ''sign relation'' of that interpreter. Understood in terms of its ''set-theoretic extension'', a sign relation <math>L\!</math> is a ''subset'' of a ''cartesian product'' <math>O \times S \times I.\!</math> Here, <math>O, S, I\!</math> are three sets that are known as the ''object domain'', the ''sign domain'', and the ''interpretant domain'', respectively, of the sign relation <math>L \subseteq O \times S \times I.\!</math> Broadly speaking, the three domains of a sign relation can be any sets whatsoever, but the kinds of sign relations typically contemplated in formal settings are usually constrained to having <math>I \subseteq S\!</math>. In this case interpretants are just a special variety of signs and this makes it convenient to lump signs and interpretants together into a single class called the ''syntactic domain''. In the forthcoming examples <math>S\!</math> and <math>I\!</math> are identical as sets, so the same elements manifest themselves in two different roles of the sign relations in question. When it is necessary to refer to the whole set of objects and signs in the union of the domains <math>O, S, I\!</math> for a given sign relation <math>L,\!</math> one may refer to this set as the ''World'' of <math>L\!</math> and write <math>W = W_L = O \cup S \cup I.\!</math> To facilitate an interest in the abstract structures of sign relations, and to keep the notations as brief as possible as the examples become more complicated, it serves to introduce the following general notations: {| align="center" cellspacing="6" width="90%" | <math>\begin{array}{ccl} O & = & \text{Object Domain} \\[6pt] S & = & \text{Sign Domain} \\[6pt] I & = & \text{Interpretant Domain} \end{array}</math> |} Introducing a few abbreviations for use in considering the present Example, we have the following data: {| align="center" cellspacing="6" width="90%" | <math>\begin{array}{cclcl} O & = & \{ \text{Ann}, \text{Bob} \} & = & \{ \mathrm{A}, \mathrm{B} \} \\[6pt] S & = & \{ {}^{\backprime\backprime} \text{Ann} {}^{\prime\prime}, {}^{\backprime\backprime} \text{Bob} {}^{\prime\prime}, {}^{\backprime\backprime} \text{I} {}^{\prime\prime}, {}^{\backprime\backprime} \text{you} {}^{\prime\prime} \} & = & \{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \} \\[6pt] I & = & \{ {}^{\backprime\backprime} \text{Ann} {}^{\prime\prime}, {}^{\backprime\backprime} \text{Bob} {}^{\prime\prime}, {}^{\backprime\backprime} \text{I} {}^{\prime\prime}, {}^{\backprime\backprime} \text{you} {}^{\prime\prime} \} & = & \{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \} \end{array}</math> |} In the present example, <math>S = I = \text{Syntactic Domain}\!</math>. The next two Tables give the sign relations associated with the interpreters <math>\mathrm{A}\!</math> and <math>\mathrm{B},\!</math> respectively, putting them in the form of ''relational databases''. Thus, the rows of each Table list the ordered triples of the form <math>(o, s, i)\!</math> that make up the corresponding sign relations, <math>L_\mathrm{A}, L_\mathrm{B} \subseteq O \times S \times I.\!</math> <br> {| align="center" style="width:92%" | width="50%" | {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 1a.} ~~ {L_\mathrm{A}} = \text{Sign Relation of Interpreter A}\!</math> |- style="height:40px; background:ghostwhite" | style="width:33%" | <math>\text{Object}\!</math> | style="width:33%" | <math>\text{Sign}\!</math> | style="width:33%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} | width="50%" | {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 1b.} ~~ {L_\mathrm{B}} = \text{Sign Relation of Interpreter B}\!</math> |- style="height:40px; background:ghostwhite" | style="width:33%" | <math>\text{Object}\!</math> | style="width:33%" | <math>\text{Sign}\!</math> | style="width:33%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- |<math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} |} <br> These Tables codify a rudimentary level of interpretive practice for the agents <math>\mathrm{A}\!</math> and <math>\mathrm{B}\!</math> and provide a basis for formalizing the initial semantics that is appropriate to their common syntactic domain. Each row of a Table names an object and two co-referent signs, making up an ordered triple of the form <math>(o, s, i)\!</math> called an ''elementary relation'', that is, one element of the relation's set-theoretic extension. Already in this elementary context, there are several different meanings that might attach to the project of a ''formal semiotics'', or a formal theory of meaning for signs. In the process of discussing these alternatives, it is useful to introduce a few terms that are occasionally used in the philosophy of language to point out the needed distinctions. ==Dyadic aspects of sign relations== For an arbitrary triadic relation <math>L \subseteq O \times S \times I,\!</math> whether it is a sign relation or not, there are six dyadic relations that can be obtained by ''projecting'' <math>L\!</math> on one of the planes of the <math>OSI\!</math>-space <math>O \times S \times I.\!</math> The six dyadic projections of a triadic relation <math>L\!</math> are defined and notated as follows: <br> {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%" |+ style="height:30px" | <math>\text{Table 2.} ~~ \text{Dyadic Projections of Triadic Relations}\!</math> | <math>\begin{matrix} L_{OS} & = & \mathrm{proj}_{OS}(L) & = & \{ (o, s) \in O \times S ~:~ (o, s, i) \in L ~\text{for some}~ i \in I \} \\[6pt] L_{SO} & = & \mathrm{proj}_{SO}(L) & = & \{ (s, o) \in S \times O ~:~ (o, s, i) \in L ~\text{for some}~ i \in I \} \\[6pt] L_{IS} & = & \mathrm{proj}_{IS}(L) & = & \{ (i, s) \in I \times S ~:~ (o, s, i) \in L ~\text{for some}~ o \in O \} \\[6pt] L_{SI} & = & \mathrm{proj}_{SI}(L) & = & \{ (s, i) \in S \times I ~:~ (o, s, i) \in L ~\text{for some}~ o \in O \} \\[6pt] L_{OI} & = & \mathrm{proj}_{OI}(L) & = & \{ (o, i) \in O \times I ~:~ (o, s, i) \in L ~\text{for some}~ s \in S \} \\[6pt] L_{IO} & = & \mathrm{proj}_{IO}(L) & = & \{ (i, o) \in I \times O ~:~ (o, s, i) \in L ~\text{for some}~ s \in S \} \end{matrix}</math> |} <br> By way of unpacking the set-theoretic notation, here is what the first definition says in ordinary language. {| align="center" cellpadding="6" width="90%" | <p>The dyadic relation that results from the projection of <math>L\!</math> on the <math>OS\!</math>-plane <math>O \times S\!</math> is written briefly as <math>L_{OS}\!</math> or written more fully as <math>\mathrm{proj}_{OS}(L),\!</math> and it is defined as the set of all ordered pairs <math>(o, s)\!</math> in the cartesian product <math>O \times S\!</math> for which there exists an ordered triple <math>(o, s, i)\!</math> in <math>L\!</math> for some interpretant <math>i\!</math> in the interpretant domain <math>I.\!</math></p> |} In the case where <math>L\!</math> is a sign relation, which it becomes by satisfying one of the definitions of a sign relation, some of the dyadic aspects of <math>L\!</math> can be recognized as formalizing aspects of sign meaning that have received their share of attention from students of signs over the centuries, and thus they can be associated with traditional concepts and terminology. Of course, traditions may vary as to the precise formation and usage of such concepts and terms. Other aspects of meaning have not received their fair share of attention, and thus remain anonymous on the contemporary scene of sign studies. ===Denotation=== One aspect of a sign's complete meaning is concerned with the reference that a sign has to its objects, which objects are collectively known as the ''denotation'' of the sign. In the pragmatic theory of sign relations, denotative references fall within the projection of the sign relation on the plane that is spanned by its object domain and its sign domain. The dyadic relation that makes up the ''denotative'', ''referential'', or ''semantic'' aspect or component of a sign relation <math>L\!</math> is notated as <math>\mathrm{Den}(L).\!</math> Information about the denotative aspect of meaning is obtained from <math>L\!</math> by taking its projection on the object-sign plane, in other words, on the 2-dimensional space that is generated by the object domain <math>O\!</math> and the sign domain <math>S.\!</math> This component of a sign relation <math>L\!</math> can be written in any of the forms, <math>\mathrm{proj}_{OS} L,\!</math> &nbsp; <math>L_{OS},\!</math> &nbsp; <math>\mathrm{proj}_{12} L,\!</math> &nbsp; <math>L_{12},\!</math> and it is defined as follows: {| align="center" cellpadding="6" width="90%" | <math>\begin{matrix} \mathrm{Den}(L) & = & \mathrm{proj}_{OS} L & = & \{ (o, s) \in O \times S ~:~ (o, s, i) \in L ~\text{for some}~ i \in I \}. \end{matrix}</math> |} Looking to the denotative aspects of <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B},\!</math> various rows of the Tables specify, for example, that <math>\mathrm{A}\!</math> uses <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> to denote <math>\mathrm{A}\!</math> and <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> to denote <math>\mathrm{B},\!</math> whereas <math>\mathrm{B}\!</math> uses <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> to denote <math>\mathrm{B}\!</math> and <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> to denote <math>\mathrm{A}.\!</math> All of these denotative references are summed up in the projections on the <math>OS\!</math>-plane, as shown in the following Tables: <br> {| align="center" style="width:90%" | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 3a.} ~~ \mathrm{Den}(L_\mathrm{A}) = \mathrm{proj}_{OS}(L_\mathrm{A})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Object}\!</math> | width="50%" | <math>\text{Sign}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 3b.} ~~ \mathrm{Den}(L_\mathrm{B}) = \mathrm{proj}_{OS}(L_\mathrm{B})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Object}\!</math> | width="50%" | <math>\text{Sign}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} |} <br> ===Connotation=== Another aspect of meaning concerns the connection that a sign has to its interpretants within a given sign relation. As before, this type of connection can be vacuous, singular, or plural in its collection of terminal points, and it can be formalized as the dyadic relation that is obtained as a planar projection of the triadic sign relation in question. The connection that a sign makes to an interpretant is here referred to as its ''connotation''. In the full theory of sign relations, this aspect of meaning includes the links that a sign has to affects, concepts, ideas, impressions, intentions, and the whole realm of an agent's mental states and allied activities, broadly encompassing intellectual associations, emotional impressions, motivational impulses, and real conduct. Taken at the full, in the natural setting of semiotic phenomena, this complex system of references is unlikely ever to find itself mapped in much detail, much less completely formalized, but the tangible warp of its accumulated mass is commonly alluded to as the connotative import of language. Formally speaking, however, the connotative aspect of meaning presents no additional difficulty. For a given sign relation <math>L,\!</math> the dyadic relation that constitutes the ''connotative aspect'' or ''connotative component'' of <math>L\!</math> is notated as <math>\mathrm{Con}(L).\!</math> The connotative aspect of a sign relation <math>L\!</math> is given by its projection on the plane of signs and interpretants, and is therefore defined as follows: {| align="center" cellpadding="6" width="90%" | <math>\begin{matrix} \mathrm{Con}(L) & = & \mathrm{proj}_{SI} L & = & \{ (s, i) \in S \times I ~:~ (o, s, i) \in L ~\text{for some}~ o \in O \}. \end{matrix}</math> |} All of these connotative references are summed up in the projections on the <math>SI\!</math>-plane, as shown in the following Tables: <br> {| align="center" style="width:90%" | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 4a.} ~~ \mathrm{Con}(L_\mathrm{A}) = \mathrm{proj}_{SI}(L_\mathrm{A})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Sign}\!</math> | width="50%" | <math>\text{Interpretant}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 4b.} ~~ \mathrm{Con}(L_\mathrm{B}) = \mathrm{proj}_{SI}(L_\mathrm{B})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Sign}\!</math> | width="50%" | <math>\text{Interpretant}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} |} <br> ===Ennotation=== The aspect of a sign's meaning that arises from the dyadic relation of its objects to its interpretants has no standard name. If an interpretant is considered to be a sign in its own right, then its independent reference to an object can be taken as belonging to another moment of denotation, but this neglects the mediational character of the whole transaction in which this occurs. Denotation and connotation have to do with dyadic relations in which the sign plays an active role, but here we have to consider a dyadic relation between objects and interpretants that is mediated by the sign from an off-stage position, as it were. As a relation between objects and interpretants that is mediated by a sign, this aspect of meaning may be referred to as the ''ennotation'' of a sign, and the dyadic relation that constitutes the ''ennotative aspect'' of a sign relation <math>L\!</math> may be notated as <math>\mathrm{Enn}(L).\!</math> The ennotational component of meaning for a sign relation <math>L\!</math> is captured by its projection on the plane of the object and interpretant domains, and it is thus defined as follows: {| align="center" cellpadding="6" width="90%" | <math>\begin{matrix} \mathrm{Enn}(L) & = & \mathrm{proj}_{OI} L & = & \{ (o, i) \in O \times I ~:~ (o, s, i) \in L ~\text{for some}~ s \in S \}. \end{matrix}</math> |} As it happens, the sign relations <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> are fully symmetric with respect to exchanging signs and interpretants, so all the data of <math>\mathrm{proj}_{OS} L_\mathrm{A}\!</math> is echoed unchanged in <math>\mathrm{proj}_{OI} L_\mathrm{A}\!</math> and all the data of <math>\mathrm{proj}_{OS} L_\mathrm{B}\!</math> is echoed unchanged in <math>\mathrm{proj}_{OI} L_\mathrm{B}.\!</math> <br> {| align="center" style="width:90%" | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 5a.} ~~ \mathrm{Enn}(L_\mathrm{A}) = \mathrm{proj}_{OI}(L_\mathrm{A})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Object}\!</math> | width="50%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 5b.} ~~ \mathrm{Enn}(L_\mathrm{B}) = \mathrm{proj}_{OI}(L_\mathrm{B})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Object}\!</math> | width="50%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} |} <br> ==Semiotic equivalence relations== A ''semiotic equivalence relation'' (SER) is a special type of equivalence relation that arises in the analysis of sign relations. As a general rule, any equivalence relation is closely associated with a family of equivalence classes that partition the underlying set of elements, frequently called the ''domain'' or ''space'' of the relation. In the case of a SER, the equivalence classes are called ''semiotic equivalence classes'' (SECs) and the partition is called a ''semiotic partition'' (SEP). The sign relations <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> have many interesting properties that are not possessed by sign relations in general. Some of these properties have to do with the relation between signs and their interpretant signs, as reflected in the projections of <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> on the <math>SI\!</math>-plane, notated as <math>\mathrm{proj}_{SI} L_\mathrm{A}\!</math> and <math>\mathrm{proj}_{SI} L_\mathrm{B},\!</math> respectively. The 2-adic relations on <math>S \times I\!</math> induced by these projections are also referred to as the ''connotative components'' of the corresponding sign relations, notated as <math>\mathrm{Con}(L_\mathrm{A})\!</math> and <math>\mathrm{Con}(L_\mathrm{B}),\!</math> respectively. Tables&nbsp;6a and 6b show the corresponding connotative components. <br> {| align="center" style="width:90%" | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 6a.} ~~ \mathrm{Con}(L_\mathrm{A}) = \mathrm{proj}_{SI}(L_\mathrm{A})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>S\!</math> | width="50%" | <math>I\!</math> |- | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \end{matrix}</math> | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \end{matrix}</math> |- | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \end{matrix}</math> | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \end{matrix}</math> |} | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 6b.} ~~ \mathrm{Con}(L_\mathrm{B}) = \mathrm{proj}_{SI}(L_\mathrm{B})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>S\!</math> | width="50%" | <math>I\!</math> |- | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \end{matrix}</math> | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \end{matrix}</math> |- | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \end{matrix}</math> | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \end{matrix}</math> |} |} <br> One nice property possessed by the sign relations <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> is that their connotative components <math>\mathrm{Con}(L_\mathrm{A})\!</math> and <math>\mathrm{Con}(L_\mathrm{B})\!</math> form a pair of equivalence relations on their common syntactic domain <math>S = I.\!</math> It is convenient to refer to such a structure as a ''semiotic equivalence relation'' (SER) since it equates signs that mean the same thing to some interpreter. Each of the SERs, <math>\mathrm{Con}(L_\mathrm{A}), \mathrm{Con}(L_\mathrm{B}) \subseteq S \times I \cong S \times S\!</math> partitions the whole collection of signs into ''semiotic equivalence classes'' (SECs). This makes for a strong form of representation in that the structure of the interpreters' common object domain <math>\{ \mathrm{A}, \mathrm{B} \}\!</math> is reflected or reconstructed, part for part, in the structure of each of their ''semiotic partitions'' (SEPs) of the syntactic domain <math>\{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}.\!</math> But it needs to be observed that the semiotic partitions for interpreters <math>\mathrm{A}\!</math> and <math>\mathrm{B}\!</math> are not the same, indeed, they are orthogonal to each other. This makes it difficult to interpret either one of the partitions or equivalence relations on the syntactic domain as corresponding to any sort of objective structure or invariant reality, independent of the individual interpreter's point of view. Information about the contrasting patterns of semiotic equivalence induced by the interpreters <math>\mathrm{A}\!</math> and <math>\mathrm{B}\!</math> is summarized in Tables&nbsp;7a and 7b. The form of these Tables should suffice to explain what is meant by saying that the SEPs for <math>\mathrm{A}\!</math> and <math>\mathrm{B}\!</math> are orthogonal to each other. <br> {| align="center" style="width:92%" | width="50%" | {| align="center" cellpadding="20" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 7a.} ~~ \text{Semiotic Partition for Interpreter A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | style="border-top:1px solid black" | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | style="border-top:1px solid black" | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |} | width="50%" | {| align="center" cellpadding="20" cellspacing="1" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 7b.} ~~ \text{Semiotic Partition for Interpreter B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | style="border-left:1px solid black" | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | style="border-left:1px solid black" | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |} |} <br> A few items of notation are useful in discussing equivalence relations in general and semiotic equivalence relations in particular. As a general consideration, if <math>E\!</math> is an equivalence relation on a set <math>X,\!</math> then every element <math>x\!</math> of <math>X\!</math> belongs to a unique equivalence class under <math>E\!</math> called ''the equivalence class of <math>x\!</math> under <math>E\!</math>''. Convention provides the ''square bracket notation'' for denoting this equivalence class, either in the subscripted form <math>[x]_E\!</math> or in the simpler form <math>[x]\!</math> when the subscript <math>E\!</math> is understood. A statement that the elements <math>x\!</math> and <math>y\!</math> are equivalent under <math>E\!</math> is called an ''equation'' or an ''equivalence'' and may be expressed in any of the following ways: {| align="center" style="text-align:center; width:100%" | <math>\begin{array}{clc} (x, y) & \in & E \\[4pt] x & \in & [y]_E \\[4pt] y & \in & [x]_E \\[4pt] [x]_E & = & [y]_E \\[4pt] x & =_E & y \end{array}</math> |} Thus we have the following definitions: {| align="center" style="text-align:center; width:100%" | <math>\begin{array}{ccc} [x]_E & = & \{ y \in X : (x, y) \in E \} \\[6pt] x =_E y & \Leftrightarrow & (x, y) \in E \end{array}</math> |} In the application to sign relations it is useful to extend the square bracket notation in the following ways. If <math>L\!</math> is a sign relation whose connotative component or syntactic projection <math>L_{SI}\!</math> is an equivalence relation on <math>S = I,\!</math> let <math>[s]_L\!</math> be the equivalence class of <math>s\!</math> under <math>L_{SI}.\!</math> That is to say, <math>[s]_L = [s]_{L_{SI}}.\!</math> A statement that the signs <math>x\!</math> and <math>y\!</math> are equivalent under a semiotic equivalence relation <math>L_{SI}\!</math> is called a ''semiotic equation'' (SEQ) and may be written in either of the following equivalent forms: {| align="center" style="text-align:center; width:100%" | <math>\begin{array}{clc} [x]_L & = & [y]_L \\[6pt] x & =_L & y \end{array}</math> |} In many situations there is one further adaptation of the square bracket notation for semiotic equivalence classes that can be useful. Namely, when there is known to exist a particular triple <math>(o, s, i)\!</math> in a sign relation <math>L,\!</math> it is permissible to let <math>[o]_L\!</math> be defined as <math>[s]_L.\!</math> These modifications are designed to make the notation for semiotic equivalence classes harmonize as well as possible with the frequent use of similar devices for the denotations of signs and expressions. The semiotic equivalence relation for interpreter <math>\mathrm{A}\!</math> yields the following semiotic equations: {| align="center" style="text-align:center; width:100%" | <math>\begin{matrix} [ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} ]_{L_\mathrm{A}} & = & [ {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} ]_{L_\mathrm{A}} \\[6pt] [ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} ]_{L_\mathrm{A}} & = & [ {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} ]_{L_\mathrm{A}} \end{matrix}</math> |} or {| align="center" style="text-align:center; width:100%" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} & =_{L_\mathrm{A}} & {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} & =_{L_\mathrm{A}} & {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \end{matrix}</math> |} Thus it induces the semiotic partition: {| align="center" cellpadding="12" style="text-align:center; width:100%" | <math>\{ \{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \}, \{ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \} \}.\!</math> |} The semiotic equivalence relation for interpreter <math>\mathrm{B}\!</math> yields the following semiotic equations: {| align="center" style="text-align:center; width:100%" | <math>\begin{matrix} [ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} ]_{L_\mathrm{B}} & = & [ {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} ]_{L_\mathrm{B}} \\[6pt] [ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} ]_{L_\mathrm{B}} & = & [ {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} ]_{L_\mathrm{B}} \end{matrix}</math> |} or {| align="center" style="text-align:center; width:100%" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} & =_{L_\mathrm{B}} & {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} & =_{L_\mathrm{B}} & {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \end{matrix}</math> |} Thus it induces the semiotic partition: {| align="center" cellpadding="12" style="text-align:center; width:100%" | <math>\{ \{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}, \{ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \} \}.\!</math> |} ==Six ways of looking at a sign relation== In the context of 3-adic relations in general, Peirce provides the following illustration of the six ''converses'' of a 3-adic relation, that is, the six differently ordered ways of stating what is logically the same 3-adic relation: : So in a triadic fact, say, the example <br> {| align="center" cellspacing="8" style="width:72%" | align="center" | ''A'' gives ''B'' to ''C'' |} : we make no distinction in the ordinary logic of relations between the ''subject nominative'', the ''direct object'', and the ''indirect object''. We say that the proposition has three ''logical subjects''. We regard it as a mere affair of English grammar that there are six ways of expressing this: <br> {| align="center" cellspacing="8" style="width:72%" | style="width:36%" | ''A'' gives ''B'' to ''C'' | style="width:36%" | ''A'' benefits ''C'' with ''B'' |- | ''B'' enriches ''C'' at expense of ''A'' | ''C'' receives ''B'' from ''A'' |- | ''C'' thanks ''A'' for ''B'' | ''B'' leaves ''A'' for ''C'' |} : These six sentences express one and the same indivisible phenomenon. (C.S. Peirce, &ldquo;The Categories Defended&rdquo;, MS&nbsp;308 (1903), EP&nbsp;2, 170&ndash;171). ===IOS=== (Text in Progress) ===ISO=== (Text in Progress) ===OIS=== {| align="center" cellpadding="6" width="90%" | <p>Words spoken are symbols or signs (σύμβολα) of affections or impressions (παθήματα) of the soul (ψυχή); written words are the signs of words spoken. As writing, so also is speech not the same for all races of men. But the mental affections themselves, of which these words are primarily signs (σημεια), are the same for the whole of mankind, as are also the objects (πράγματα) of which those affections are representations or likenesses, images, copies (ομοιώματα). (Aristotle, ''De Interpretatione'', 1.16^a4).</p> |} ===OSI=== (Text in Progress) ===SIO=== {| align="center" cellpadding="6" width="90%" | <p>Logic will here be defined as ''formal semiotic''. A definition of a sign will be given which no more refers to human thought than does the definition of a line as the place which a particle occupies, part by part, during a lapse of time. Namely, a sign is something, ''A'', which brings something, ''B'', its ''interpretant'' sign determined or created by it, into the same sort of correspondence with something, ''C'', its ''object'', as that in which itself stands to ''C''. It is from this definition, together with a definition of &ldquo;formal&rdquo;, that I deduce mathematically the principles of logic. I also make a historical review of all the definitions and conceptions of logic, and show, not merely that my definition is no novelty, but that my non-psychological conception of logic has ''virtually'' been quite generally held, though not generally recognized. (C.S. Peirce, &ldquo;Application to the Carnegie Institution&rdquo;, L75 (1902), NEM 4, 20&ndash;21).</p> |} ===SOI=== {| align="center" cellpadding="6" width="90%" | <p>A ''Sign'' is anything which is related to a Second thing, its ''Object'', in respect to a Quality, in such a way as to bring a Third thing, its ''Interpretant'', into relation to the same Object, and that in such a way as to bring a Fourth into relation to that Object in the same form, ''ad infinitum''. (CP&nbsp;2.92, quoted in Fisch 1986, p.&nbsp;274)</p> |} ==References== ==Bibliography== ===Primary sources=== * [[Charles Sanders Peirce (Bibliography)]] ===Secondary sources=== * Awbrey, J.L., and Awbrey, S.M. (1995), &ldquo;Interpretation as Action : The Risk of Inquiry&rdquo;, ''Inquiry : Critical Thinking Across the Disciplines'' 15(1), pp. 40&ndash;52. [http://web.archive.org/web/19970626071826/http://chss.montclair.edu/inquiry/fall95/awbrey.html Archive]. [http://independent.academia.edu/JonAwbrey/Papers/1302117/Interpretation_as_Action_The_Risk_of_Inquiry Online]. * Deledalle, Gérard (2000), ''C.S. Peirce's Philosophy of Signs'', Indiana University Press, Bloomington, IN. * Eisele, Carolyn (1979), in ''Studies in the Scientific and Mathematical Philosophy of C.S. Peirce'', Richard Milton Martin (ed.), Mouton, The Hague. * Esposito, Joseph (1980), ''Evolutionary Metaphysics : The Development of Peirce's Theory of Categories'', Ohio University Press (?). * Fisch, Max (1986), ''Peirce, Semeiotic, and Pragmatism'', Indiana University Press, Bloomington, IN. * Houser, N., Roberts, D.D., and Van Evra, J. (eds.)(1997), ''Studies in the Logic of C.S. Peirce'', Indiana University Press, Bloomington, IN. * Liszka, J.J. (1996), ''A General Introduction to the Semeiotic of C.S. Peirce'', Indiana University Press, Bloomington, IN. * Misak, C. (ed.)(2004), ''Cambridge Companion to C.S. Peirce'', Cambridge University Press. * Moore, E., and Robin, R. (1964), ''Studies in the Philosophy of C.S. Peirce, Second Series'', University of Massachusetts Press, Amherst, MA. * Murphey, M. (1961), ''The Development of Peirce's Thought''. Reprinted, Hackett, Indianapolis, IN, 1993. * Percy, Walker (2000), pp. 271&ndash;291 in ''Signposts in a Strange Land'', P. Samway (ed.), Saint Martin's Press. ==Resources== * [http://www.helsinki.fi/science/commens/dictionary.html The Commens Dictionary of Peirce's Terms] ** [http://www.helsinki.fi/science/commens/terms/semeiotic.html Semeiotic, Semiotic, Semeotic, Semeiotics] * [http://forum.wolframscience.com/archive/ A New Kind Of Science &bull; Forum Archive] ** [http://forum.wolframscience.com/archive/topic/647.html Excerpts from Peirce on Sign Relations] ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Sign_relation Sign Relation @ InterSciWiki] * [http://mywikibiz.com/Sign_relation Sign Relation @ MyWikiBiz] {{col-break}} * [http://beta.wikiversity.org/wiki/Sign_relation Sign Relation @ Wikiversity Beta] * [http://ref.subwiki.org/wiki/Sign_relation Sign Relation @ Subject Wikis] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Sign_relation Sign Relation], [http://mywikibiz.com/ MyWikiBiz] * [http://mathweb.org/wiki/Sign_relation Sign Relation], [http://mathweb.org/ MathWeb Wiki] {{col-break}} * [http://planetmath.org/SignRelation Sign Relation], [http://planetmath.org/ PlanetMath] * [http://beta.wikiversity.org/wiki/Sign_relation Sign Relation], [http://beta.wikiversity.org/ Wikiversity Beta] {{col-break}} * [http://getwiki.net/-Sign_Relation Sign Relation], [http://getwiki.net/ GetWiki] * [http://en.wikipedia.org/w/index.php?title=Sign_relation&oldid=161631069 Sign Relation], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Artificial Intelligence]] [[Category:Charles Sanders Peirce]] [[Category:Cognitive Sciences]] [[Category:Computer Science]] [[Category:Formal Sciences]] [[Category:Hermeneutics]] [[Category:Information Systems]] [[Category:Information Theory]] [[Category:Inquiry]] [[Category:Intelligence Amplification]] [[Category:Knowledge Representation]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Philosophy]] [[Category:Pragmatics]] [[Category:Semantics]] [[Category:Semiotics]] [[Category:Syntax]] f9029aabb461a646604926b365093479ed20ee9e 665 664 2013-11-04T20:22:06Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''sign relation''' is the basic construct in the theory of signs, also known as [[semeiotic]] or [[semiotics]], as developed by Charles Sanders Peirce. ==Anthesis== {| align="center" cellpadding="6" width="90%" | <p>Thus, if a sunflower, in turning towards the sun, becomes by that very act fully capable, without further condition, of reproducing a sunflower which turns in precisely corresponding ways toward the sun, and of doing so with the same reproductive power, the sunflower would become a Representamen of the sun. (C.S. Peirce, &ldquo;Syllabus&rdquo; (''c''.&nbsp;1902), ''Collected Papers'', CP&nbsp;2.274).</p> |} In his picturesque illustration of a sign relation, along with his tracing of a corresponding sign process, or ''[[semiosis]]'', Peirce uses the technical term ''representamen'' for his concept of a sign, but the shorter word is precise enough, so long as one recognizes that its meaning in a particular theory of signs is given by a specific definition of what it means to be a sign. ==Definition== One of Peirce's clearest and most complete definitions of a sign is one that he gives in the context of providing a definition for ''logic'', and so it is informative to view it in that setting. {| align="center" cellpadding="6" width="90%" | <p>Logic will here be defined as ''formal semiotic''. A definition of a sign will be given which no more refers to human thought than does the definition of a line as the place which a particle occupies, part by part, during a lapse of time. Namely, a sign is something, ''A'', which brings something, ''B'', its ''interpretant'' sign determined or created by it, into the same sort of correspondence with something, ''C'', its ''object'', as that in which itself stands to ''C''. It is from this definition, together with a definition of &ldquo;formal&rdquo;, that I deduce mathematically the principles of logic. I also make a historical review of all the definitions and conceptions of logic, and show, not merely that my definition is no novelty, but that my non-psychological conception of logic has ''virtually'' been quite generally held, though not generally recognized. (C.S. Peirce, NEM&nbsp;4, 20&ndash;21).</p> |} In the general discussion of diverse theories of signs, the question frequently arises whether signhood is an absolute, essential, indelible, or ''ontological'' property of a thing, or whether it is a relational, interpretive, and mutable role that a thing can be said to have only within a particular context of relationships. Peirce's definition of a ''sign'' defines it in relation to its ''object'' and its ''interpretant sign'', and thus it defines signhood in ''[[logic of relatives|relative terms]]'', by means of a predicate with three places. In this definition, signhood is a role in a [[triadic relation]], a role that a thing bears or plays in a given context of relationships &mdash; it is not as an ''absolute'', ''non-relative'' property of a thing-in-itself, one that it possesses independently of all relationships to other things. Some of the terms that Peirce uses in his definition of a sign may need to be elaborated for the contemporary reader. :* '''Correspondence.''' From the way that Peirce uses this term throughout his work, it is clear that he means what he elsewhere calls a &ldquo;triple correspondence&rdquo;, and thus this is just another way of referring to the whole triadic sign relation itself. In particular, his use of this term should not be taken to imply a dyadic correspondence, like the kinds of &ldquo;mirror image&rdquo; correspondence between realities and representations that are bandied about in contemporary controversies about &ldquo;correspondence theories of truth&rdquo;. :* '''Determination.''' Peirce's concept of determination is broader in several directions than the sense of the word that refers to strictly deterministic causal-temporal processes. First, and especially in this context, he is invoking a more general concept of determination, what is called a ''formal'' or ''informational'' determination, as in saying &ldquo;two points determine a line&rdquo;, rather than the more special cases of causal and temporal determinisms. Second, he characteristically allows for what is called ''determination in measure'', that is, an order of determinism that admits a full spectrum of more and less determined relationships. :* '''Non-psychological.''' Peirce's &ldquo;non-psychological conception of logic&rdquo; must be distinguished from any variety of ''anti-psychologism''. He was quite interested in matters of psychology and had much of import to say about them. But logic and psychology operate on different planes of study even when they have occasion to view the same data, as logic is a ''[[normative science]]'' where psychology is a ''[[descriptive science]]'', and so they have very different aims, methods, and rationales. ==Signs and inquiry== : ''Main article : [[Inquiry]]'' There is a close relationship between the pragmatic theory of signs and the pragmatic theory of [[inquiry]]. In fact, the correspondence between the two studies exhibits so many congruences and parallels that it is often best to treat them as integral parts of one and the same subject. In a very real sense, inquiry is the process by which sign relations come to be established and continue to evolve. In other words, inquiry, &ldquo;thinking&rdquo; in its best sense, &ldquo;is a term denoting the various ways in which things acquire significance&rdquo; (John Dewey). Thus, there is an active and intricate form of cooperation that needs to be appreciated and maintained between these converging modes of investigation. Its proper character is best understood by realizing that the theory of inquiry is adapted to study the developmental aspects of sign relations, a subject which the theory of signs is specialized to treat from structural and comparative points of view. ==Examples of sign relations== Because the examples to follow have been artificially constructed to be as simple as possible, their detailed elaboration can run the risk of trivializing the whole theory of sign relations. Despite their simplicity, however, these examples have subtleties of their own, and their careful treatment will serve to illustrate many important issues in the general theory of signs. Imagine a discussion between two people, Ann and Bob, and attend only to that aspect of their interpretive practice that involves the use of the following nouns and pronouns: &ldquo;Ann&rdquo;, &ldquo;Bob&rdquo;, &ldquo;I&rdquo;, &ldquo;you&rdquo;. :* The ''object domain'' of this discussion fragment is the set of two people <math>\{ \text{Ann}, \text{Bob} \}.\!</math> :* The ''syntactic domain'' or the ''sign system'' of their discussion is limited to the set of four signs <math>\{ {}^{\backprime\backprime} \text{Ann} {}^{\prime\prime}, {}^{\backprime\backprime} \text{Bob} {}^{\prime\prime}, {}^{\backprime\backprime} \text{I} {}^{\prime\prime}, {}^{\backprime\backprime} \text{you} {}^{\prime\prime} \}.\!</math> In their discussion, Ann and Bob are not only the passive objects of nominative and accusative references but also the active interpreters of the language that they use. The ''system of interpretation'' (SOI) associated with each language user can be represented in the form of an individual [[three-place relation]] called the ''sign relation'' of that interpreter. Understood in terms of its ''set-theoretic extension'', a sign relation <math>L\!</math> is a ''subset'' of a ''cartesian product'' <math>O \times S \times I.\!</math> Here, <math>O, S, I\!</math> are three sets that are known as the ''object domain'', the ''sign domain'', and the ''interpretant domain'', respectively, of the sign relation <math>L \subseteq O \times S \times I.\!</math> Broadly speaking, the three domains of a sign relation can be any sets whatsoever, but the kinds of sign relations typically contemplated in formal settings are usually constrained to having <math>I \subseteq S\!</math>. In this case interpretants are just a special variety of signs and this makes it convenient to lump signs and interpretants together into a single class called the ''syntactic domain''. In the forthcoming examples <math>S\!</math> and <math>I\!</math> are identical as sets, so the same elements manifest themselves in two different roles of the sign relations in question. When it is necessary to refer to the whole set of objects and signs in the union of the domains <math>O, S, I\!</math> for a given sign relation <math>L,\!</math> one may refer to this set as the ''World'' of <math>L\!</math> and write <math>W = W_L = O \cup S \cup I.\!</math> To facilitate an interest in the abstract structures of sign relations, and to keep the notations as brief as possible as the examples become more complicated, it serves to introduce the following general notations: {| align="center" cellspacing="6" width="90%" | <math>\begin{array}{ccl} O & = & \text{Object Domain} \\[6pt] S & = & \text{Sign Domain} \\[6pt] I & = & \text{Interpretant Domain} \end{array}</math> |} Introducing a few abbreviations for use in considering the present Example, we have the following data: {| align="center" cellspacing="6" width="90%" | <math>\begin{array}{cclcl} O & = & \{ \text{Ann}, \text{Bob} \} & = & \{ \mathrm{A}, \mathrm{B} \} \\[6pt] S & = & \{ {}^{\backprime\backprime} \text{Ann} {}^{\prime\prime}, {}^{\backprime\backprime} \text{Bob} {}^{\prime\prime}, {}^{\backprime\backprime} \text{I} {}^{\prime\prime}, {}^{\backprime\backprime} \text{you} {}^{\prime\prime} \} & = & \{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \} \\[6pt] I & = & \{ {}^{\backprime\backprime} \text{Ann} {}^{\prime\prime}, {}^{\backprime\backprime} \text{Bob} {}^{\prime\prime}, {}^{\backprime\backprime} \text{I} {}^{\prime\prime}, {}^{\backprime\backprime} \text{you} {}^{\prime\prime} \} & = & \{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \} \end{array}</math> |} In the present example, <math>S = I = \text{Syntactic Domain}\!</math>. The next two Tables give the sign relations associated with the interpreters <math>\mathrm{A}\!</math> and <math>\mathrm{B},\!</math> respectively, putting them in the form of ''relational databases''. Thus, the rows of each Table list the ordered triples of the form <math>(o, s, i)\!</math> that make up the corresponding sign relations, <math>L_\mathrm{A}, L_\mathrm{B} \subseteq O \times S \times I.\!</math> <br> {| align="center" style="width:92%" | width="50%" | {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 1a.} ~~ {L_\mathrm{A}} = \text{Sign Relation of Interpreter A}\!</math> |- style="height:40px; background:ghostwhite" | style="width:33%" | <math>\text{Object}\!</math> | style="width:33%" | <math>\text{Sign}\!</math> | style="width:33%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} | width="50%" | {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 1b.} ~~ {L_\mathrm{B}} = \text{Sign Relation of Interpreter B}\!</math> |- style="height:40px; background:ghostwhite" | style="width:33%" | <math>\text{Object}\!</math> | style="width:33%" | <math>\text{Sign}\!</math> | style="width:33%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- |<math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} |} <br> These Tables codify a rudimentary level of interpretive practice for the agents <math>\mathrm{A}\!</math> and <math>\mathrm{B}\!</math> and provide a basis for formalizing the initial semantics that is appropriate to their common syntactic domain. Each row of a Table names an object and two co-referent signs, making up an ordered triple of the form <math>(o, s, i)\!</math> called an ''elementary relation'', that is, one element of the relation's set-theoretic extension. Already in this elementary context, there are several different meanings that might attach to the project of a ''formal semiotics'', or a formal theory of meaning for signs. In the process of discussing these alternatives, it is useful to introduce a few terms that are occasionally used in the philosophy of language to point out the needed distinctions. ==Dyadic aspects of sign relations== For an arbitrary triadic relation <math>L \subseteq O \times S \times I,\!</math> whether it is a sign relation or not, there are six dyadic relations that can be obtained by ''projecting'' <math>L\!</math> on one of the planes of the <math>OSI\!</math>-space <math>O \times S \times I.\!</math> The six dyadic projections of a triadic relation <math>L\!</math> are defined and notated as follows: <br> {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%" |+ style="height:30px" | <math>\text{Table 2.} ~~ \text{Dyadic Projections of Triadic Relations}\!</math> | <math>\begin{matrix} L_{OS} & = & \mathrm{proj}_{OS}(L) & = & \{ (o, s) \in O \times S ~:~ (o, s, i) \in L ~\text{for some}~ i \in I \} \\[6pt] L_{SO} & = & \mathrm{proj}_{SO}(L) & = & \{ (s, o) \in S \times O ~:~ (o, s, i) \in L ~\text{for some}~ i \in I \} \\[6pt] L_{IS} & = & \mathrm{proj}_{IS}(L) & = & \{ (i, s) \in I \times S ~:~ (o, s, i) \in L ~\text{for some}~ o \in O \} \\[6pt] L_{SI} & = & \mathrm{proj}_{SI}(L) & = & \{ (s, i) \in S \times I ~:~ (o, s, i) \in L ~\text{for some}~ o \in O \} \\[6pt] L_{OI} & = & \mathrm{proj}_{OI}(L) & = & \{ (o, i) \in O \times I ~:~ (o, s, i) \in L ~\text{for some}~ s \in S \} \\[6pt] L_{IO} & = & \mathrm{proj}_{IO}(L) & = & \{ (i, o) \in I \times O ~:~ (o, s, i) \in L ~\text{for some}~ s \in S \} \end{matrix}</math> |} <br> By way of unpacking the set-theoretic notation, here is what the first definition says in ordinary language. {| align="center" cellpadding="6" width="90%" | <p>The dyadic relation that results from the projection of <math>L\!</math> on the <math>OS\!</math>-plane <math>O \times S\!</math> is written briefly as <math>L_{OS}\!</math> or written more fully as <math>\mathrm{proj}_{OS}(L),\!</math> and it is defined as the set of all ordered pairs <math>(o, s)\!</math> in the cartesian product <math>O \times S\!</math> for which there exists an ordered triple <math>(o, s, i)\!</math> in <math>L\!</math> for some interpretant <math>i\!</math> in the interpretant domain <math>I.\!</math></p> |} In the case where <math>L\!</math> is a sign relation, which it becomes by satisfying one of the definitions of a sign relation, some of the dyadic aspects of <math>L\!</math> can be recognized as formalizing aspects of sign meaning that have received their share of attention from students of signs over the centuries, and thus they can be associated with traditional concepts and terminology. Of course, traditions may vary as to the precise formation and usage of such concepts and terms. Other aspects of meaning have not received their fair share of attention, and thus remain anonymous on the contemporary scene of sign studies. ===Denotation=== One aspect of a sign's complete meaning is concerned with the reference that a sign has to its objects, which objects are collectively known as the ''denotation'' of the sign. In the pragmatic theory of sign relations, denotative references fall within the projection of the sign relation on the plane that is spanned by its object domain and its sign domain. The dyadic relation that makes up the ''denotative'', ''referential'', or ''semantic'' aspect or component of a sign relation <math>L\!</math> is notated as <math>\mathrm{Den}(L).\!</math> Information about the denotative aspect of meaning is obtained from <math>L\!</math> by taking its projection on the object-sign plane, in other words, on the 2-dimensional space that is generated by the object domain <math>O\!</math> and the sign domain <math>S.\!</math> This component of a sign relation <math>L\!</math> can be written in any of the forms, <math>\mathrm{proj}_{OS} L,\!</math> &nbsp; <math>L_{OS},\!</math> &nbsp; <math>\mathrm{proj}_{12} L,\!</math> &nbsp; <math>L_{12},\!</math> and it is defined as follows: {| align="center" cellpadding="6" width="90%" | <math>\begin{matrix} \mathrm{Den}(L) & = & \mathrm{proj}_{OS} L & = & \{ (o, s) \in O \times S ~:~ (o, s, i) \in L ~\text{for some}~ i \in I \}. \end{matrix}</math> |} Looking to the denotative aspects of <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B},\!</math> various rows of the Tables specify, for example, that <math>\mathrm{A}\!</math> uses <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> to denote <math>\mathrm{A}\!</math> and <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> to denote <math>\mathrm{B},\!</math> whereas <math>\mathrm{B}\!</math> uses <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> to denote <math>\mathrm{B}\!</math> and <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> to denote <math>\mathrm{A}.\!</math> All of these denotative references are summed up in the projections on the <math>OS\!</math>-plane, as shown in the following Tables: <br> {| align="center" style="width:90%" | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 3a.} ~~ \mathrm{Den}(L_\mathrm{A}) = \mathrm{proj}_{OS}(L_\mathrm{A})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Object}\!</math> | width="50%" | <math>\text{Sign}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 3b.} ~~ \mathrm{Den}(L_\mathrm{B}) = \mathrm{proj}_{OS}(L_\mathrm{B})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Object}\!</math> | width="50%" | <math>\text{Sign}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} |} <br> ===Connotation=== Another aspect of meaning concerns the connection that a sign has to its interpretants within a given sign relation. As before, this type of connection can be vacuous, singular, or plural in its collection of terminal points, and it can be formalized as the dyadic relation that is obtained as a planar projection of the triadic sign relation in question. The connection that a sign makes to an interpretant is here referred to as its ''connotation''. In the full theory of sign relations, this aspect of meaning includes the links that a sign has to affects, concepts, ideas, impressions, intentions, and the whole realm of an agent's mental states and allied activities, broadly encompassing intellectual associations, emotional impressions, motivational impulses, and real conduct. Taken at the full, in the natural setting of semiotic phenomena, this complex system of references is unlikely ever to find itself mapped in much detail, much less completely formalized, but the tangible warp of its accumulated mass is commonly alluded to as the connotative import of language. Formally speaking, however, the connotative aspect of meaning presents no additional difficulty. For a given sign relation <math>L,\!</math> the dyadic relation that constitutes the ''connotative aspect'' or ''connotative component'' of <math>L\!</math> is notated as <math>\mathrm{Con}(L).\!</math> The connotative aspect of a sign relation <math>L\!</math> is given by its projection on the plane of signs and interpretants, and is therefore defined as follows: {| align="center" cellpadding="6" width="90%" | <math>\begin{matrix} \mathrm{Con}(L) & = & \mathrm{proj}_{SI} L & = & \{ (s, i) \in S \times I ~:~ (o, s, i) \in L ~\text{for some}~ o \in O \}. \end{matrix}</math> |} All of these connotative references are summed up in the projections on the <math>SI\!</math>-plane, as shown in the following Tables: <br> {| align="center" style="width:90%" | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 4a.} ~~ \mathrm{Con}(L_\mathrm{A}) = \mathrm{proj}_{SI}(L_\mathrm{A})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Sign}\!</math> | width="50%" | <math>\text{Interpretant}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 4b.} ~~ \mathrm{Con}(L_\mathrm{B}) = \mathrm{proj}_{SI}(L_\mathrm{B})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Sign}\!</math> | width="50%" | <math>\text{Interpretant}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} |} <br> ===Ennotation=== The aspect of a sign's meaning that arises from the dyadic relation of its objects to its interpretants has no standard name. If an interpretant is considered to be a sign in its own right, then its independent reference to an object can be taken as belonging to another moment of denotation, but this neglects the mediational character of the whole transaction in which this occurs. Denotation and connotation have to do with dyadic relations in which the sign plays an active role, but here we have to consider a dyadic relation between objects and interpretants that is mediated by the sign from an off-stage position, as it were. As a relation between objects and interpretants that is mediated by a sign, this aspect of meaning may be referred to as the ''ennotation'' of a sign, and the dyadic relation that constitutes the ''ennotative aspect'' of a sign relation <math>L\!</math> may be notated as <math>\mathrm{Enn}(L).\!</math> The ennotational component of meaning for a sign relation <math>L\!</math> is captured by its projection on the plane of the object and interpretant domains, and it is thus defined as follows: {| align="center" cellpadding="6" width="90%" | <math>\begin{matrix} \mathrm{Enn}(L) & = & \mathrm{proj}_{OI} L & = & \{ (o, i) \in O \times I ~:~ (o, s, i) \in L ~\text{for some}~ s \in S \}. \end{matrix}</math> |} As it happens, the sign relations <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> are fully symmetric with respect to exchanging signs and interpretants, so all the data of <math>\mathrm{proj}_{OS} L_\mathrm{A}\!</math> is echoed unchanged in <math>\mathrm{proj}_{OI} L_\mathrm{A}\!</math> and all the data of <math>\mathrm{proj}_{OS} L_\mathrm{B}\!</math> is echoed unchanged in <math>\mathrm{proj}_{OI} L_\mathrm{B}.\!</math> <br> {| align="center" style="width:90%" | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 5a.} ~~ \mathrm{Enn}(L_\mathrm{A}) = \mathrm{proj}_{OI}(L_\mathrm{A})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Object}\!</math> | width="50%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 5b.} ~~ \mathrm{Enn}(L_\mathrm{B}) = \mathrm{proj}_{OI}(L_\mathrm{B})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Object}\!</math> | width="50%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} |} <br> ==Semiotic equivalence relations== A ''semiotic equivalence relation'' (SER) is a special type of equivalence relation that arises in the analysis of sign relations. As a general rule, any equivalence relation is closely associated with a family of equivalence classes that partition the underlying set of elements, frequently called the ''domain'' or ''space'' of the relation. In the case of a SER, the equivalence classes are called ''semiotic equivalence classes'' (SECs) and the partition is called a ''semiotic partition'' (SEP). The sign relations <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> have many interesting properties that are not possessed by sign relations in general. Some of these properties have to do with the relation between signs and their interpretant signs, as reflected in the projections of <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> on the <math>SI\!</math>-plane, notated as <math>\mathrm{proj}_{SI} L_\mathrm{A}\!</math> and <math>\mathrm{proj}_{SI} L_\mathrm{B},\!</math> respectively. The 2-adic relations on <math>S \times I\!</math> induced by these projections are also referred to as the ''connotative components'' of the corresponding sign relations, notated as <math>\mathrm{Con}(L_\mathrm{A})\!</math> and <math>\mathrm{Con}(L_\mathrm{B}),\!</math> respectively. Tables&nbsp;6a and 6b show the corresponding connotative components. <br> {| align="center" style="width:90%" | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 6a.} ~~ \mathrm{Con}(L_\mathrm{A}) = \mathrm{proj}_{SI}(L_\mathrm{A})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>S\!</math> | width="50%" | <math>I\!</math> |- | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \end{matrix}</math> | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \end{matrix}</math> |- | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \end{matrix}</math> | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \end{matrix}</math> |} | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 6b.} ~~ \mathrm{Con}(L_\mathrm{B}) = \mathrm{proj}_{SI}(L_\mathrm{B})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>S\!</math> | width="50%" | <math>I\!</math> |- | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \end{matrix}</math> | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \end{matrix}</math> |- | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \end{matrix}</math> | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \end{matrix}</math> |} |} <br> One nice property possessed by the sign relations <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> is that their connotative components <math>\mathrm{Con}(L_\mathrm{A})\!</math> and <math>\mathrm{Con}(L_\mathrm{B})\!</math> form a pair of equivalence relations on their common syntactic domain <math>S = I.\!</math> It is convenient to refer to such a structure as a ''semiotic equivalence relation'' (SER) since it equates signs that mean the same thing to some interpreter. Each of the SERs, <math>\mathrm{Con}(L_\mathrm{A}), \mathrm{Con}(L_\mathrm{B}) \subseteq S \times I \cong S \times S\!</math> partitions the whole collection of signs into ''semiotic equivalence classes'' (SECs). This makes for a strong form of representation in that the structure of the interpreters' common object domain <math>\{ \mathrm{A}, \mathrm{B} \}\!</math> is reflected or reconstructed, part for part, in the structure of each of their ''semiotic partitions'' (SEPs) of the syntactic domain <math>\{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}.\!</math> But it needs to be observed that the semiotic partitions for interpreters <math>\mathrm{A}\!</math> and <math>\mathrm{B}\!</math> are not the same, indeed, they are orthogonal to each other. This makes it difficult to interpret either one of the partitions or equivalence relations on the syntactic domain as corresponding to any sort of objective structure or invariant reality, independent of the individual interpreter's point of view. Information about the contrasting patterns of semiotic equivalence induced by the interpreters <math>\mathrm{A}\!</math> and <math>\mathrm{B}\!</math> is summarized in Tables&nbsp;7a and 7b. The form of these Tables should suffice to explain what is meant by saying that the SEPs for <math>\mathrm{A}\!</math> and <math>\mathrm{B}\!</math> are orthogonal to each other. <br> {| align="center" style="width:92%" | width="50%" | {| align="center" cellpadding="20" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 7a.} ~~ \text{Semiotic Partition for Interpreter A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | style="border-top:1px solid black" | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | style="border-top:1px solid black" | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |} | width="50%" | {| align="center" cellpadding="20" cellspacing="1" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 7b.} ~~ \text{Semiotic Partition for Interpreter B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | style="border-left:1px solid black" | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | style="border-left:1px solid black" | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |} |} <br> A few items of notation are useful in discussing equivalence relations in general and semiotic equivalence relations in particular. As a general consideration, if <math>E\!</math> is an equivalence relation on a set <math>X,\!</math> then every element <math>x\!</math> of <math>X\!</math> belongs to a unique equivalence class under <math>E\!</math> called ''the equivalence class of <math>x\!</math> under <math>E\!</math>''. Convention provides the ''square bracket notation'' for denoting this equivalence class, either in the subscripted form <math>[x]_E\!</math> or in the simpler form <math>[x]\!</math> when the subscript <math>E\!</math> is understood. A statement that the elements <math>x\!</math> and <math>y\!</math> are equivalent under <math>E\!</math> is called an ''equation'' or an ''equivalence'' and may be expressed in any of the following ways: {| align="center" style="text-align:center; width:100%" | <math>\begin{array}{clc} (x, y) & \in & E \\[4pt] x & \in & [y]_E \\[4pt] y & \in & [x]_E \\[4pt] [x]_E & = & [y]_E \\[4pt] x & =_E & y \end{array}</math> |} Thus we have the following definitions: {| align="center" style="text-align:center; width:100%" | <math>\begin{array}{ccc} [x]_E & = & \{ y \in X : (x, y) \in E \} \\[6pt] x =_E y & \Leftrightarrow & (x, y) \in E \end{array}</math> |} In the application to sign relations it is useful to extend the square bracket notation in the following ways. If <math>L\!</math> is a sign relation whose connotative component or syntactic projection <math>L_{SI}\!</math> is an equivalence relation on <math>S = I,\!</math> let <math>[s]_L\!</math> be the equivalence class of <math>s\!</math> under <math>L_{SI}.\!</math> That is to say, <math>[s]_L = [s]_{L_{SI}}.\!</math> A statement that the signs <math>x\!</math> and <math>y\!</math> are equivalent under a semiotic equivalence relation <math>L_{SI}\!</math> is called a ''semiotic equation'' (SEQ) and may be written in either of the following equivalent forms: {| align="center" style="text-align:center; width:100%" | <math>\begin{array}{clc} [x]_L & = & [y]_L \\[6pt] x & =_L & y \end{array}</math> |} In many situations there is one further adaptation of the square bracket notation for semiotic equivalence classes that can be useful. Namely, when there is known to exist a particular triple <math>(o, s, i)\!</math> in a sign relation <math>L,\!</math> it is permissible to let <math>[o]_L\!</math> be defined as <math>[s]_L.\!</math> These modifications are designed to make the notation for semiotic equivalence classes harmonize as well as possible with the frequent use of similar devices for the denotations of signs and expressions. The semiotic equivalence relation for interpreter <math>\mathrm{A}\!</math> yields the following semiotic equations: {| align="center" style="text-align:center; width:100%" | <math>\begin{matrix} [ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} ]_{L_\mathrm{A}} & = & [ {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} ]_{L_\mathrm{A}} \\[6pt] [ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} ]_{L_\mathrm{A}} & = & [ {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} ]_{L_\mathrm{A}} \end{matrix}</math> |} or {| align="center" style="text-align:center; width:100%" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} & =_{L_\mathrm{A}} & {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} & =_{L_\mathrm{A}} & {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \end{matrix}</math> |} Thus it induces the semiotic partition: {| align="center" cellpadding="12" style="text-align:center; width:100%" | <math>\{ \{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \}, \{ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \} \}.\!</math> |} The semiotic equivalence relation for interpreter <math>\mathrm{B}\!</math> yields the following semiotic equations: {| align="center" style="text-align:center; width:100%" | <math>\begin{matrix} [ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} ]_{L_\mathrm{B}} & = & [ {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} ]_{L_\mathrm{B}} \\[6pt] [ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} ]_{L_\mathrm{B}} & = & [ {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} ]_{L_\mathrm{B}} \end{matrix}</math> |} or {| align="center" style="text-align:center; width:100%" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} & =_{L_\mathrm{B}} & {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} & =_{L_\mathrm{B}} & {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \end{matrix}</math> |} Thus it induces the semiotic partition: {| align="center" cellpadding="12" style="text-align:center; width:100%" | <math>\{ \{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}, \{ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \} \}.\!</math> |} ==Graphical representations== The dyadic components of sign relations can be given graph-theoretic representations, as ''digraphs'' (or ''directed graphs''), that provide concise pictures of their structural and potential dynamic properties. By way of terminology, a directed edge <math>(x, y)\!</math> is called an ''arc'' from point <math>x\!</math> to point <math>y,\!</math> and a self-loop <math>(x, x)\!</math> is called a ''sling'' at <math>x.\!</math> The denotative components <math>\mathrm{Den}(L_\mathrm{A})\!</math> and <math>\mathrm{Den}(L_\mathrm{B})\!</math> can be represented as digraphs on the six points of their common world set <math>W = O \cup S \cup I =\!</math> <math>\{ \mathrm{A}, \mathrm{B}, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}.\!</math> The arcs are given as follows: {| align="center" cellspacing="6" width="90%" | <p><math>\mathrm{Den}(L_\mathrm{A})\!</math> has an arc from each point of <math>\{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \}\!</math> to <math>\mathrm{A}\!</math> and an arc from each point of <math>\{ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}\!</math> to <math>\mathrm{B}.\!</math></p> |- | <p><math>\mathrm{Den}(L_\mathrm{B})\!</math> has an arc from each point of <math>\{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}\!</math> to <math>\mathrm{A}\!</math> and an arc from each point of <math>\{ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \}\!</math> to <math>\mathrm{B}.\!</math></p> |} <math>\mathrm{Den}(L_\mathrm{A})\!</math> and <math>\mathrm{Den}(L_\mathrm{B})\!</math> can be interpreted as ''transition digraphs'' that chart the succession of steps or the connection of states in a computational process. If the graphs are read this way, the denotational arcs summarize the ''upshots'' of the computations that are involved when the interpreters <math>\mathrm{A}\!</math> and <math>\mathrm{B}\!</math> evaluate the signs in <math>S\!</math> according to their own frames of reference. The connotative components <math>\mathrm{Con}(L_\mathrm{A})\!</math> and <math>\mathrm{Con}(L_\mathrm{B})\!</math> can be represented as digraphs on the four points of their common syntactic domain <math>S = I =\!</math> <math>\{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}.\!</math> Since <math>\mathrm{Con}(L_\mathrm{A})\!</math> and <math>\mathrm{Con}(L_\mathrm{B})\!</math> are semiotic equivalence relations, their digraphs conform to the pattern that is manifested by all digraphs of equivalence relations. In general, a digraph of an equivalence relation falls into connected components that correspond to the parts of the associated partition, with a complete digraph on the points of each part, and no other arcs. In the present case, the arcs are given as follows: {| align="center" cellspacing="6" width="90%" | <p><math>\mathrm{Con}(L_\mathrm{A})\!</math> has the structure of a semiotic equivalence relation on <math>S,\!</math> with a sling at each point of <math>S,\!</math> arcs in both directions between the points of <math>\{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \},\!</math> and arcs in both directions between the points of <math>\{ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}.\!</math></p> |- | <p><math>\mathrm{Con}(L_\mathrm{B})\!</math> has the structure of a semiotic equivalence relation on <math>S,\!</math> with a sling at each point of <math>S,\!</math> arcs in both directions between the points of <math>\{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \},\!</math> and arcs in both directions between the points of <math>\{ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \}.\!</math></p> |} Taken as transition digraphs, <math>\mathrm{Con}(L_\mathrm{A})\!</math> and <math>\mathrm{Con}(L_\mathrm{B})\!</math> highlight the associations that are permitted between equivalent signs, as this equivalence is judged by the interpreters <math>\mathrm{A}\!</math> and <math>\mathrm{B},\!</math> respectively. ==Six ways of looking at a sign relation== In the context of 3-adic relations in general, Peirce provides the following illustration of the six ''converses'' of a 3-adic relation, that is, the six differently ordered ways of stating what is logically the same 3-adic relation: : So in a triadic fact, say, the example <br> {| align="center" cellspacing="8" style="width:72%" | align="center" | ''A'' gives ''B'' to ''C'' |} : we make no distinction in the ordinary logic of relations between the ''subject nominative'', the ''direct object'', and the ''indirect object''. We say that the proposition has three ''logical subjects''. We regard it as a mere affair of English grammar that there are six ways of expressing this: <br> {| align="center" cellspacing="8" style="width:72%" | style="width:36%" | ''A'' gives ''B'' to ''C'' | style="width:36%" | ''A'' benefits ''C'' with ''B'' |- | ''B'' enriches ''C'' at expense of ''A'' | ''C'' receives ''B'' from ''A'' |- | ''C'' thanks ''A'' for ''B'' | ''B'' leaves ''A'' for ''C'' |} : These six sentences express one and the same indivisible phenomenon. (C.S. Peirce, &ldquo;The Categories Defended&rdquo;, MS&nbsp;308 (1903), EP&nbsp;2, 170&ndash;171). ===IOS=== (Text in Progress) ===ISO=== (Text in Progress) ===OIS=== {| align="center" cellpadding="6" width="90%" | <p>Words spoken are symbols or signs (σύμβολα) of affections or impressions (παθήματα) of the soul (ψυχή); written words are the signs of words spoken. As writing, so also is speech not the same for all races of men. But the mental affections themselves, of which these words are primarily signs (σημεια), are the same for the whole of mankind, as are also the objects (πράγματα) of which those affections are representations or likenesses, images, copies (ομοιώματα). (Aristotle, ''De Interpretatione'', 1.16^a4).</p> |} ===OSI=== (Text in Progress) ===SIO=== {| align="center" cellpadding="6" width="90%" | <p>Logic will here be defined as ''formal semiotic''. A definition of a sign will be given which no more refers to human thought than does the definition of a line as the place which a particle occupies, part by part, during a lapse of time. Namely, a sign is something, ''A'', which brings something, ''B'', its ''interpretant'' sign determined or created by it, into the same sort of correspondence with something, ''C'', its ''object'', as that in which itself stands to ''C''. It is from this definition, together with a definition of &ldquo;formal&rdquo;, that I deduce mathematically the principles of logic. I also make a historical review of all the definitions and conceptions of logic, and show, not merely that my definition is no novelty, but that my non-psychological conception of logic has ''virtually'' been quite generally held, though not generally recognized. (C.S. Peirce, &ldquo;Application to the Carnegie Institution&rdquo;, L75 (1902), NEM 4, 20&ndash;21).</p> |} ===SOI=== {| align="center" cellpadding="6" width="90%" | <p>A ''Sign'' is anything which is related to a Second thing, its ''Object'', in respect to a Quality, in such a way as to bring a Third thing, its ''Interpretant'', into relation to the same Object, and that in such a way as to bring a Fourth into relation to that Object in the same form, ''ad infinitum''. (CP&nbsp;2.92, quoted in Fisch 1986, p.&nbsp;274)</p> |} ==References== ==Bibliography== ===Primary sources=== * [[Charles Sanders Peirce (Bibliography)]] ===Secondary sources=== * Awbrey, J.L., and Awbrey, S.M. (1995), &ldquo;Interpretation as Action : The Risk of Inquiry&rdquo;, ''Inquiry : Critical Thinking Across the Disciplines'' 15(1), pp. 40&ndash;52. [http://web.archive.org/web/19970626071826/http://chss.montclair.edu/inquiry/fall95/awbrey.html Archive]. [http://independent.academia.edu/JonAwbrey/Papers/1302117/Interpretation_as_Action_The_Risk_of_Inquiry Online]. * Deledalle, Gérard (2000), ''C.S. Peirce's Philosophy of Signs'', Indiana University Press, Bloomington, IN. * Eisele, Carolyn (1979), in ''Studies in the Scientific and Mathematical Philosophy of C.S. Peirce'', Richard Milton Martin (ed.), Mouton, The Hague. * Esposito, Joseph (1980), ''Evolutionary Metaphysics : The Development of Peirce's Theory of Categories'', Ohio University Press (?). * Fisch, Max (1986), ''Peirce, Semeiotic, and Pragmatism'', Indiana University Press, Bloomington, IN. * Houser, N., Roberts, D.D., and Van Evra, J. (eds.)(1997), ''Studies in the Logic of C.S. Peirce'', Indiana University Press, Bloomington, IN. * Liszka, J.J. (1996), ''A General Introduction to the Semeiotic of C.S. Peirce'', Indiana University Press, Bloomington, IN. * Misak, C. (ed.)(2004), ''Cambridge Companion to C.S. Peirce'', Cambridge University Press. * Moore, E., and Robin, R. (1964), ''Studies in the Philosophy of C.S. Peirce, Second Series'', University of Massachusetts Press, Amherst, MA. * Murphey, M. (1961), ''The Development of Peirce's Thought''. Reprinted, Hackett, Indianapolis, IN, 1993. * Percy, Walker (2000), pp. 271&ndash;291 in ''Signposts in a Strange Land'', P. Samway (ed.), Saint Martin's Press. ==Resources== * [http://www.helsinki.fi/science/commens/dictionary.html The Commens Dictionary of Peirce's Terms] ** [http://www.helsinki.fi/science/commens/terms/semeiotic.html Semeiotic, Semiotic, Semeotic, Semeiotics] * [http://forum.wolframscience.com/archive/ A New Kind Of Science &bull; Forum Archive] ** [http://forum.wolframscience.com/archive/topic/647.html Excerpts from Peirce on Sign Relations] ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Sign_relation Sign Relation @ InterSciWiki] * [http://mywikibiz.com/Sign_relation Sign Relation @ MyWikiBiz] {{col-break}} * [http://beta.wikiversity.org/wiki/Sign_relation Sign Relation @ Wikiversity Beta] * [http://ref.subwiki.org/wiki/Sign_relation Sign Relation @ Subject Wikis] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Sign_relation Sign Relation], [http://mywikibiz.com/ MyWikiBiz] * [http://mathweb.org/wiki/Sign_relation Sign Relation], [http://mathweb.org/ MathWeb Wiki] {{col-break}} * [http://planetmath.org/SignRelation Sign Relation], [http://planetmath.org/ PlanetMath] * [http://beta.wikiversity.org/wiki/Sign_relation Sign Relation], [http://beta.wikiversity.org/ Wikiversity Beta] {{col-break}} * [http://getwiki.net/-Sign_Relation Sign Relation], [http://getwiki.net/ GetWiki] * [http://en.wikipedia.org/w/index.php?title=Sign_relation&oldid=161631069 Sign Relation], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Artificial Intelligence]] [[Category:Charles Sanders Peirce]] [[Category:Cognitive Sciences]] [[Category:Computer Science]] [[Category:Formal Sciences]] [[Category:Hermeneutics]] [[Category:Information Systems]] [[Category:Information Theory]] [[Category:Inquiry]] [[Category:Intelligence Amplification]] [[Category:Knowledge Representation]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Philosophy]] [[Category:Pragmatics]] [[Category:Semantics]] [[Category:Semiotics]] [[Category:Syntax]] 75f2ed6c24dc9b319cf12ded1bc72cb27cae7e59 0 344 656 566 2013-10-28T14:34:06Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. '''''Semeiotic''''' is one of the terms that Charles Sanders Peirce used to describe his theory of [[triadic relation|triadic]] [[sign relations]], along with ''semiotic'' and the plural variants of both terms. The form ''semeiotic'' is often used to distinguish Peirce's theory, since it is less often used by other writers to denote their particular approaches to the subject. ==Types of signs== There are three principal ways that a sign can denote its objects. These are usually described as ''kinds'', ''species'', or ''types'' of signs, but it is important to recognize that these are not ontological species, that is, they are not mutually exclusive features of description, since the same thing can be a sign in several different ways. Beginning very roughly, the three main ways of being a sign can be described as follows: :* An ''icon'' is a sign that denotes its objects by virtue of a quality that it shares with its objects. :* An ''index'' is a sign that denotes its objects by virtue of an existential connection that it has with its objects. :* A ''symbol'' is a sign that denotes its objects solely by virtue of the fact that it is interpreted to do so. One of Peirce's early delineations of the three types of signs is still quite useful as a first approach to understanding their differences and their relationships to each other: {| align="center" cellpadding="8" width="90%" | <p>In the first place there are likenesses or copies &mdash; such as ''statues'', ''pictures'', ''emblems'', ''hieroglyphics'', and the like. Such representations stand for their objects only so far as they have an actual resemblance to them &mdash; that is agree with them in some characters. The peculiarity of such representations is that they do not determine their objects &mdash; they stand for anything more or less; for they stand for whatever they resemble and they resemble everything more or less.</p> <p>The second kind of representations are such as are set up by a convention of men or a decree of God. Such are ''tallies'', ''proper names'', &c. The peculiarity of these ''conventional signs'' is that they represent no character of their objects. Likenesses denote nothing in particular; ''conventional signs'' connote nothing in particular.</p> <p>The third and last kind of representations are ''symbols'' or general representations. They connote attributes and so connote them as to determine what they denote. To this class belong all ''words'' and all ''conceptions''. Most combinations of words are also symbols. A proposition, an argument, even a whole book may be, and should be, a single symbol. (Peirce 1866, &ldquo;Lowell Lecture 7&rdquo;, CE&nbsp;1, 467&ndash;468).</p> |} ==References== * Peirce, C.S., [[Charles Sanders Peirce (Bibliography)|Bibliography]]. * Peirce, C.S., ''Writings of Charles S. Peirce : A Chronological Edition, Volume&nbsp;1, 1857&ndash;1866'', Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1982. Cited as CE&nbsp;1. * Peirce, C.S. (1865), "On the Logic of Science", Harvard University Lectures, CE&nbsp;1, 161&ndash;302. * Peirce, C.S. (1866), "The Logic of Science, or, Induction and Hypothesis", Lowell Institute Lectures, CE&nbsp;1, 357&ndash;504. ==Readings== * Awbrey, J.L., and Awbrey, S.M. (1995), &ldquo;Interpretation as Action : The Risk of Inquiry&rdquo;, ''Inquiry : Critical Thinking Across the Disciplines'' 15(1), pp. 40&ndash;52. [http://web.archive.org/web/19970626071826/http://chss.montclair.edu/inquiry/fall95/awbrey.html Archive]. [http://independent.academia.edu/JonAwbrey/Papers/1302117/Interpretation_as_Action_The_Risk_of_Inquiry Online]. ==Resources== * [http://vectors.usc.edu/thoughtmesh/publish/142.php Semeiotic &rarr; ThoughtMesh] * Bergman & Paavola (eds.), ''Commens Dictionary of Peirce's Terms'', [http://www.helsinki.fi/science/commens/dictionary.html Webpage] ** ''[http://www.helsinki.fi/science/commens/terms/semeiotic.html Semeiotic]'' ** ''[http://www.helsinki.fi/science/commens/terms/icon.html Icon]'' ** ''[http://www.helsinki.fi/science/commens/terms/index2.html Index]'' ** ''[http://www.helsinki.fi/science/commens/terms/symbol.html Symbol]'' ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Semeiotic Semeiotic @ InterSciWiki] * [http://mywikibiz.com/Semeiotic Semeiotic @ MyWikiBiz] {{col-break}} * [http://beta.wikiversity.org/wiki/Semeiotic Semeiotic @ Wikiversity Beta] * [http://ref.subwiki.org/wiki/Semeiotic Semeiotic @ Subject Wikis] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Semiotic_Information Semiotic Information] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Semeiotic Semeiotic], [http://mywikibiz.com/ MyWikiBiz] * [http://mathweb.org/wiki/Semeiotic Semeiotic], [http://mathweb.org/wiki/ MathWeb Wiki] {{col-break}} * [http://semanticweb.org/wiki/Semeiotic Semeiotic], [http://semanticweb.org/ Semantic Web] * [http://vectors.usc.edu/thoughtmesh/publish/142.php Semeiotic], [http://vectors.usc.edu/thoughtmesh/ ThoughtMesh] {{col-break}} * [http://getwiki.net/-Semeiotic Semeiotic], [http://getwiki.net/ GetWiki] * [http://en.wikipedia.org/w/index.php?title=Semeiotic&oldid=246563989 Semeiotic], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Artificial Intelligence]] [[Category:Charles Sanders Peirce]] [[Category:Critical Thinking]] [[Category:Cybernetics]] [[Category:Education]] [[Category:Hermeneutics]] [[Category:Information Systems]] [[Category:Inquiry]] [[Category:Intelligence Amplification]] [[Category:Learning Organizations]] [[Category:Knowledge Representation]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Philosophy]] [[Category:Pragmatics]] [[Category:Semantics]] [[Category:Semiotics]] [[Category:Systems Science]] 4c7611eb245b274ade58ea6dd6797886d8d941ab 699 656 2015-10-29T04:12:40Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. '''''Semeiotic''''' is one of the terms that Charles Sanders Peirce used to describe his theory of [[triadic relation|triadic]] [[sign relations]], along with ''semiotic'' and the plural variants of both terms. The form ''semeiotic'' is often used to distinguish Peirce's theory, since it is less often used by other writers to denote their particular approaches to the subject. ==Types of signs== There are three principal ways that a sign can denote its objects. These are usually described as ''kinds'', ''species'', or ''types'' of signs, but it is important to recognize that these are not ontological species, that is, they are not mutually exclusive features of description, since the same thing can be a sign in several different ways. Beginning very roughly, the three main ways of being a sign can be described as follows: :* An ''icon'' is a sign that denotes its objects by virtue of a quality that it shares with its objects. :* An ''index'' is a sign that denotes its objects by virtue of an existential connection that it has with its objects. :* A ''symbol'' is a sign that denotes its objects solely by virtue of the fact that it is interpreted to do so. One of Peirce's early delineations of the three types of signs is still quite useful as a first approach to understanding their differences and their relationships to each other: {| align="center" cellpadding="8" width="90%" | <p>In the first place there are likenesses or copies &mdash; such as ''statues'', ''pictures'', ''emblems'', ''hieroglyphics'', and the like. Such representations stand for their objects only so far as they have an actual resemblance to them &mdash; that is agree with them in some characters. The peculiarity of such representations is that they do not determine their objects &mdash; they stand for anything more or less; for they stand for whatever they resemble and they resemble everything more or less.</p> <p>The second kind of representations are such as are set up by a convention of men or a decree of God. Such are ''tallies'', ''proper names'', &c. The peculiarity of these ''conventional signs'' is that they represent no character of their objects. Likenesses denote nothing in particular; ''conventional signs'' connote nothing in particular.</p> <p>The third and last kind of representations are ''symbols'' or general representations. They connote attributes and so connote them as to determine what they denote. To this class belong all ''words'' and all ''conceptions''. Most combinations of words are also symbols. A proposition, an argument, even a whole book may be, and should be, a single symbol. (Peirce 1866, &ldquo;Lowell Lecture 7&rdquo;, CE&nbsp;1, 467&ndash;468).</p> |} ==References== * Peirce, C.S., [[Charles Sanders Peirce (Bibliography)|Bibliography]]. * Peirce, C.S., ''Writings of Charles S. Peirce : A Chronological Edition, Volume&nbsp;1, 1857&ndash;1866'', Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1982. Cited as CE&nbsp;1. * Peirce, C.S. (1865), "On the Logic of Science", Harvard University Lectures, CE&nbsp;1, 161&ndash;302. * Peirce, C.S. (1866), "The Logic of Science, or, Induction and Hypothesis", Lowell Institute Lectures, CE&nbsp;1, 357&ndash;504. ==Readings== * Awbrey, J.L., and Awbrey, S.M. (1995), &ldquo;Interpretation as Action : The Risk of Inquiry&rdquo;, ''Inquiry : Critical Thinking Across the Disciplines'' 15(1), pp. 40&ndash;52. [http://web.archive.org/web/19970626071826/http://chss.montclair.edu/inquiry/fall95/awbrey.html Archive]. [http://independent.academia.edu/JonAwbrey/Papers/1302117/Interpretation_as_Action_The_Risk_of_Inquiry Online]. ==Resources== * [http://vectors.usc.edu/thoughtmesh/publish/142.php Semeiotic &rarr; ThoughtMesh] * Bergman & Paavola (eds.), ''Commens Dictionary of Peirce's Terms'', [http://www.helsinki.fi/science/commens/dictionary.html Webpage] ** ''[http://www.helsinki.fi/science/commens/terms/semeiotic.html Semeiotic]'' ** ''[http://www.helsinki.fi/science/commens/terms/icon.html Icon]'' ** ''[http://www.helsinki.fi/science/commens/terms/index2.html Index]'' ** ''[http://www.helsinki.fi/science/commens/terms/symbol.html Symbol]'' ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Semeiotic Semeiotic @ InterSciWiki] * [http://mywikibiz.com/Semeiotic Semeiotic @ MyWikiBiz] {{col-break}} * [http://beta.wikiversity.org/wiki/Semeiotic Semeiotic @ Wikiversity Beta] * [http://ref.subwiki.org/wiki/Semeiotic Semeiotic @ Subject Wikis] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Semiotic_Information Semiotic Information] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Semeiotic Semeiotic], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Semeiotic Semeiotic], [http://mywikibiz.com/ MyWikiBiz] * [http://ref.subwiki.org/wiki/Semeiotic Semeiotic], [http://ref.subwiki.org/ Subject Wikis] {{col-break}} * [http://wikinfo.org/w/index.php/Semeiotic Semeiotic], [http://wikinfo.org/w/ Wikinfo] * [https://en.wikiversity.org/wiki/Semeiotic Semeiotic], [https://en.wikiversity.org/ Wikiversity] * [https://beta.wikiversity.org/wiki/Semeiotic Semeiotic], [https://beta.wikiversity.org/ Wikiversity Beta] {{col-break}} * [http://semanticweb.org/wiki/Semeiotic Semeiotic], [http://semanticweb.org/ Semantic Web] * [http://vectors.usc.edu/thoughtmesh/publish/142.php Semeiotic], [http://vectors.usc.edu/thoughtmesh/ ThoughtMesh] * [http://en.wikipedia.org/w/index.php?title=Semeiotic&oldid=246563989 Semeiotic], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Artificial Intelligence]] [[Category:Charles Sanders Peirce]] [[Category:Critical Thinking]] [[Category:Cybernetics]] [[Category:Education]] [[Category:Hermeneutics]] [[Category:Information Systems]] [[Category:Inquiry]] [[Category:Intelligence Amplification]] [[Category:Learning Organizations]] [[Category:Knowledge Representation]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Philosophy]] [[Category:Pragmatics]] [[Category:Semantics]] [[Category:Semiotics]] [[Category:Systems Science]] 39f7c7962ed3125fb58f0ed227372a4c606bf077 0 315 657 646 2013-10-28T15:36:18Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''minimal negation operator''' <math>(\texttt{Mno})</math> is a logical connective that says &ldquo;just one false&rdquo; of its logical arguments. If the list of arguments is empty, as expressed in the form <math>\texttt{Mno}(),</math> then it cannot be true that exactly one of the arguments is false, so <math>\texttt{Mno}() = \texttt{False}.</math> If <math>p\!</math> is the only argument, then <math>\texttt{Mno}(p)</math> says that <math>p\!</math> is false, so <math>\texttt{Mno}(p)</math> expresses the logical negation of the proposition <math>p.\!</math> Wrtten in several different notations, <math>\texttt{Mno}(p) = \texttt{Not}(p) = \lnot p = \tilde{p} = p^\prime.</math> If <math>p\!</math> and <math>q\!</math> are the only two arguments, then <math>\texttt{Mno}(p, q)</math> says that exactly one of <math>p, q\!</math> is false, so <math>\texttt{Mno}(p, q)</math> says the same thing as <math>p \neq q.\!</math> Expressing <math>\texttt{Mno}(p, q)</math> in terms of ands <math>(\cdot),</math> ors <math>(\lor),</math> and nots <math>(\tilde{~})</math> gives the following form. {| align="center" cellpadding="8" | <math>\texttt{Mno}(p, q) = \tilde{p} \cdot q \lor p \cdot \tilde{q}.</math> |} As usual, one drops the dots <math>(\cdot)</math> in contexts where they are understood, giving the following form. {| align="center" cellpadding="8" | <math>\texttt{Mno}(p, q) = \tilde{p}q \lor p\tilde{q}.</math> |} The venn diagram for <math>\texttt{Mno}(p, q)</math> is shown in Figure&nbsp;1. {| align="center" cellpadding="8" style="text-align:center" | <p>[[Image:Venn Diagram (P,Q).jpg|500px]]</p> <p><math>\text{Figure 1.}~~\texttt{Mno}(p, q)</math></p> |} The venn diagram for <math>\texttt{Mno}(p, q, r)</math> is shown in Figure&nbsp;2. {| align="center" cellpadding="8" style="text-align:center" | <p>[[Image:Venn Diagram (P,Q,R).jpg|500px]]</p> <p><math>\text{Figure 2.}~~\texttt{Mno}(p, q, r)</math></p> |} The center cell is the region where all three arguments <math>p, q, r\!</math> hold true, so <math>\texttt{Mno}(p, q, r)</math> holds true in just the three neighboring cells. In other words, <math>\texttt{Mno}(p, q, r) = \tilde{p}qr \lor p\tilde{q}r \lor pq\tilde{r}.</math> ==Initial definition== The '''minimal negation operator''' <math>\nu\!</math> is a [[multigrade operator]] <math>(\nu_k)_{k \in \mathbb{N}}</math> where each <math>\nu_k\!</math> is a <math>k\!</math>-ary [[boolean function]] defined in such a way that <math>\nu_k (x_1, \ldots , x_k) = 1</math> in just those cases where exactly one of the arguments <math>x_j\!</math> is <math>0.\!</math> In contexts where the initial letter <math>\nu\!</math> is understood, the minimal negation operators can be indicated by argument lists in parentheses. In the following text a distinctive typeface will be used for logical expressions based on minimal negation operators, for example, <math>\texttt{(x, y, z)}</math> = <math>\nu (x, y, z).\!</math> The first four members of this family of operators are shown below, with paraphrases in a couple of other notations, where tildes and primes, respectively, indicate logical negation. {| align="center" cellpadding="8" style="text-align:center" | <math>\begin{matrix} \texttt{()} & = & \nu_0 & = & 0 & = & \operatorname{false} \\[6pt] \texttt{(x)} & = & \nu_1 (x) & = & \tilde{x} & = & x^\prime \\[6pt] \texttt{(x, y)} & = & \nu_2 (x, y) & = & \tilde{x}y \lor x\tilde{y} & = & x^\prime y \lor x y^\prime \\[6pt] \texttt{(x, y, z)} & = & \nu_3 (x, y, z) & = & \tilde{x}yz \lor x\tilde{y}z \lor xy\tilde{z} & = & x^\prime y z \lor x y^\prime z \lor x y z^\prime \end{matrix}</math> |} ==Formal definition== To express the general case of <math>\nu_k\!</math> in terms of familiar operations, it helps to introduce an intermediary concept: '''Definition.''' Let the function <math>\lnot_j : \mathbb{B}^k \to \mathbb{B}</math> be defined for each integer <math>j\!</math> in the interval <math>[1, k]\!</math> by the following equation: {| align="center" cellpadding="8" | <math>\lnot_j (x_1, \ldots, x_j, \ldots, x_k) ~=~ x_1 \land \ldots \land x_{j-1} \land \lnot x_j \land x_{j+1} \land \ldots \land x_k.</math> |} Then <math>\nu_k : \mathbb{B}^k \to \mathbb{B}</math> is defined by the following equation: {| align="center" cellpadding="8" | <math>\nu_k (x_1, \ldots, x_k) ~=~ \lnot_1 (x_1, \ldots, x_k) \lor \ldots \lor \lnot_j (x_1, \ldots, x_k) \lor \ldots \lor \lnot_k (x_1, \ldots, x_k).</math> |} If we think of the point <math>x = (x_1, \ldots, x_k) \in \mathbb{B}^k</math> as indicated by the boolean product <math>x_1 \cdot \ldots \cdot x_k</math> or the logical conjunction <math>x_1 \land \ldots \land x_k,</math> then the minimal negation <math>\texttt{(} x_1, \ldots, x_k \texttt{)}</math> indicates the set of points in <math>\mathbb{B}^k</math> that differ from <math>x\!</math> in exactly one coordinate. This makes <math>\texttt{(} x_1, \ldots, x_k \texttt{)}</math> a discrete functional analogue of a ''point omitted neighborhood'' in analysis, more exactly, a ''point omitted distance one neighborhood''. In this light, the minimal negation operator can be recognized as a differential construction, an observation that opens a very wide field. It also serves to explain a variety of other names for the same concept, for example, ''logical boundary operator'', ''limen operator'', ''least action operator'', or ''hedge operator'', to name but a few. The rationale for these names is visible in the venn diagrams of the corresponding operations on sets. The remainder of this discussion proceeds on the ''algebraic boolean convention'' that the plus sign <math>(+)\!</math> and the summation symbol <math>(\textstyle\sum)</math> both refer to addition modulo 2. Unless otherwise noted, the boolean domain <math>\mathbb{B} = \{ 0, 1 \}</math> is interpreted so that <math>0 = \operatorname{false}</math> and <math>1 = \operatorname{true}.</math> This has the following consequences: {| align="center" cellpadding="4" width="90%" | valign="top" | <big>&bull;</big> | The operation <math>x + y\!</math> is a function equivalent to the exclusive disjunction of <math>x\!</math> and <math>y,\!</math> while its fiber of 1 is the relation of inequality between <math>x\!</math> and <math>y.\!</math> |- | valign="top" | <big>&bull;</big> | The operation <math>\textstyle\sum_{j=1}^k x_j</math> maps the bit sequence <math>(x_1, \ldots, x_k)\!</math> to its ''parity''. |} The following properties of the minimal negation operators <math>\nu_k : \mathbb{B}^k \to \mathbb{B}</math> may be noted: {| align="center" cellpadding="4" width="90%" | valign="top" | <big>&bull;</big> | The function <math>\texttt{(x, y)}</math> is the same as that associated with the operation <math>x + y\!</math> and the relation <math>x \ne y.</math> |- | valign="top" | <big>&bull;</big> | In contrast, <math>\texttt{(x, y, z)}</math> is not identical to <math>x + y + z.\!</math> |- | valign="top" | <big>&bull;</big> | More generally, the function <math>\nu_k (x_1, \dots, x_k)</math> for <math>k > 2\!</math> is not identical to the boolean sum <math>\textstyle\sum_{j=1}^k x_j.</math> |- | valign="top" | <big>&bull;</big> | The inclusive disjunctions indicated for the <math>\nu_k\!</math> of more than one argument may be replaced with exclusive disjunctions without affecting the meaning, since the terms disjoined are already disjoint. |} ==Truth tables== Table&nbsp;3 is a [[truth table]] for the sixteen boolean functions of type <math>f : \mathbb{B}^3 \to \mathbb{B}</math> whose fibers of 1 are either the boundaries of points in <math>\mathbb{B}^3</math> or the complements of those boundaries. <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:70%" |+ <math>\text{Table 3.} ~~ \text{Logical Boundaries and Their Complements}\!</math> |- style="background:ghostwhite" | <math>\mathcal{L}_1</math> | <math>\mathcal{L}_2</math> | <math>\mathcal{L}_3</math> | <math>\mathcal{L}_4</math> |- style="background:ghostwhite" | &nbsp; | align="right" | <math>p\colon\!</math> | <math>1~1~1~1~0~0~0~0</math> | &nbsp; |- style="background:ghostwhite" | &nbsp; | align="right" | <math>q\colon\!</math> | <math>1~1~0~0~1~1~0~0</math> | &nbsp; |- style="background:ghostwhite" | &nbsp; | align="right" | <math>r\colon\!</math> | <math>1~0~1~0~1~0~1~0</math> | &nbsp; |- | <math>\begin{matrix} f_{104} \\[4pt] f_{148} \\[4pt] f_{146} \\[4pt] f_{97} \\[4pt] f_{134} \\[4pt] f_{73} \\[4pt] f_{41} \\[4pt] f_{22} \end{matrix}</math> | <math>\begin{matrix} f_{01101000} \\[4pt] f_{10010100} \\[4pt] f_{10010010} \\[4pt] f_{01100001} \\[4pt] f_{10000110} \\[4pt] f_{01001001} \\[4pt] f_{00101001} \\[4pt] f_{00010110} \end{matrix}</math> | <math>\begin{matrix} 0~1~1~0~1~0~0~0 \\[4pt] 1~0~0~1~0~1~0~0 \\[4pt] 1~0~0~1~0~0~1~0 \\[4pt] 0~1~1~0~0~0~0~1 \\[4pt] 1~0~0~0~0~1~1~0 \\[4pt] 0~1~0~0~1~0~0~1 \\[4pt] 0~0~1~0~1~0~0~1 \\[4pt] 0~0~0~1~0~1~1~0 \end{matrix}</math> | <math>\begin{matrix} \texttt{(~p~,~q~,~r~)} \\[4pt] \texttt{(~p~,~q~,(r))} \\[4pt] \texttt{(~p~,(q),~r~)} \\[4pt] \texttt{(~p~,(q),(r))} \\[4pt] \texttt{((p),~q~,~r~)} \\[4pt] \texttt{((p),~q~,(r))} \\[4pt] \texttt{((p),(q),~r~)} \\[4pt] \texttt{((p),(q),(r))} \end{matrix}</math> |- | <math>\begin{matrix} f_{233} \\[4pt] f_{214} \\[4pt] f_{182} \\[4pt] f_{121} \\[4pt] f_{158} \\[4pt] f_{109} \\[4pt] f_{107} \\[4pt] f_{151} \end{matrix}</math> | <math>\begin{matrix} f_{11101001} \\[4pt] f_{11010110} \\[4pt] f_{10110110} \\[4pt] f_{01111001} \\[4pt] f_{10011110} \\[4pt] f_{01101101} \\[4pt] f_{01101011} \\[4pt] f_{10010111} \end{matrix}</math> | <math>\begin{matrix} 1~1~1~0~1~0~0~1 \\[4pt] 1~1~0~1~0~1~1~0 \\[4pt] 1~0~1~1~0~1~1~0 \\[4pt] 0~1~1~1~1~0~0~1 \\[4pt] 1~0~0~1~1~1~1~0 \\[4pt] 0~1~1~0~1~1~0~1 \\[4pt] 0~1~1~0~1~0~1~1 \\[4pt] 1~0~0~1~0~1~1~1 \end{matrix}</math> | <math>\begin{matrix} \texttt{(((p),(q),(r)))} \\[4pt] \texttt{(((p),(q),~r~))} \\[4pt] \texttt{(((p),~q~,(r)))} \\[4pt] \texttt{(((p),~q~,~r~))} \\[4pt] \texttt{((~p~,(q),(r)))} \\[4pt] \texttt{((~p~,(q),~r~))} \\[4pt] \texttt{((~p~,~q~,(r)))} \\[4pt] \texttt{((~p~,~q~,~r~))} \end{matrix}</math> |} <br> ==Charts and graphs== This Section focuses on visual representations of minimal negation operators. A few bits of terminology are useful in describing the pictures, but the formal details are tedious reading, and may be familiar to many readers, so the full definitions of the terms marked in ''italics'' are relegated to a Glossary at the end of the article. Two ways of visualizing the space <math>\mathbb{B}^k</math> of <math>2^k\!</math> points are the [[hypercube]] picture and the [[venn diagram]] picture. The hypercube picture associates each point of <math>\mathbb{B}^k</math> with a unique point of the <math>k\!</math>-dimensional hypercube. The venn diagram picture associates each point of <math>\mathbb{B}^k</math> with a unique "cell" of the venn diagram on <math>k\!</math> "circles". In addition, each point of <math>\mathbb{B}^k</math> is the unique point in the ''[[fiber (mathematics)|fiber]] of truth'' <math>[|s|]\!</math> of a ''singular proposition'' <math>s : \mathbb{B}^k \to \mathbb{B},</math> and thus it is the unique point where a ''singular conjunction'' of <math>k\!</math> ''literals'' is <math>1.\!</math> For example, consider two cases at opposite vertices of the cube: {| align="center" cellpadding="4" width="90%" | valign="top" | <big>&bull;</big> | The point <math>(1, 1, \ldots , 1, 1)</math> with all 1's as coordinates is the point where the conjunction of all posited variables evaluates to <math>1,\!</math> namely, the point where: |- | &nbsp; | align="center" | <math>x_1 ~ x_2 ~\ldots~ x_{n-1} ~ x_n ~=~ 1.</math> |- | valign="top" | <big>&bull;</big> | The point <math>(0, 0, \ldots , 0, 0)</math> with all 0's as coordinates is the point where the conjunction of all negated variables evaluates to <math>1,\!</math> namely, the point where: |- | &nbsp; | align="center" | <math>\texttt{(} x_1 \texttt{)(} x_2 \texttt{)} \ldots \texttt{(} x_{n-1} \texttt{)(} x_n \texttt{)} ~=~ 1.</math> |} To pass from these limiting examples to the general case, observe that a singular proposition <math>s : \mathbb{B}^k \to \mathbb{B}</math> can be given canonical expression as a conjunction of literals, <math>s = e_1 e_2 \ldots e_{k-1} e_k</math>. Then the proposition <math>\nu (e_1, e_2, \ldots, e_{k-1}, e_k)</math> is <math>1\!</math> on the points adjacent to the point where <math>s\!</math> is <math>1,\!</math> and 0 everywhere else on the cube. For example, consider the case where <math>k = 3.\!</math> Then the minimal negation operation <math>\nu (p, q, r)\!</math> &mdash; written more simply as <math>\texttt{(p, q, r)}</math> &mdash; has the following venn diagram: {| align="center" cellpadding="8" style="text-align:center" | <p>[[Image:Venn Diagram (P,Q,R).jpg|500px]]</p> <p><math>\text{Figure 4.}~~\texttt{(p, q, r)}</math></p> |} For a contrasting example, the boolean function expressed by the form <math>\texttt{((p),(q),(r))}</math> has the following venn diagram: {| align="center" cellpadding="8" style="text-align:center" | <p>[[Image:Venn Diagram ((P),(Q),(R)).jpg|500px]]</p> <p><math>\text{Figure 5.}~~\texttt{((p),(q),(r))}</math></p> |} ==Glossary of basic terms== ; Boolean domain : A ''[[boolean domain]]'' <math>\mathbb{B}</math> is a generic 2-element set, for example, <math>\mathbb{B} = \{ 0, 1 \},</math> whose elements are interpreted as logical values, usually but not invariably with <math>0 = \operatorname{false}</math> and <math>1 = \operatorname{true}.</math> ; Boolean variable : A ''[[boolean variable]]'' <math>x\!</math> is a variable that takes its value from a boolean domain, as <math>x \in \mathbb{B}.</math> ; Proposition : In situations where boolean values are interpreted as logical values, a [[boolean-valued function]] <math>f : X \to \mathbb{B}</math> or a [[boolean function]] <math>g : \mathbb{B}^k \to \mathbb{B}</math> is frequently called a ''[[proposition]]''. ; Basis element, Coordinate projection : Given a sequence of <math>k\!</math> boolean variables, <math>x_1, \ldots, x_k,</math> each variable <math>x_j\!</math> may be treated either as a ''basis element'' of the space <math>\mathbb{B}^k</math> or as a ''coordinate projection'' <math>x_j : \mathbb{B}^k \to \mathbb{B}.</math> ; Basic proposition : This means that the set of objects <math>\{ x_j : 1 \le j \le k \}</math> is a set of boolean functions <math>\{ x_j : \mathbb{B}^k \to \mathbb{B} \}</math> subject to logical interpretation as a set of ''basic propositions'' that collectively generate the complete set of <math>2^{2^k}</math> propositions over <math>\mathbb{B}^k.</math> ; Literal : A ''literal'' is one of the <math>2k\!</math> propositions <math>x_1, \ldots, x_k, \texttt{(} x_1 \texttt{)}, \ldots, \texttt{(} x_k \texttt{)},</math> in other words, either a ''posited'' basic proposition <math>x_j\!</math> or a ''negated'' basic proposition <math>\texttt{(} x_j \texttt{)},</math> for some <math>j = 1 ~\text{to}~ k.</math> ; Fiber : In mathematics generally, the ''[[fiber (mathematics)|fiber]]'' of a point <math>y \in Y</math> under a function <math>f : X \to Y</math> is defined as the inverse image <math>f^{-1}(y) \subseteq X.</math> : In the case of a boolean function <math>f : \mathbb{B}^k \to \mathbb{B},</math> there are just two fibers: : The fiber of <math>0\!</math> under <math>f,\!</math> defined as <math>f^{-1}(0),\!</math> is the set of points where the value of <math>f\!</math> is <math>0.\!</math> : The fiber of <math>1\!</math> under <math>f,\!</math> defined as <math>f^{-1}(1),\!</math> is the set of points where the value of <math>f\!</math> is <math>1.\!</math> ; Fiber of truth : When <math>1\!</math> is interpreted as the logical value <math>\operatorname{true},</math> then <math>f^{-1}(1)\!</math> is called the ''fiber of truth'' in the proposition <math>f.\!</math> Frequent mention of this fiber makes it useful to have a shorter way of referring to it. This leads to the definition of the notation <math>[|f|] = f^{-1}(1)\!</math> for the fiber of truth in the proposition <math>f.\!</math> ; Singular boolean function : A ''singular boolean function'' <math>s : \mathbb{B}^k \to \mathbb{B}</math> is a boolean function whose fiber of <math>1\!</math> is a single point of <math>\mathbb{B}^k.</math> ; Singular proposition : In the interpretation where <math>1\!</math> equals <math>\operatorname{true},</math> a singular boolean function is called a ''singular proposition''. : Singular boolean functions and singular propositions serve as functional or logical representatives of the points in <math>\mathbb{B}^k.</math> ; Singular conjunction : A ''singular conjunction'' in <math>\mathbb{B}^k \to \mathbb{B}</math> is a conjunction of <math>k\!</math> literals that includes just one conjunct of the pair <math>\{ x_j, ~\nu(x_j) \}</math> for each <math>j = 1 ~\text{to}~ k.</math> : A singular proposition <math>s : \mathbb{B}^k \to \mathbb{B}</math> can be expressed as a singular conjunction: {| align="center" cellspacing"10" width="90%" | height="36" | <math>s ~=~ e_1 e_2 \ldots e_{k-1} e_k</math>, |- | <math>\begin{array}{llll} \text{where} & e_j & = & x_j \\[6pt] \text{or} & e_j & = & \nu (x_j), \\[6pt] \text{for} & j & = & 1 ~\text{to}~ k. \end{array}</math> |} ==Resources== * [http://planetmath.org/MinimalNegationOperator Minimal Negation Operator] @ [http://planetmath.org/ PlanetMath] * [http://planetphysics.us/encyclopedia/MinimalNegationOperator.html Minimal Negation Operator] @ [http://planetphysics.us/ PlanetPhysics] * [http://www.proofwiki.org/wiki/Definition:Minimal_Negation_Operator Minimal Negation Operator] @ [http://www.proofwiki.org/ ProofWiki] * [http://atlas.wolfram.com/ Wolfram Atlas of Simple Programs] ** [http://atlas.wolfram.com/01/01/ Elementary Cellular Automata Rules (ECARs)] ** [http://atlas.wolfram.com/01/01/rulelist.html ECAR Index] ** [http://atlas.wolfram.com/01/01/views/3/TableView.html ECAR Icons] ** [http://atlas.wolfram.com/01/01/views/87/TableView.html ECAR Examples] ** [http://atlas.wolfram.com/01/01/views/172/TableView.html ECAR Formulas] ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Minimal_negation_operator Minimal Negation Operator @ InterSciWiki] * [http://mywikibiz.com/Minimal_negation_operator Minimal Negation Operator @ MyWikiBiz] {{col-break}} * [http://beta.wikiversity.org/wiki/Minimal_negation_operator Minimal Negation Operator @ Wikiversity Beta] * [http://ref.subwiki.org/wiki/Minimal_negation_operator Minimal Negation Operator @ Subject Wikis] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Minimal_negation_operator Minimal Negation Operator], [http://mywikibiz.com/ MyWikiBiz] * [http://planetmath.org/MinimalNegationOperator Minimal Negation Operator], [http://planetmath.org/ PlanetMath] * [http://proofwiki.org/wiki/Definition:Minimal_Negation_Operator Minimal Negation Operator], [http://proofwiki.org/ ProofWiki] * [http://beta.wikiversity.org/wiki/Minimal_negation_operator Minimal Negation Operator], [http://beta.wikiversity.org/ Wikiversity Beta] {{col-break}} * [http://getwiki.net/-Minimal_Negation_Operator Minimal Negation Operator], [http://getwiki.net/ GetWiki] * [http://web.archive.org/web/20070703045600/http://wikinfo.org/index.php/Minimal_negation_operator Minimal Negation Operator], [http://wikinfo.org/ Wikinfo] * [http://textop.org/wiki/index.php?title=Minimal_negation_operator Minimal Negation Operator], [http://textop.org/wiki/ Textop Wiki] * [http://en.wikipedia.org/w/index.php?title=Minimal_negation_operator&oldid=75156728 Minimal Negation Operator], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Automata Theory]] [[Category:Charles Sanders Peirce]] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Differential Logic]] [[Category:Equational Reasoning]] [[Category:Formal Languages]] [[Category:Formal Sciences]] [[Category:Formal Systems]] [[Category:Inquiry]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Neural Networks]] [[Category:Philosophy]] [[Category:Semiotics]] e98fa1141813ca1065adbb972f647c2626c49b39 667 657 2013-12-04T16:28:12Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''minimal negation operator''' <math>(\texttt{Mno})</math> is a logical connective that says &ldquo;just one false&rdquo; of its logical arguments. If the list of arguments is empty, as expressed in the form <math>\texttt{Mno}(),</math> then it cannot be true that exactly one of the arguments is false, so <math>\texttt{Mno}() = \texttt{False}.</math> If <math>p\!</math> is the only argument, then <math>\texttt{Mno}(p)</math> says that <math>p\!</math> is false, so <math>\texttt{Mno}(p)</math> expresses the logical negation of the proposition <math>p.\!</math> Wrtten in several different notations, <math>\texttt{Mno}(p) = \texttt{Not}(p) = \lnot p = \tilde{p} = p^\prime.</math> If <math>p\!</math> and <math>q\!</math> are the only two arguments, then <math>\texttt{Mno}(p, q)</math> says that exactly one of <math>p, q\!</math> is false, so <math>\texttt{Mno}(p, q)</math> says the same thing as <math>p \neq q.\!</math> Expressing <math>\texttt{Mno}(p, q)</math> in terms of ands <math>(\cdot),</math> ors <math>(\lor),</math> and nots <math>(\tilde{~})</math> gives the following form. {| align="center" cellpadding="8" | <math>\texttt{Mno}(p, q) = \tilde{p} \cdot q \lor p \cdot \tilde{q}.</math> |} As usual, one drops the dots <math>(\cdot)</math> in contexts where they are understood, giving the following form. {| align="center" cellpadding="8" | <math>\texttt{Mno}(p, q) = \tilde{p}q \lor p\tilde{q}.</math> |} The venn diagram for <math>\texttt{Mno}(p, q)</math> is shown in Figure&nbsp;1. {| align="center" cellpadding="8" style="text-align:center" | <p>[[Image:Venn Diagram (P,Q).jpg|500px]]</p> <p><math>\text{Figure 1.}~~\texttt{Mno}(p, q)</math></p> |} The venn diagram for <math>\texttt{Mno}(p, q, r)</math> is shown in Figure&nbsp;2. {| align="center" cellpadding="8" style="text-align:center" | <p>[[Image:Venn Diagram (P,Q,R).jpg|500px]]</p> <p><math>\text{Figure 2.}~~\texttt{Mno}(p, q, r)</math></p> |} The center cell is the region where all three arguments <math>p, q, r\!</math> hold true, so <math>\texttt{Mno}(p, q, r)</math> holds true in just the three neighboring cells. In other words, <math>\texttt{Mno}(p, q, r) = \tilde{p}qr \lor p\tilde{q}r \lor pq\tilde{r}.</math> ==Initial definition== The '''minimal negation operator''' <math>\nu\!</math> is a [[multigrade operator]] <math>(\nu_k)_{k \in \mathbb{N}}</math> where each <math>\nu_k\!</math> is a <math>k\!</math>-ary [[boolean function]] defined in such a way that <math>\nu_k (x_1, \ldots , x_k) = 1</math> in just those cases where exactly one of the arguments <math>x_j\!</math> is <math>0.\!</math> In contexts where the initial letter <math>\nu\!</math> is understood, the minimal negation operators can be indicated by argument lists in parentheses. In the following text a distinctive typeface will be used for logical expressions based on minimal negation operators, for example, <math>\texttt{(x, y, z)}</math> = <math>\nu (x, y, z).\!</math> The first four members of this family of operators are shown below, with paraphrases in a couple of other notations, where tildes and primes, respectively, indicate logical negation. {| align="center" cellpadding="8" style="text-align:center" | <math>\begin{matrix} \texttt{()} & = & \nu_0 & = & 0 & = & \operatorname{false} \\[6pt] \texttt{(x)} & = & \nu_1 (x) & = & \tilde{x} & = & x^\prime \\[6pt] \texttt{(x, y)} & = & \nu_2 (x, y) & = & \tilde{x}y \lor x\tilde{y} & = & x^\prime y \lor x y^\prime \\[6pt] \texttt{(x, y, z)} & = & \nu_3 (x, y, z) & = & \tilde{x}yz \lor x\tilde{y}z \lor xy\tilde{z} & = & x^\prime y z \lor x y^\prime z \lor x y z^\prime \end{matrix}</math> |} ==Formal definition== To express the general case of <math>\nu_k\!</math> in terms of familiar operations, it helps to introduce an intermediary concept: '''Definition.''' Let the function <math>\lnot_j : \mathbb{B}^k \to \mathbb{B}</math> be defined for each integer <math>j\!</math> in the interval <math>[1, k]\!</math> by the following equation: {| align="center" cellpadding="8" | <math>\lnot_j (x_1, \ldots, x_j, \ldots, x_k) ~=~ x_1 \land \ldots \land x_{j-1} \land \lnot x_j \land x_{j+1} \land \ldots \land x_k.</math> |} Then <math>\nu_k : \mathbb{B}^k \to \mathbb{B}</math> is defined by the following equation: {| align="center" cellpadding="8" | <math>\nu_k (x_1, \ldots, x_k) ~=~ \lnot_1 (x_1, \ldots, x_k) \lor \ldots \lor \lnot_j (x_1, \ldots, x_k) \lor \ldots \lor \lnot_k (x_1, \ldots, x_k).</math> |} If we think of the point <math>x = (x_1, \ldots, x_k) \in \mathbb{B}^k</math> as indicated by the boolean product <math>x_1 \cdot \ldots \cdot x_k</math> or the logical conjunction <math>x_1 \land \ldots \land x_k,</math> then the minimal negation <math>\texttt{(} x_1, \ldots, x_k \texttt{)}</math> indicates the set of points in <math>\mathbb{B}^k</math> that differ from <math>x\!</math> in exactly one coordinate. This makes <math>\texttt{(} x_1, \ldots, x_k \texttt{)}</math> a discrete functional analogue of a ''point omitted neighborhood'' in analysis, more exactly, a ''point omitted distance one neighborhood''. In this light, the minimal negation operator can be recognized as a differential construction, an observation that opens a very wide field. It also serves to explain a variety of other names for the same concept, for example, ''logical boundary operator'', ''limen operator'', ''least action operator'', or ''hedge operator'', to name but a few. The rationale for these names is visible in the venn diagrams of the corresponding operations on sets. The remainder of this discussion proceeds on the ''algebraic boolean convention'' that the plus sign <math>(+)\!</math> and the summation symbol <math>(\textstyle\sum)</math> both refer to addition modulo 2. Unless otherwise noted, the boolean domain <math>\mathbb{B} = \{ 0, 1 \}</math> is interpreted so that <math>0 = \operatorname{false}</math> and <math>1 = \operatorname{true}.</math> This has the following consequences: {| align="center" cellpadding="4" width="90%" | valign="top" | <big>&bull;</big> | The operation <math>x + y\!</math> is a function equivalent to the exclusive disjunction of <math>x\!</math> and <math>y,\!</math> while its fiber of 1 is the relation of inequality between <math>x\!</math> and <math>y.\!</math> |- | valign="top" | <big>&bull;</big> | The operation <math>\textstyle\sum_{j=1}^k x_j</math> maps the bit sequence <math>(x_1, \ldots, x_k)\!</math> to its ''parity''. |} The following properties of the minimal negation operators <math>\nu_k : \mathbb{B}^k \to \mathbb{B}</math> may be noted: {| align="center" cellpadding="4" width="90%" | valign="top" | <big>&bull;</big> | The function <math>\texttt{(x, y)}</math> is the same as that associated with the operation <math>x + y\!</math> and the relation <math>x \ne y.</math> |- | valign="top" | <big>&bull;</big> | In contrast, <math>\texttt{(x, y, z)}</math> is not identical to <math>x + y + z.\!</math> |- | valign="top" | <big>&bull;</big> | More generally, the function <math>\nu_k (x_1, \dots, x_k)</math> for <math>k > 2\!</math> is not identical to the boolean sum <math>\textstyle\sum_{j=1}^k x_j.</math> |- | valign="top" | <big>&bull;</big> | The inclusive disjunctions indicated for the <math>\nu_k\!</math> of more than one argument may be replaced with exclusive disjunctions without affecting the meaning, since the terms disjoined are already disjoint. |} ==Truth tables== Table&nbsp;3 is a [[truth table]] for the sixteen boolean functions of type <math>f : \mathbb{B}^3 \to \mathbb{B}</math> whose fibers of 1 are either the boundaries of points in <math>\mathbb{B}^3</math> or the complements of those boundaries. <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:70%" |+ <math>\text{Table 3.} ~~ \text{Logical Boundaries and Their Complements}\!</math> |- style="background:ghostwhite" | <math>\mathcal{L}_1</math> | <math>\mathcal{L}_2</math> | <math>\mathcal{L}_3</math> | <math>\mathcal{L}_4</math> |- style="background:ghostwhite" | &nbsp; | align="right" | <math>p\colon\!</math> | <math>1~1~1~1~0~0~0~0</math> | &nbsp; |- style="background:ghostwhite" | &nbsp; | align="right" | <math>q\colon\!</math> | <math>1~1~0~0~1~1~0~0</math> | &nbsp; |- style="background:ghostwhite" | &nbsp; | align="right" | <math>r\colon\!</math> | <math>1~0~1~0~1~0~1~0</math> | &nbsp; |- | <math>\begin{matrix} f_{104} \\[4pt] f_{148} \\[4pt] f_{146} \\[4pt] f_{97} \\[4pt] f_{134} \\[4pt] f_{73} \\[4pt] f_{41} \\[4pt] f_{22} \end{matrix}</math> | <math>\begin{matrix} f_{01101000} \\[4pt] f_{10010100} \\[4pt] f_{10010010} \\[4pt] f_{01100001} \\[4pt] f_{10000110} \\[4pt] f_{01001001} \\[4pt] f_{00101001} \\[4pt] f_{00010110} \end{matrix}</math> | <math>\begin{matrix} 0~1~1~0~1~0~0~0 \\[4pt] 1~0~0~1~0~1~0~0 \\[4pt] 1~0~0~1~0~0~1~0 \\[4pt] 0~1~1~0~0~0~0~1 \\[4pt] 1~0~0~0~0~1~1~0 \\[4pt] 0~1~0~0~1~0~0~1 \\[4pt] 0~0~1~0~1~0~0~1 \\[4pt] 0~0~0~1~0~1~1~0 \end{matrix}</math> | <math>\begin{matrix} \texttt{(~p~,~q~,~r~)} \\[4pt] \texttt{(~p~,~q~,(r))} \\[4pt] \texttt{(~p~,(q),~r~)} \\[4pt] \texttt{(~p~,(q),(r))} \\[4pt] \texttt{((p),~q~,~r~)} \\[4pt] \texttt{((p),~q~,(r))} \\[4pt] \texttt{((p),(q),~r~)} \\[4pt] \texttt{((p),(q),(r))} \end{matrix}</math> |- | <math>\begin{matrix} f_{233} \\[4pt] f_{214} \\[4pt] f_{182} \\[4pt] f_{121} \\[4pt] f_{158} \\[4pt] f_{109} \\[4pt] f_{107} \\[4pt] f_{151} \end{matrix}</math> | <math>\begin{matrix} f_{11101001} \\[4pt] f_{11010110} \\[4pt] f_{10110110} \\[4pt] f_{01111001} \\[4pt] f_{10011110} \\[4pt] f_{01101101} \\[4pt] f_{01101011} \\[4pt] f_{10010111} \end{matrix}</math> | <math>\begin{matrix} 1~1~1~0~1~0~0~1 \\[4pt] 1~1~0~1~0~1~1~0 \\[4pt] 1~0~1~1~0~1~1~0 \\[4pt] 0~1~1~1~1~0~0~1 \\[4pt] 1~0~0~1~1~1~1~0 \\[4pt] 0~1~1~0~1~1~0~1 \\[4pt] 0~1~1~0~1~0~1~1 \\[4pt] 1~0~0~1~0~1~1~1 \end{matrix}</math> | <math>\begin{matrix} \texttt{(((p),(q),(r)))} \\[4pt] \texttt{(((p),(q),~r~))} \\[4pt] \texttt{(((p),~q~,(r)))} \\[4pt] \texttt{(((p),~q~,~r~))} \\[4pt] \texttt{((~p~,(q),(r)))} \\[4pt] \texttt{((~p~,(q),~r~))} \\[4pt] \texttt{((~p~,~q~,(r)))} \\[4pt] \texttt{((~p~,~q~,~r~))} \end{matrix}</math> |} <br> ==Charts and graphs== This Section focuses on visual representations of minimal negation operators. A few bits of terminology are useful in describing the pictures, but the formal details are tedious reading, and may be familiar to many readers, so the full definitions of the terms marked in ''italics'' are relegated to a Glossary at the end of the article. Two ways of visualizing the space <math>\mathbb{B}^k</math> of <math>2^k\!</math> points are the [[hypercube]] picture and the [[venn diagram]] picture. The hypercube picture associates each point of <math>\mathbb{B}^k</math> with a unique point of the <math>k\!</math>-dimensional hypercube. The venn diagram picture associates each point of <math>\mathbb{B}^k</math> with a unique "cell" of the venn diagram on <math>k\!</math> "circles". In addition, each point of <math>\mathbb{B}^k</math> is the unique point in the ''[[fiber (mathematics)|fiber]] of truth'' <math>[|s|]\!</math> of a ''singular proposition'' <math>s : \mathbb{B}^k \to \mathbb{B},</math> and thus it is the unique point where a ''singular conjunction'' of <math>k\!</math> ''literals'' is <math>1.\!</math> For example, consider two cases at opposite vertices of the cube: {| align="center" cellpadding="4" width="90%" | valign="top" | <big>&bull;</big> | The point <math>(1, 1, \ldots , 1, 1)</math> with all 1's as coordinates is the point where the conjunction of all posited variables evaluates to <math>1,\!</math> namely, the point where: |- | &nbsp; | align="center" | <math>x_1 ~ x_2 ~\ldots~ x_{n-1} ~ x_n ~=~ 1.</math> |- | valign="top" | <big>&bull;</big> | The point <math>(0, 0, \ldots , 0, 0)</math> with all 0's as coordinates is the point where the conjunction of all negated variables evaluates to <math>1,\!</math> namely, the point where: |- | &nbsp; | align="center" | <math>\texttt{(} x_1 \texttt{)(} x_2 \texttt{)} \ldots \texttt{(} x_{n-1} \texttt{)(} x_n \texttt{)} ~=~ 1.</math> |} To pass from these limiting examples to the general case, observe that a singular proposition <math>s : \mathbb{B}^k \to \mathbb{B}</math> can be given canonical expression as a conjunction of literals, <math>s = e_1 e_2 \ldots e_{k-1} e_k</math>. Then the proposition <math>\nu (e_1, e_2, \ldots, e_{k-1}, e_k)</math> is <math>1\!</math> on the points adjacent to the point where <math>s\!</math> is <math>1,\!</math> and 0 everywhere else on the cube. For example, consider the case where <math>k = 3.\!</math> Then the minimal negation operation <math>\nu (p, q, r)\!</math> &mdash; written more simply as <math>\texttt{(p, q, r)}</math> &mdash; has the following venn diagram: {| align="center" cellpadding="8" style="text-align:center" | <p>[[Image:Venn Diagram (P,Q,R).jpg|500px]]</p> <p><math>\text{Figure 4.}~~\texttt{(p, q, r)}</math></p> |} For a contrasting example, the boolean function expressed by the form <math>\texttt{((p),(q),(r))}</math> has the following venn diagram: {| align="center" cellpadding="8" style="text-align:center" | <p>[[Image:Venn Diagram ((P),(Q),(R)).jpg|500px]]</p> <p><math>\text{Figure 5.}~~\texttt{((p),(q),(r))}</math></p> |} ==Glossary of basic terms== ; Boolean domain : A ''[[boolean domain]]'' <math>\mathbb{B}</math> is a generic 2-element set, for example, <math>\mathbb{B} = \{ 0, 1 \},</math> whose elements are interpreted as logical values, usually but not invariably with <math>0 = \operatorname{false}</math> and <math>1 = \operatorname{true}.</math> ; Boolean variable : A ''[[boolean variable]]'' <math>x\!</math> is a variable that takes its value from a boolean domain, as <math>x \in \mathbb{B}.</math> ; Proposition : In situations where boolean values are interpreted as logical values, a [[boolean-valued function]] <math>f : X \to \mathbb{B}</math> or a [[boolean function]] <math>g : \mathbb{B}^k \to \mathbb{B}</math> is frequently called a ''[[proposition]]''. ; Basis element, Coordinate projection : Given a sequence of <math>k\!</math> boolean variables, <math>x_1, \ldots, x_k,</math> each variable <math>x_j\!</math> may be treated either as a ''basis element'' of the space <math>\mathbb{B}^k</math> or as a ''coordinate projection'' <math>x_j : \mathbb{B}^k \to \mathbb{B}.</math> ; Basic proposition : This means that the set of objects <math>\{ x_j : 1 \le j \le k \}</math> is a set of boolean functions <math>\{ x_j : \mathbb{B}^k \to \mathbb{B} \}</math> subject to logical interpretation as a set of ''basic propositions'' that collectively generate the complete set of <math>2^{2^k}</math> propositions over <math>\mathbb{B}^k.</math> ; Literal : A ''literal'' is one of the <math>2k\!</math> propositions <math>x_1, \ldots, x_k, \texttt{(} x_1 \texttt{)}, \ldots, \texttt{(} x_k \texttt{)},</math> in other words, either a ''posited'' basic proposition <math>x_j\!</math> or a ''negated'' basic proposition <math>\texttt{(} x_j \texttt{)},</math> for some <math>j = 1 ~\text{to}~ k.</math> ; Fiber : In mathematics generally, the ''[[fiber (mathematics)|fiber]]'' of a point <math>y \in Y</math> under a function <math>f : X \to Y</math> is defined as the inverse image <math>f^{-1}(y) \subseteq X.</math> : In the case of a boolean function <math>f : \mathbb{B}^k \to \mathbb{B},</math> there are just two fibers: : The fiber of <math>0\!</math> under <math>f,\!</math> defined as <math>f^{-1}(0),\!</math> is the set of points where the value of <math>f\!</math> is <math>0.\!</math> : The fiber of <math>1\!</math> under <math>f,\!</math> defined as <math>f^{-1}(1),\!</math> is the set of points where the value of <math>f\!</math> is <math>1.\!</math> ; Fiber of truth : When <math>1\!</math> is interpreted as the logical value <math>\operatorname{true},</math> then <math>f^{-1}(1)\!</math> is called the ''fiber of truth'' in the proposition <math>f.\!</math> Frequent mention of this fiber makes it useful to have a shorter way of referring to it. This leads to the definition of the notation <math>[|f|] = f^{-1}(1)\!</math> for the fiber of truth in the proposition <math>f.\!</math> ; Singular boolean function : A ''singular boolean function'' <math>s : \mathbb{B}^k \to \mathbb{B}</math> is a boolean function whose fiber of <math>1\!</math> is a single point of <math>\mathbb{B}^k.</math> ; Singular proposition : In the interpretation where <math>1\!</math> equals <math>\operatorname{true},</math> a singular boolean function is called a ''singular proposition''. : Singular boolean functions and singular propositions serve as functional or logical representatives of the points in <math>\mathbb{B}^k.</math> ; Singular conjunction : A ''singular conjunction'' in <math>\mathbb{B}^k \to \mathbb{B}</math> is a conjunction of <math>k\!</math> literals that includes just one conjunct of the pair <math>\{ x_j, ~\nu(x_j) \}</math> for each <math>j = 1 ~\text{to}~ k.</math> : A singular proposition <math>s : \mathbb{B}^k \to \mathbb{B}</math> can be expressed as a singular conjunction: {| align="center" cellspacing"10" width="90%" | height="36" | <math>s ~=~ e_1 e_2 \ldots e_{k-1} e_k</math>, |- | <math>\begin{array}{llll} \text{where} & e_j & = & x_j \\[6pt] \text{or} & e_j & = & \nu (x_j), \\[6pt] \text{for} & j & = & 1 ~\text{to}~ k. \end{array}</math> |} ==Resources== * [http://planetmath.org/MinimalNegationOperator Minimal Negation Operator] @ [http://planetmath.org/ PlanetMath] * [http://planetphysics.us/encyclopedia/MinimalNegationOperator.html Minimal Negation Operator] @ [http://planetphysics.us/ PlanetPhysics] * [http://www.proofwiki.org/wiki/Definition:Minimal_Negation_Operator Minimal Negation Operator] @ [http://www.proofwiki.org/ ProofWiki] * [http://atlas.wolfram.com/ Wolfram Atlas of Simple Programs] ** [http://atlas.wolfram.com/01/01/ Elementary Cellular Automata Rules (ECARs)] ** [http://atlas.wolfram.com/01/01/rulelist.html ECAR Index] ** [http://atlas.wolfram.com/01/01/views/3/TableView.html ECAR Icons] ** [http://atlas.wolfram.com/01/01/views/87/TableView.html ECAR Examples] ** [http://atlas.wolfram.com/01/01/views/172/TableView.html ECAR Formulas] ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Minimal_negation_operator Minimal Negation Operator @ InterSciWiki] * [http://mywikibiz.com/Minimal_negation_operator Minimal Negation Operator @ MyWikiBiz] {{col-break}} * [http://beta.wikiversity.org/wiki/Minimal_negation_operator Minimal Negation Operator @ Wikiversity Beta] * [http://ref.subwiki.org/wiki/Minimal_negation_operator Minimal Negation Operator @ Subject Wikis] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Minimal_negation_operator Minimal Negation Operator], [http://mywikibiz.com/ MyWikiBiz] * [http://planetmath.org/MinimalNegationOperator Minimal Negation Operator], [http://planetmath.org/ PlanetMath] * [http://beta.wikiversity.org/wiki/Minimal_negation_operator Minimal Negation Operator], [http://beta.wikiversity.org/ Wikiversity Beta] {{col-break}} * [http://web.archive.org/web/20070703045600/http://wikinfo.org/index.php/Minimal_negation_operator Minimal Negation Operator], [http://wikinfo.org/ Wikinfo] * [http://textop.org/wiki/index.php?title=Minimal_negation_operator Minimal Negation Operator], [http://textop.org/wiki/ Textop Wiki] * [http://web.archive.org/web/20060913000000/http://en.wikipedia.org/wiki/Minimal_negation_operator Minimal Negation Operator], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Automata Theory]] [[Category:Boolean Functions]] [[Category:Charles Sanders Peirce]] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Differential Logic]] [[Category:Equational Reasoning]] [[Category:Formal Languages]] [[Category:Formal Sciences]] [[Category:Formal Systems]] [[Category:Inquiry]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Logical Graphs]] [[Category:Mathematics]] [[Category:Neural Networks]] [[Category:Philosophy]] [[Category:Propositional Calculus]] [[Category:Semiotics]] b44914889bfef33ab3f89fe67a34d54a22ab78ca 0 365 659 2013-10-29T20:12:16Z Jon Awbrey 3 #REDIRECT [[Semeiotic]] wikitext text/x-wiki #REDIRECT [[Semeiotic]] 6ff4dbd091188a5e18db51d7ec4d3305a7ec225d 0 187 666 651 2013-12-01T16:30:09Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''logical graph''' is a graph-theoretic structure in one of the systems of graphical syntax that Charles Sanders Peirce developed for logic. In his papers on ''qualitative logic'', ''entitative graphs'', and ''existential graphs'', Peirce developed several versions of a graphical formalism, or a graph-theoretic formal language, designed to be interpreted for logic. In the century since Peirce initiated this line of development, a variety of formal systems have branched out from what is abstractly the same formal base of graph-theoretic structures. This article examines the common basis of these formal systems from a bird's eye view, focusing on those aspects of form that are shared by the entire family of algebras, calculi, or languages, however they happen to be viewed in a given application. ==Abstract point of view== {| width="100%" cellpadding="2" cellspacing="0" | width="60%" | &nbsp; | width="40%" | ''Wollust ward dem Wurm gegeben &hellip;'' |- | &nbsp; | align="right" | &mdash; Friedrich Schiller, ''An die Freude'' |} The bird's eye view in question is more formally known as the perspective of formal equivalence, from which remove one cannot see many distinctions that appear momentous from lower levels of abstraction. In particular, expressions of different formalisms whose syntactic structures are isomorphic from the standpoint of algebra or topology are not recognized as being different from each other in any significant sense. Though we may note in passing such historical details as the circumstance that Charles Sanders Peirce used a ''streamer-cross symbol'' where George Spencer Brown used a ''carpenter's square marker'', the theme of principal interest at the abstract level of form is neutral with regard to variations of that order. ==In lieu of a beginning== Consider the formal equations indicated in Figures&nbsp;1 and 2. {| align="center" border="0" cellpadding="10" cellspacing="0" | [[Image:Logical_Graph_Figure_1_Visible_Frame.jpg|500px]] || (1) |- | [[Image:Logical_Graph_Figure_2_Visible_Frame.jpg|500px]] || (2) |} For the time being these two forms of transformation may be referred to as ''axioms'' or ''initial equations''. ==Duality : logical and topological== There are two types of duality that have to be kept separately mind in the use of logical graphs &mdash; logical duality and topological duality. There is a standard way that graphs of the order that Peirce considered, those embedded in a continuous manifold like that commonly represented by a plane sheet of paper &mdash; with or without the paper bridges that Peirce used to augment its topological genus &mdash; can be represented in linear text as what are called ''parse strings'' or ''traversal strings'' and parsed into ''pointer structures'' in computer memory. A blank sheet of paper can be represented in linear text as a blank space, but that way of doing it tends to be confusing unless the logical expression under consideration is set off in a separate display. For example, consider the axiom or initial equation that is shown below: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_3_Visible_Frame.jpg|500px]] || (3) |} This can be written inline as <math>{}^{\backprime\backprime} \texttt{(~(~)~)} = \quad {}^{\prime\prime}\!</math> or set off in a text display as follows: {| align="center" cellpadding="10" | width="33%" | <math>\texttt{(~(~)~)}\!</math> | width="34%" | <math>=\!</math> | width="33%" | &nbsp; |} When we turn to representing the corresponding expressions in computer memory, where they can be manipulated with utmost facility, we begin by transforming the planar graphs into their topological duals. The planar regions of the original graph correspond to nodes (or points) of the dual graph, and the boundaries between planar regions in the original graph correspond to edges (or lines) between the nodes of the dual graph. For example, overlaying the corresponding dual graphs on the plane-embedded graphs shown above, we get the following composite picture: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_4_Visible_Frame.jpg|500px]] || (4) |} Though it's not really there in the most abstract topology of the matter, for all sorts of pragmatic reasons we find ourselves compelled to single out the outermost region of the plane in a distinctive way and to mark it as the ''root node'' of the corresponding dual graph. In the present style of Figure the root nodes are marked by horizontal strike-throughs. Extracting the dual graphs from their composite matrices, we get this picture: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_5_Visible_Frame.jpg|500px]] || (5) |} It is easy to see the relationship between the parenthetical expressions of Peirce's logical graphs, that somewhat clippedly picture the ordered containments of their formal contents, and the associated dual graphs, that constitute the species of rooted trees here to be described. In the case of our last example, a moment's contemplation of the following picture will lead us to see that we can get the corresponding parenthesis string by starting at the root of the tree, climbing up the left side of the tree until we reach the top, then climbing back down the right side of the tree until we return to the root, all the while reading off the symbols, in this case either <math>{}^{\backprime\backprime} \texttt{(} {}^{\prime\prime}\!</math> or <math>{}^{\backprime\backprime} \texttt{)} {}^{\prime\prime},\!</math> that we happen to encounter in our travels. {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_6_Visible_Frame.jpg|500px]] || (6) |} This ritual is called ''traversing'' the tree, and the string read off is called the ''traversal string'' of the tree. The reverse ritual, that passes from the string to the tree, is called ''parsing'' the string, and the tree constructed is called the ''parse graph'' of the string. The speakers thereof tend to be a bit loose in this language, often using ''parse string'' to mean the string that gets parsed into the associated graph. We have treated in some detail various forms of the initial equation or logical axiom that is formulated in string form as <math>{}^{\backprime\backprime} \texttt{(~(~)~)} = \quad {}^{\prime\prime}.~\!</math> For the sake of comparison, let's record the plane-embedded and topological dual forms of the axiom that is formulated in string form as <math>{}^{\backprime\backprime} \texttt{(~)(~)} = \texttt{(~)} {}^{\prime\prime}.\!</math> First the plane-embedded maps: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_7_Visible_Frame.jpg|500px]] || (7) |} Next the plane-embedded maps and their dual trees superimposed: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_8_Visible_Frame.jpg|500px]] || (8) |} Finally the dual trees by themselves: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_9_Visible_Frame.jpg|500px]] || (9) |} And here are the parse trees with their traversal strings indicated: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_10_Visible_Frame.jpg|500px]] || (10) |} We have at this point enough material to begin thinking about the forms of analogy, iconicity, metaphor, morphism, whatever you want to call them, that are pertinent to the use of logical graphs in their various logical interpretations, for instance, those that Peirce described as ''entitative graphs'' and ''existential graphs''. ==Computational representation== The parse graphs that we've been looking at so far bring us one step closer to the pointer graphs that it takes to make these maps and trees live in computer memory, but they are still a couple of steps too abstract to detail the concrete species of dynamic data structures that we need. The time has come to flesh out the skeletons that we've drawn up to this point. Nodes in a graph represent ''records'' in computer memory. A record is a collection of data that can be conceived to reside at a specific ''address''. The address of a record is analogous to a demonstrative pronoun, on which account programmers commonly describe it as a ''pointer'' and semioticians recognize it as a type of sign called an ''index''. At the next level of concreteness, a pointer-record structure is represented as follows: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_11_Visible_Frame.jpg|500px]] || (11) |} This portrays the pointer <math>\mathit{index}_0\!</math> as the address of a record that contains the following data: {| align="center" cellpadding="10" | <math>\mathit{datum}_1, \mathit{datum}_2, \mathit{datum}_3, \ldots,\!</math> and so on. |} What makes it possible to represent graph-theoretical structures as data structures in computer memory is the fact that an address is just another datum, and so we may have a state of affairs like the following: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_12_Visible_Frame.jpg|500px]] || (12) |} Returning to the abstract level, it takes three nodes to represent the three data records illustrated above: one root node connected to a couple of adjacent nodes. The items of data that do not point any further up the tree are then treated as labels on the record-nodes where they reside, as shown below: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_13_Visible_Frame.jpg|500px]] || (13) |} Notice that drawing the arrows is optional with rooted trees like these, since singling out a unique node as the root induces a unique orientation on all the edges of the tree, with ''up'' being the same direction as ''away from the root''. ==Quick tour of the neighborhood== This much preparation allows us to take up the founding axioms or initial equations that determine the entire system of logical graphs. ===Primary arithmetic as semiotic system=== Though it may not seem too exciting, logically speaking, there are many reasons to make oneself at home with the system of forms that is represented indifferently, topologically speaking, by rooted trees, balanced strings of parentheses, or finite sets of non-intersecting simple closed curves in the plane. :* One reason is that it gives us a respectable example of a sign domain on which to cut our semiotic teeth, non-trivial in the sense that it contains a countable infinity of signs. :* Another reason is that it allows us to study a simple form of computation that is recognizable as a species of ''[[semiosis]]'', or sign-transforming process. This space of forms, along with the two axioms that induce its partition into exactly two equivalence classes, is what George Spencer Brown called the ''primary arithmetic''. The axioms of the primary arithmetic are shown below, as they appear in both graph and string forms, along with pairs of names that come in handy for referring to the two opposing directions of applying the axioms. {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_14_Banner.jpg|500px]] || (14) |- | [[Image:Logical_Graph_Figure_15_Banner.jpg|500px]] || (15) |} Let <math>S\!</math> be the set of rooted trees and let <math>S_0\!</math> be the 2-element subset of <math>S\!</math> that consists of a rooted node and a rooted edge. {| align="center" cellpadding="10" style="text-align:center" | <math>S\!</math> | <math>=\!</math> | <math>\{ \text{rooted trees} \}\!</math> |- | <math>S_0\!</math> | <math>=\!</math> | <math>\{ \ominus, \vert \} = \{\!</math>[[Image:Rooted Node.jpg|16px]], [[Image:Rooted Edge.jpg|12px]]<math>\}\!</math> |} Simple intuition, or a simple inductive proof, assures us that any rooted tree can be reduced by way of the arithmetic initials either to a root node [[Image:Rooted Node.jpg|16px]] or else to a rooted edge [[Image:Rooted Edge.jpg|12px]]&nbsp;. For example, consider the reduction that proceeds as follows: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_16.jpg|500px]] || (16) |} Regarded as a semiotic process, this amounts to a sequence of signs, every one after the first serving as the ''interpretant'' of its predecessor, ending in a final sign that may be taken as the canonical sign for their common object, in the upshot being the result of the computation process. Simple as it is, this exhibits the main features of any computation, namely, a semiotic process that proceeds from an obscure sign to a clear sign of the same object, being in its aim and effect an action on behalf of clarification. ===Primary algebra as pattern calculus=== Experience teaches that complex objects are best approached in a gradual, laminar, modular fashion, one step, one layer, one piece at a time, and it's just as much the case when the complexity of the object is irreducible, that is, when the articulations of the representation are necessarily at joints that are cloven disjointedly from nature, with some assembly required in the synthetic integrity of the intuition. That's one good reason for spending so much time on the first half of [[zeroth order logic]], represented here by the primary arithmetic, a level of formal structure that C.S. Peirce verged on intuiting at numerous points and times in his work on logical graphs, and that Spencer Brown named and brought more completely to life. There is one other reason for lingering a while longer in these primitive forests, and this is that an acquaintance with &ldquo;bare trees&rdquo;, those as yet unadorned with literal or numerical labels, will provide a firm basis for understanding what's really at issue in such problems as the &ldquo;ontological status of variables&rdquo;. It is probably best to illustrate this theme in the setting of a concrete case, which we can do by reviewing the previous example of reductive evaluation shown in Figure&nbsp;16. The observation of several ''semioses'', or sign-transformations, of roughly this shape will most likely lead an observer with any observational facility whatever to notice that it doesn't really matter what sorts of branches happen to sprout from the side of the root aside from the lone edge that also grows there &mdash; the end result will always be the same. Our observer might think to summarize the results of many such observations by introducing a label or variable to signify any shape of branch whatever, writing something like the following: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_17.jpg|500px]] || (17) |} Observations like that, made about an arithmetic of any variety, germinated by their summarizations, are the root of all algebra. Speaking of algebra, and having encountered already one example of an algebraic law, we might as well introduce the axioms of the ''primary algebra'', once again deriving their substance and their name from the works of Charles Sanders Peirce and George Spencer Brown, respectively. {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_18.jpg|500px]] || (18) |- | [[Image:Logical_Graph_Figure_19.jpg|500px]] || (19) |} The choice of axioms for any formal system is to some degree a matter of aesthetics, as it is commonly the case that many different selections of formal rules will serve as axioms to derive all the rest as theorems. As it happens, the example of an algebraic law that we noticed first, <math>a(~) = (~),\!</math> as simple as it appears, proves to be provable as a theorem on the grounds of the foregoing axioms. We might also notice at this point a subtle difference between the primary arithmetic and the primary algebra with respect to the grounds of justification that we have naturally if tacitly adopted for their respective sets of axioms. The arithmetic axioms were introduced by fiat, in an ''a&nbsp;priori'' fashion, though of course it is only long prior experience with the practical uses of comparably developed generations of formal systems that would actually induce us to such a quasi-primal move. The algebraic axioms, in contrast, can be seen to derive their motive and their justice from the observation and summarization of patterns that are visible in the arithmetic spectrum. ==Formal development== What precedes this point is intended as an informal introduction to the axioms of the primary arithmetic and primary algebra, and hopefully provides the reader with an intuitive sense of their motivation and rationale. The next order of business is to give the exact forms of the axioms that are used in the following more formal development, devolving from Peirce's various systems of logical graphs via Spencer-Brown's ''Laws of Form'' (LOF). In formal proofs, a variation of the annotation scheme from LOF will be used to mark each step of the proof according to which axiom, or ''initial'', is being invoked to justify the corresponding step of syntactic transformation, whether it applies to graphs or to strings. ===Axioms=== The axioms are just four in number, divided into the ''arithmetic initials'', <math>I_1\!</math> and <math>I_2,\!</math> and the ''algebraic initials'', <math>J_1\!</math> and <math>J_2.\!</math> {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_20.jpg|500px]] || (20) |- | [[Image:Logical_Graph_Figure_21.jpg|500px]] || (21) |- | [[Image:Logical_Graph_Figure_22.jpg|500px]] || (22) |- | [[Image:Logical_Graph_Figure_23.jpg|500px]] || (23) |} One way of assigning logical meaning to the initial equations is known as the ''entitative interpretation'' (EN). Under EN, the axioms read as follows: {| align="center" cellpadding="10" | <math>\begin{array}{ccccc} I_1 & : & \operatorname{true}\ \operatorname{or}\ \operatorname{true} & = & \operatorname{true} \\ I_2 & : & \operatorname{not}\ \operatorname{true}\ & = & \operatorname{false} \\ J_1 & : & a\ \operatorname{or}\ \operatorname{not}\ a & = & \operatorname{true} \\ J_2 & : & (a\ \operatorname{or}\ b)\ \operatorname{and}\ (a\ \operatorname{or}\ c) & = & a\ \operatorname{or}\ (b\ \operatorname{and}\ c) \\ \end{array}</math> |} Another way of assigning logical meaning to the initial equations is known as the ''existential interpretation'' (EX). Under EX, the axioms read as follows: {| align="center" cellpadding="10" | <math>\begin{array}{ccccc} I_1 & : & \operatorname{false}\ \operatorname{and}\ \operatorname{false} & = & \operatorname{false} \\ I_2 & : & \operatorname{not}\ \operatorname{false} & = & \operatorname{true} \\ J_1 & : & a\ \operatorname{and}\ \operatorname{not}\ a & = & \operatorname{false} \\ J_2 & : & (a\ \operatorname{and}\ b)\ \operatorname{or}\ (a\ \operatorname{and}\ c) & = & a\ \operatorname{and}\ (b\ \operatorname{or}\ c) \\ \end{array}</math> |} All of the axioms in this set have the form of equations. This means that all of the inference steps that they allow are reversible. The proof annotation scheme employed below makes use of a double bar <math>\overline{\underline{~~~~~~}}\!</math> to mark this fact, although it will often be left to the reader to decide which of the two possible directions is the one required for applying the indicated axiom. ===Frequently used theorems=== The actual business of proof is a far more strategic affair than the simple cranking of inference rules might suggest. Part of the reason for this lies in the circumstance that the usual brands of inference rules combine the moving forward of a state of inquiry with the losing of information along the way that doesn't appear to be immediately relevant, at least, not as viewed in the local focus and the short run of the moment to moment proceedings of the proof in question. Over the long haul, this has the pernicious side-effect that one is forever strategically required to reconstruct much of the information that one had strategically thought to forget in earlier stages of the proof, if "before the proof started" can be counted as an earlier stage of the proof in view. For this reason, among others, it is very instructive to study equational inference rules of the sort that our axioms have just provided. Although equational forms of reasoning are paramount in mathematics, they are less familiar to the student of conventional logic textbooks, who may find a few surprises here. By way of gaining a minimal experience with how equational proofs look in the present forms of syntax, let us examine the proofs of a few essential theorems in the primary algebra. ====C₁. Double negation==== The first theorem goes under the names of ''Consequence&nbsp;1'' <math>(C_1)\!</math>, the ''double negation theorem'' (DNT), or ''Reflection''. {| align="center" cellpadding="10" | [[Image:Double Negation 1.0 Splash Page.png|500px]] || (24) |} The proof that follows is adapted from the one that was given by George Spencer Brown in his book ''Laws of Form'' (LOF) and credited to two of his students, John Dawes and D.A. Utting. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Double Negation 1.0 Marquee Title.png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 1.png|500px]] |- | [[Image:Equational Inference I2 Elicit (( )).png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 2.png|500px]] |- | [[Image:Equational Inference J1 Insert (a).png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 3.png|500px]] |- | [[Image:Equational Inference J2 Distribute ((a)).png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 4.png|500px]] |- | [[Image:Equational Inference J1 Delete (a).png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 5.png|500px]] |- | [[Image:Equational Inference J1 Insert a.png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 6.png|500px]] |- | [[Image:Equational Inference J2 Collect a.png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 7.png|500px]] |- | [[Image:Equational Inference J1 Delete ((a)).png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 8.png|500px]] |- | [[Image:Equational Inference I2 Cancel (( )).png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 9.png|500px]] |- | [[Image:Equational Inference Marquee QED.png|500px]] |} | (25) |} The steps of this proof are replayed in the following animation. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Double Negation 2.0 Animation.gif]] |} | (26) |} ====C₂. Generation theorem==== One theorem of frequent use goes under the nickname of the ''weed and seed theorem'' (WAST). The proof is just an exercise in mathematical induction, once a suitable basis is laid down, and it will be left as an exercise for the reader. What the WAST says is that a label can be freely distributed or freely erased anywhere in a subtree whose root is labeled with that label. The second in our list of frequently used theorems is in fact the base case of this weed and seed theorem. In LOF, it goes by the names of ''Consequence&nbsp;2'' <math>(C_2)\!</math> or ''Generation''. {| align="center" cellpadding="10" | [[Image:Generation Theorem 1.0 Splash Page.png|500px]] || (27) |} Here is a proof of the Generation Theorem. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Generation Theorem 1.0 Marquee Title.png|500px]] |- | [[Image:Generation Theorem 1.0 Storyboard 1.png|500px]] |- | [[Image:Equational Inference C1 Reflect a(b).png|500px]] |- | [[Image:Generation Theorem 1.0 Storyboard 2.png|500px]] |- | [[Image:Equational Inference I2 Elicit (( )).png|500px]] |- | [[Image:Generation Theorem 1.0 Storyboard 3.png|500px]] |- | [[Image:Equational Inference J1 Insert a.png|500px]] |- | [[Image:Generation Theorem 1.0 Storyboard 4.png|500px]] |- | [[Image:Equational Inference J2 Collect a.png|500px]] |- | [[Image:Generation Theorem 1.0 Storyboard 5.png|500px]] |- | [[Image:Equational Inference C1 Reflect a, b.png|500px]] |- | [[Image:Generation Theorem 1.0 Storyboard 6.png|500px]] |- | [[Image:Equational Inference Marquee QED.png|500px]] |} | (28) |} The steps of this proof are replayed in the following animation. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Generation Theorem 2.0 Animation.gif]] |} | (29) |} ====C₃. Dominant form theorem==== The third of the frequently used theorems of service to this survey is one that Spencer-Brown annotates as ''Consequence&nbsp;3'' <math>(C_3)\!</math> or ''Integration''. A better mnemonic might be ''dominance and recession theorem'' (DART), but perhaps the brevity of ''dominant form theorem'' (DFT) is sufficient reminder of its double-edged role in proofs. {| align="center" cellpadding="10" | [[Image:Dominant Form 1.0 Splash Page.png|500px]] || (30) |} Here is a proof of the Dominant Form Theorem. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Dominant Form 1.0 Marquee Title.png|500px]] |- | [[Image:Dominant Form 1.0 Storyboard 1.png|500px]] |- | [[Image:Equational Inference C2 Regenerate a.png|500px]] |- | [[Image:Dominant Form 1.0 Storyboard 2.png|500px]] |- | [[Image:Equational Inference J1 Delete a.png|500px]] |- | [[Image:Dominant Form 1.0 Storyboard 3.png|500px]] |- | [[Image:Equational Inference Marquee QED.png|500px]] |} | (31) |} The following animation provides an instant re*play. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Dominant Form 2.0 Animation.gif]] |} | (32) |} ===Exemplary proofs=== Based on the axioms given at the outest, and aided by the theorems recorded so far, it is possible to prove a multitude of much more complex theorems. A couple of all-time favorites are given next. ====Peirce's law==== : ''Main article'' : [[Peirce's law]] Peirce's law is commonly written in the following form: {| align="center" cellpadding="10" | <math>((p \Rightarrow q) \Rightarrow p) \Rightarrow p\!</math> |} The existential graph representation of Peirce's law is shown in Figure&nbsp;33. {| align="center" cellpadding="10" | [[Image:Peirce's Law 1.0 Splash Page.png|500px]] || (33) |} A graphical proof of Peirce's law is shown in Figure&nbsp;34. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Peirce's Law 1.0 Marquee Title.png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 1.png|500px]] |- | [[Image:Equational Inference Band Collect p.png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 2.png|500px]] |- | [[Image:Equational Inference Band Quit ((q)).png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 3.png|500px]] |- | [[Image:Equational Inference Band Cancel (( )).png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 4.png|500px]] |- | [[Image:Equational Inference Band Delete p.png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 5.png|500px]] |- | [[Image:Equational Inference Band Cancel (( )).png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 6.png|500px]] |- | [[Image:Equational Inference Marquee QED.png|500px]] |} | (34) |} The following animation replays the steps of the proof. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Peirce's Law 2.0 Animation.gif]] |} | (35) |} ====Praeclarum theorema==== An illustrious example of a propositional theorem is the ''praeclarum theorema'', the ''admirable'', ''shining'', or ''splendid'' theorem of Leibniz. {| align="center" cellspacing="6" width="90%" <!--QUOTE--> | <p>If ''a'' is ''b'' and ''d'' is ''c'', then ''ad'' will be ''bc''.</p> <p>This is a fine theorem, which is proved in this way:</p> <p>''a'' is ''b'', therefore ''ad'' is ''bd'' (by what precedes),</p> <p>''d'' is ''c'', therefore ''bd'' is ''bc'' (again by what precedes),</p> <p>''ad'' is ''bd'', and ''bd'' is ''bc'', therefore ''ad'' is ''bc''. Q.E.D.</p> <p>(Leibniz, ''Logical Papers'', p. 41).</p> |} Under the existential interpretation, the praeclarum theorema is represented by means of the following logical graph. {| align="center" cellpadding="10" | [[Image:Praeclarum Theorema 1.0 Splash Page.png|500px]] || (36) |} And here's a neat proof of that nice theorem. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Praeclarum Theorema 1.0 Marquee Title.png|500px]] |- | [[Image:Praeclarum Theorema 1.0 Storyboard 1.png|500px]] |- | [[Image:Equational Inference Rule Reflect ad(bc).png|500px]] |- | [[Image:Praeclarum Theorema 1.0 Storyboard 2.png|500px]] |- | [[Image:Equational Inference Rule Weed a, d.png|500px]] |- | [[Image:Praeclarum Theorema 1.0 Storyboard 3.png|500px]] |- | [[Image:Equational Inference Rule Reflect b, c.png|500px]] |- | [[Image:Praeclarum Theorema 1.0 Storyboard 4.png|500px]] |- | [[Image:Equational Inference Rule Weed bc.png|500px]] |- | [[Image:Praeclarum Theorema 1.0 Storyboard 5.png|500px]] |- | [[Image:Equational Inference Rule Quit abcd.png|500px]] |- | [[Image:Praeclarum Theorema 1.0 Storyboard 6.png|500px]] |- | [[Image:Equational Inference Rule Cancel (( )).png|500px]] |- | [[Image:Praeclarum Theorema 1.0 Storyboard 7.png|500px]] |- | [[Image:Equational Inference Marquee QED.png|500px]] |} | (37) |} The steps of the proof are replayed in the following animation. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Praeclarum Theorema 2.0 Animation.gif]] |} | (38) |} ====Two-thirds majority function==== Consider the following equation in boolean algebra, posted as a [http://mathoverflow.net/questions/9292/newbie-boolean-algebra-question problem for proof] at [http://mathoverflow.net/ MathOverFlow]. {| align="center" cellpadding="20" | <math>\begin{matrix} a b \bar{c} + a \bar{b} c + \bar{a} b c + a b c \\[6pt] \iff \\[6pt] a b + a c + b c \end{matrix}</math> | &nbsp;&nbsp;&nbsp;&nbsp; |} The required equation can be proven in the medium of logical graphs as shown in the following Figure. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Two-Thirds Majority Eq 1 Pf 1 Banner Title.png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 1.png|500px]] |- | [[Image:Equational Inference Bar Reflect ab, ac, bc.png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 2.png|500px]] |- | [[Image:Equational Inference Bar Distribute (abc).png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 3.png|500px]] |- | [[Image:Equational Inference Bar Collect ab, ac, bc.png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 4.png|500px]] |- | [[Image:Equational Inference Bar Quit (a), (b), (c).png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 5.png|500px]] |- | [[Image:Equational Inference Bar Cancel (( )).png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 6.png|500px]] |- | [[Image:Equational Inference Bar Weed ab, ac, bc.png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 7.png|500px]] |- | [[Image:Equational Inference Bar Delete a, b, c.png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 8.png|500px]] |- | [[Image:Equational Inference Bar Cancel (( )).png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 9.png|500px]] |- | [[Image:Equational Inference Banner QED.png|500px]] |} | (39) |} Here's an animated recap of the graphical transformations that occur in the above proof: {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Two-Thirds Majority Function 500 x 250 Animation.gif]] |} | (40) |} ==Bibliography== * Leibniz, G.W. (1679&ndash;1686 ?), &ldquo;Addenda to the Specimen of the Universal Calculus&rdquo;, pp. 40&ndash;46 in G.H.R. Parkinson (ed. and trans., 1966), ''Leibniz : Logical Papers'', Oxford University Press, London, UK. * Peirce, C.S. (1931&ndash;1935, 1958), ''Collected Papers of Charles Sanders Peirce'', vols. 1&ndash;6, Charles Hartshorne and Paul Weiss (eds.), vols. 7&ndash;8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA. Cited as (CP&nbsp;volume.paragraph). * Peirce, C.S. (1981&ndash;), ''Writings of Charles S. Peirce : A Chronological Edition'', Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN. Cited as (CE&nbsp;volume, page). * Peirce, C.S. (1885), &ldquo;On the Algebra of Logic : A Contribution to the Philosophy of Notation&rdquo;, ''American Journal of Mathematics'' 7 (1885), 180&ndash;202. Reprinted as CP&nbsp;3.359&ndash;403 and CE&nbsp;5, 162&ndash;190. * Peirce, C.S. (''c.'' 1886), &ldquo;Qualitative Logic&rdquo;, MS 736. Published as pp. 101&ndash;115 in Carolyn Eisele (ed., 1976), ''The New Elements of Mathematics by Charles S. Peirce, Volume 4, Mathematical Philosophy'', Mouton, The Hague. * Peirce, C.S. (1886 a), &ldquo;Qualitative Logic&rdquo;, MS 582. Published as pp. 323&ndash;371 in ''Writings of Charles S. Peirce : A Chronological Edition, Volume 5, 1884&ndash;1886'', Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1993. * Peirce, C.S. (1886 b), &ldquo;The Logic of Relatives : Qualitative and Quantitative&rdquo;, MS 584. Published as pp. 372&ndash;378 in ''Writings of Charles S. Peirce : A Chronological Edition, Volume 5, 1884&ndash;1886'', Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1993. * Spencer Brown, George (1969), ''Laws of Form'', George Allen and Unwin, London, UK. ==Resources== * [http://planetmath.org/ PlanetMath] ** [http://planetmath.org/LogicalGraphIntroduction Logical Graph : Introduction] ** [http://planetmath.org/LogicalGraphFormalDevelopment Logical Graph : Formal Development] * Bergman and Paavola (eds.), [http://www.helsinki.fi/science/commens/dictionary.html Commens Dictionary of Peirce's Terms] ** [http://www.helsinki.fi/science/commens/terms/graphexis.html Existential Graph] ** [http://www.helsinki.fi/science/commens/terms/graphlogi.html Logical Graph] * [http://dr-dau.net/index.shtml Dau, Frithjof] ** [http://dr-dau.net/eg_readings.shtml Peirce's Existential Graphs : Readings and Links] ** [http://web.archive.org/web/20070706192257/http://dr-dau.net/pc.shtml Existential Graphs as Moving Pictures of Thought] &mdash; Computer Animated Proof of Leibniz's Praeclarum Theorema * [http://www.math.uic.edu/~kauffman/ Kauffman, Louis H.] ** [http://www.math.uic.edu/~kauffman/Arithmetic.htm Box Algebra, Boundary Mathematics, Logic, and Laws of Form] * [http://mathworld.wolfram.com/ MathWorld : A Wolfram Web Resource] ** [http://mathworld.wolfram.com/about/author.html Weisstein, Eric W.], [http://mathworld.wolfram.com/Spencer-BrownForm.html Spencer-Brown Form] * Shoup, Richard (ed.), [http://www.lawsofform.org/ Laws of Form Web Site] ** Spencer-Brown, George (1973), [http://www.lawsofform.org/aum/session1.html Transcript Session One], [http://www.lawsofform.org/aum/ AUM Conference], Esalen, CA. ==Translations== * [http://pt.wikipedia.org/wiki/Grafo_l%C3%B3gico Grafo lógico], [http://pt.wikipedia.org/ Portuguese Wikipedia]. ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Logical_graph Logical Graph @ InterSciWiki] * [http://mywikibiz.com/Logical_graph Logical Graph @ MyWikiBiz] {{col-break}} * [http://beta.wikiversity.org/wiki/Logical_graph Logical Graph @ Wikiversity Beta] * [http://ref.subwiki.org/wiki/Logical_graph Logical Graph @ Subject Wikis] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Logical_graph Logical Graph], [http://mywikibiz.com/ MyWikiBiz] * [http://mathweb.org/wiki/Logical_graph Logical Graph], [http://mathweb.org/wiki/ MathWeb Wiki] * [http://semanticweb.org/wiki/Logical_graph Logical Graph], [http://semanticweb.org/ Semantic Web] {{col-break}} * [http://planetmath.org/LogicalGraphIntroduction Logical Graph 1], [http://planetmath.org/LogicalGraphFormalDevelopment Logical Graph 2], [http://planetmath.org/ PlanetMath] * [http://beta.wikiversity.org/wiki/Logical_graph Logical Graph], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://web.archive.org/web/20100527220558/http://www.proofwiki.org/wiki/Definition:Logical_Graph Logical Graph], [http://proofwiki.org/ ProofWiki] {{col-break}} * [http://web.archive.org/web/20100528070412/http://wikinfo.org/index.php/Logical_graph Logical Graph], [http://wikinfo.org/ Wikinfo] * [http://textop.org/wiki/index.php?title=Logical_graph Logical Graph], [http://textop.org/wiki/ Textop Wiki] * [http://en.wikipedia.org/w/index.php?title=Logical_graph&oldid=67277491 Logical Graph], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Artificial Intelligence]] [[Category:Boolean Functions]] [[Category:Charles Sanders Peirce]] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Cybernetics]] [[Category:Equational Reasoning]] [[Category:Formal Languages]] [[Category:Formal Systems]] [[Category:George Spencer Brown]] [[Category:Graph Theory]] [[Category:History of Logic]] [[Category:History of Mathematics]] [[Category:Inquiry]] [[Category:Knowledge Representation]] [[Category:Laws of Form]] [[Category:Logic]] [[Category:Logical Graphs]] [[Category:Mathematics]] [[Category:Philosophy]] [[Category:Propositional Calculus]] [[Category:Semiotics]] [[Category:Visualization]] b5f39cf5db483002f1e7bf9002b8aa1eed1e0626 0 1 668 312 2013-12-17T22:08:47Z Vipul 2 wikitext text/x-wiki Here is our list of subject wikis: {{:Ref:Subwiki list}} 487789b25f06480cfee999c87a75ea8fcf4a172c 673 668 2013-12-17T22:47:02Z Vipul 2 wikitext text/x-wiki {{DISPLAYTITLE:The Subject Wikis Reference Guide}} Here is our list of subject wikis: {{:Ref:Subwiki list}} a95a98d79a855f95d048a29b156c4a83c28fa530 681 673 2013-12-19T00:00:40Z Vipul 2 wikitext text/x-wiki {{DISPLAYTITLE:The Subject Wikis Reference Guide}} Here is our list of subject wikis: {{:Ref:Subwiki list}} You can access the history of subject wikis [[Subwiki:History of subject wikis|here]]. a5552324a34fd1a8f1f57449bff9f32a607f2832 692 681 2013-12-19T00:34:18Z Vipul 2 wikitext text/x-wiki {{DISPLAYTITLE:The Subject Wikis Reference Guide}} Here is our list of subject wikis: {{:Subwiki:Subwiki list}} You can access the history of subject wikis [[Subwiki:History of subject wikis|here]]. 258b2f4b795ec9c72f732e9f4d23505d2a74f486 693 692 2013-12-19T00:34:29Z Vipul 2 wikitext text/x-wiki {{DISPLAYTITLE:The Subject Wikis Reference Guide}} Here is our list of subject wikis: {{:Subwiki:List of subject wikis}} You can access the history of subject wikis [[Subwiki:History of subject wikis|here]]. 5bf96aae8f1c6d2a6a52861b7b23758296edfea9 694 693 2013-12-30T04:21:23Z Vipul 2 wikitext text/x-wiki {{DISPLAYTITLE:The Subject Wikis Reference Guide}} Here is our list of subject wikis: {{:Subwiki:List of subject wikis}} You can access the history of subject wikis [[Subwiki:History of subject wikis|here]]. 476696848789b383bceee5097781542935506490 106 22 669 627 2013-12-17T22:18:25Z Vipul 2 wikitext text/x-wiki {| class="sortable" border="1" ! Wiki !! Main topic !! Number of pages (lower bound) !! Creation month !! State of development !! Content actively being added? !! Error log maintained? |- | [[Groupprops:Main Page|Groupprops]] || Group theory || 7200 || December 2006 || beta || Occasionally || [[Groupprops:Groupprops:Error log|here]] |- | [[Topospaces:Main Page|Topospaces]] || Topology (point set, algebraic) || 600 || May 2007 || alpha || No || [[Topospaces:Topospaces:Error log|here]] |- | [[Commalg:Main Page|Commalg]] || Commutative algebra || 400 || January 2007 || alpha || No || [[Commalg:Commalg:Error log|here]] |- | [[Diffgeom:Main Page|Diffgeom]] || Differential geometry || 400 || February 2007 || alpha || No || [[Diffgeom:Diffgeom:Error log|here]] |- | [[Number:Main Page|Number]] || Number theory || 200 || March 2009 || alpha || No || [[Number:Number:Error log|here]] |- | [[Market:Main Page|Market]] || Economic theory of markets, choices, and prices || 100 || January 2009 || alpha || Occasionally || [[Market:Market:Error log|here]] |- | [[Mech:Main Page|Mech]] || Classical mechanics || 50 || January 2009 || alpha || No || [[Mech:Mech:Error log|here]] |- | [[Calculus:Main Page|Calculus]] || Calculus || 50 || August 2011 || alpha || No || [[Calculus:Calculus:Error log|here]] |- | [[Learning:Main Page|Learning]] || Learning, teaching, and pedagogy || 25 || alpha || Occasionally || -- |} '''NOTE''': The creation month need not coincide with the month of MediaWiki installation. In cases where the wiki was ported from elsewhere, the creation month may be earlier; in other cases, it may be later if the first edits happened long after installation. a063ead6088d93a152c864e73e4e5775e7a76aba 670 669 2013-12-17T22:18:39Z Vipul 2 wikitext text/x-wiki {| class="sortable" border="1" ! Wiki !! Main topic !! Number of pages (lower bound) !! Creation month !! State of development !! Content actively being added? !! Error log maintained? |- | [[Groupprops:Main Page|Groupprops]] || Group theory || 7200 || December 2006 || beta || Occasionally || [[Groupprops:Groupprops:Error log|here]] |- | [[Topospaces:Main Page|Topospaces]] || Topology (point set, algebraic) || 600 || May 2007 || alpha || No || [[Topospaces:Topospaces:Error log|here]] |- | [[Commalg:Main Page|Commalg]] || Commutative algebra || 400 || January 2007 || alpha || No || [[Commalg:Commalg:Error log|here]] |- | [[Diffgeom:Main Page|Diffgeom]] || Differential geometry || 400 || February 2007 || alpha || No || [[Diffgeom:Diffgeom:Error log|here]] |- | [[Number:Main Page|Number]] || Number theory || 200 || March 2009 || alpha || No || [[Number:Number:Error log|here]] |- | [[Market:Main Page|Market]] || Economic theory of markets, choices, and prices || 100 || January 2009 || alpha || Occasionally || [[Market:Market:Error log|here]] |- | [[Mech:Main Page|Mech]] || Classical mechanics || 50 || January 2009 || alpha || No || [[Mech:Mech:Error log|here]] |- | [[Calculus:Main Page|Calculus]] || Calculus || 50 || August 2011 || alpha || No || [[Calculus:Calculus:Error log|here]] |- | [[Learning:Main Page|Learning]] || Learning, teaching, and pedagogy || 25 || September 2013 || alpha || Occasionally || -- |} '''NOTE''': The creation month need not coincide with the month of MediaWiki installation. In cases where the wiki was ported from elsewhere, the creation month may be earlier; in other cases, it may be later if the first edits happened long after installation. 8d12826b2ba7df8bc98d0bb661481a64e4763b7a 671 670 2013-12-17T22:19:59Z Vipul 2 wikitext text/x-wiki {| class="sortable" border="1" ! Wiki !! Main topic !! Number of pages (lower bound) !! Creation month !! State of development !! Content actively being added? !! Error log maintained? |- | [[Groupprops:Main Page|Groupprops]] || Group theory || 7200 || December 2006 || beta || Occasionally || [[Groupprops:Groupprops:Error log|here]] |- | [[Topospaces:Main Page|Topospaces]] || Topology (point set, algebraic) || 600 || May 2007 || alpha || No || [[Topospaces:Topospaces:Error log|here]] |- | [[Commalg:Main Page|Commalg]] || Commutative algebra || 400 || January 2007 || alpha || No || No errors reported yet |- | [[Diffgeom:Main Page|Diffgeom]] || Differential geometry || 400 || February 2007 || alpha || No || No errors reported yet |- | [[Number:Main Page|Number]] || Number theory || 200 || March 2009 || alpha || No || No errors reported yet |- | [[Market:Main Page|Market]] || Economic theory of markets, choices, and prices || 100 || January 2009 || alpha || Occasionally || [[Market:Market:Error log|here]] |- | [[Mech:Main Page|Mech]] || Classical mechanics || 50 || January 2009 || alpha || No || [[Mech:Mech:Error log|here]] |- | [[Calculus:Main Page|Calculus]] || Calculus || 50 || August 2011 || alpha || No || [[Calculus:Calculus:Error log|here]] |- | [[Learning:Main Page|Learning]] || Learning, teaching, and pedagogy || 25 || September 2013 || alpha || Occasionally || -- |} '''NOTE''': The creation month need not coincide with the month of MediaWiki installation. In cases where the wiki was ported from elsewhere, the creation month may be earlier; in other cases, it may be later if the first edits happened long after installation. 45caa37f4d3dec6eff84a35d27deacdc56167e10 672 671 2013-12-17T22:28:48Z Vipul 2 wikitext text/x-wiki {| class="sortable" border="1" ! Wiki !! Main topic !! Number of pages (lower bound) !! Creation month !! State of development !! Content actively being added? !! Error log maintained? |- | [[Groupprops:Main Page|Groupprops]] || Group theory || 7200 || December 2006 || beta || Occasionally || [[Groupprops:Groupprops:Error log|here]] |- | [[Topospaces:Main Page|Topospaces]] || Topology (point set, algebraic) || 600 || May 2007 || alpha || No || [[Topospaces:Topospaces:Error log|here]] |- | [[Commalg:Main Page|Commalg]] || Commutative algebra || 400 || January 2007 || alpha || No || No errors reported yet |- | [[Diffgeom:Main Page|Diffgeom]] || Differential geometry || 400 || February 2007 || alpha || No || No errors reported yet |- | [[Number:Main Page|Number]] || Number theory || 200 || March 2009 || alpha || No || No errors reported yet |- | [[Market:Main Page|Market]] || Economic theory of markets, choices, and prices || 100 || January 2009 || alpha || Occasionally || [[Market:Market:Error log|here]] |- | [[Mech:Main Page|Mech]] || Classical mechanics || 50 || January 2009 || alpha || No || [[Mech:Mech:Error log|here]] |- | [[Calculus:Main Page|Calculus]] || Calculus || 50 || August 2011 || alpha || No || [[Calculus:Calculus:Error log|here]] |- | [[Learning:Main Page|Learning]] || Learning, teaching, and mastery || 25 || September 2013 || alpha || Occasionally || -- |} '''NOTE''': The creation month need not coincide with the month of MediaWiki installation. In cases where the wiki was ported from elsewhere, the creation month may be earlier; in other cases, it may be later if the first edits happened long after installation. 0f0b39b69fe3e874b3a3e043279622b5f3eb70c3 674 672 2013-12-18T03:53:03Z Vipul 2 wikitext text/x-wiki {| class="sortable" border="1" ! Wiki !! Main topic !! Number of pages (lower bound) !! Creation month !! State of development !! Content being added? !! Error log |- | [[Groupprops:Main Page|Groupprops]] || Group theory || 7200 || December 2006 || beta || Occasionally || [[Groupprops:Groupprops:Error log|here]] |- | [[Topospaces:Main Page|Topospaces]] || Topology (point set, algebraic) || 600 || May 2007 || alpha || No || [[Topospaces:Topospaces:Error log|here]] |- | [[Commalg:Main Page|Commalg]] || Commutative algebra || 400 || January 2007 || alpha || No || Not yet |- | [[Diffgeom:Main Page|Diffgeom]] || Differential geometry || 400 || February 2007 || alpha || No || Not yet |- | [[Number:Main Page|Number]] || Number theory || 200 || March 2009 || alpha || No || Not yet |- | [[Market:Main Page|Market]] || Economic theory of markets, choices, and prices || 100 || January 2009 || alpha || Occasionally || [[Market:Market:Error log|here]] |- | [[Mech:Main Page|Mech]] || Classical mechanics || 50 || January 2009 || alpha || No || [[Mech:Mech:Error log|here]] |- | [[Calculus:Main Page|Calculus]] || Calculus || 50 || August 2011 || alpha || Occasionally || [[Calculus:Calculus:Error log|here]] |- | [[Learning:Main Page|Learning]] || Learning, teaching, and mastery || 25 || September 2013 || alpha || Occasionally || -- |} '''NOTE''': The creation month need not coincide with the month of MediaWiki installation. In cases where the wiki was ported from elsewhere, the creation month may be earlier; in other cases, it may be later if the first edits happened long after installation. fdbbbfc4cfa50c480f511a6fb36de1a1ee0bb45c 675 674 2013-12-18T03:53:22Z Vipul 2 wikitext text/x-wiki {| class="sortable" border="1" ! Wiki !! Main topic !! Pagecount (lower bound) !! Creation month !! State of development !! Content being added? !! Error log |- | [[Groupprops:Main Page|Groupprops]] || Group theory || 7200 || December 2006 || beta || Occasionally || [[Groupprops:Groupprops:Error log|here]] |- | [[Topospaces:Main Page|Topospaces]] || Topology (point set, algebraic) || 600 || May 2007 || alpha || No || [[Topospaces:Topospaces:Error log|here]] |- | [[Commalg:Main Page|Commalg]] || Commutative algebra || 400 || January 2007 || alpha || No || Not yet |- | [[Diffgeom:Main Page|Diffgeom]] || Differential geometry || 400 || February 2007 || alpha || No || Not yet |- | [[Number:Main Page|Number]] || Number theory || 200 || March 2009 || alpha || No || Not yet |- | [[Market:Main Page|Market]] || Economic theory of markets, choices, and prices || 100 || January 2009 || alpha || Occasionally || [[Market:Market:Error log|here]] |- | [[Mech:Main Page|Mech]] || Classical mechanics || 50 || January 2009 || alpha || No || [[Mech:Mech:Error log|here]] |- | [[Calculus:Main Page|Calculus]] || Calculus || 50 || August 2011 || alpha || Occasionally || [[Calculus:Calculus:Error log|here]] |- | [[Learning:Main Page|Learning]] || Learning, teaching, and mastery || 25 || September 2013 || alpha || Occasionally || -- |} '''NOTE''': The creation month need not coincide with the month of MediaWiki installation. In cases where the wiki was ported from elsewhere, the creation month may be earlier; in other cases, it may be later if the first edits happened long after installation. b2b10d10ecb091d18027eac9027302803030f4f1 676 675 2013-12-18T04:03:06Z Vipul 2 wikitext text/x-wiki {| class="sortable" border="1" ! Wiki !! Main topic !! Page count (lower bound) !! Creation month !! State of development !! Content being added? !! Error log |- | [[Groupprops:Main Page|Groupprops]] || Group theory || 7200 || December 2006 || beta || Occasionally || [[Groupprops:Groupprops:Error log|here]] |- | [[Topospaces:Main Page|Topospaces]] || Topology (point set, algebraic) || 600 || May 2007 || alpha || No || [[Topospaces:Topospaces:Error log|here]] |- | [[Commalg:Main Page|Commalg]] || Commutative algebra || 400 || January 2007 || alpha || No || Not yet |- | [[Diffgeom:Main Page|Diffgeom]] || Differential geometry || 400 || February 2007 || alpha || No || Not yet |- | [[Number:Main Page|Number]] || Number theory || 200 || March 2009 || alpha || No || Not yet |- | [[Market:Main Page|Market]] || Economic theory of markets, choices, and prices || 100 || January 2009 || alpha || Occasionally || [[Market:Market:Error log|here]] |- | [[Mech:Main Page|Mech]] || Classical mechanics || 50 || January 2009 || alpha || No || [[Mech:Mech:Error log|here]] |- | [[Calculus:Main Page|Calculus]] || Calculus || 50 || August 2011 || alpha || Occasionally || [[Calculus:Calculus:Error log|here]] |- | [[Learning:Main Page|Learning]] || Learning, teaching, and mastery || 25 || September 2013 || alpha || Occasionally || -- |} '''NOTE''': The creation month need not coincide with the month of MediaWiki installation. In cases where the wiki was ported from elsewhere, the creation month may be earlier; in other cases, it may be later if the first edits happened long after installation. 70aeac932ce07eeabf325447e77e1aca9c9c50b5 677 676 2013-12-18T04:08:08Z Vipul 2 wikitext text/x-wiki {| class="sortable" border="1" ! Wiki !! Main topic !! Page count (lower bound) !! Creation month !! State of development !! Content being added? !! Error log !! Pageview count (2013) (estimate) |- | [[Groupprops:Main Page|Groupprops]] || Group theory || 7200 || December 2006 || beta || Occasionally || [[Groupprops:Groupprops:Error log|here]] || 675,000 |- | [[Topospaces:Main Page|Topospaces]] || Topology (point set, algebraic) || 600 || May 2007 || alpha || No || [[Topospaces:Topospaces:Error log|here]] || 8,000 |- | [[Commalg:Main Page|Commalg]] || Commutative algebra || 400 || January 2007 || alpha || No || Not yet || 14,000 |- | [[Diffgeom:Main Page|Diffgeom]] || Differential geometry || 400 || February 2007 || alpha || No || Not yet || 5,000 |- | [[Number:Main Page|Number]] || Number theory || 200 || March 2009 || alpha || No || Not yet || 7,000 |- | [[Market:Main Page|Market]] || Economic theory of markets, choices, and prices || 100 || January 2009 || alpha || Occasionally || [[Market:Market:Error log|here]] || 195,000 |- | [[Mech:Main Page|Mech]] || Classical mechanics || 50 || January 2009 || alpha || No || [[Mech:Mech:Error log|here]] || 35,000 |- | [[Calculus:Main Page|Calculus]] || Calculus || 50 || August 2011 || alpha || Occasionally || [[Calculus:Calculus:Error log|here]] || 150,000 |- | [[Learning:Main Page|Learning]] || Learning, teaching, and mastery || 25 || September 2013 || alpha || Occasionally || Not yet || NA |} '''NOTE''': The creation month need not coincide with the month of MediaWiki installation. In cases where the wiki was ported from elsewhere, the creation month may be earlier; in other cases, it may be later if the first edits happened long after installation. afcf6bd4da387189dad0fa8be4e12310f2461b94 678 677 2013-12-18T04:09:21Z Vipul 2 wikitext text/x-wiki {| class="sortable" border="1" ! Wiki !! Main topic !! Page count (lower bound) !! Creation month !! State of development !! Content being added? !! Error log !! Pageview count (2013) (estimate) |- | [[Groupprops:Main Page|Groupprops]] || Group theory || 7200 || December 2006 || beta || Occasionally || [[Groupprops:Groupprops:Error log|here]] || 675,000 |- | [[Topospaces:Main Page|Topospaces]] || Topology (point set, algebraic) || 600 || May 2007 || alpha || No || [[Topospaces:Topospaces:Error log|here]] || 85,000 |- | [[Commalg:Main Page|Commalg]] || Commutative algebra || 400 || January 2007 || alpha || No || Not yet || 14,000 |- | [[Diffgeom:Main Page|Diffgeom]] || Differential geometry || 400 || February 2007 || alpha || No || Not yet || 5,000 |- | [[Number:Main Page|Number]] || Number theory || 200 || March 2009 || alpha || No || Not yet || 7,000 |- | [[Market:Main Page|Market]] || Economic theory of markets, choices, and prices || 100 || January 2009 || alpha || Occasionally || [[Market:Market:Error log|here]] || 195,000 |- | [[Mech:Main Page|Mech]] || Classical mechanics || 50 || January 2009 || alpha || No || [[Mech:Mech:Error log|here]] || 35,000 |- | [[Calculus:Main Page|Calculus]] || Calculus || 50 || August 2011 || alpha || Occasionally || [[Calculus:Calculus:Error log|here]] || 150,000 |- | [[Learning:Main Page|Learning]] || Learning, teaching, and mastery || 25 || September 2013 || alpha || Occasionally || Not yet || NA |} '''NOTE''': The creation month need not coincide with the month of MediaWiki installation. In cases where the wiki was ported from elsewhere, the creation month may be earlier; in other cases, it may be later if the first edits happened long after installation. 0b4dbfd41872ec23e4bd38d6a58fc34db5f4d4de 679 678 2013-12-18T04:10:30Z Vipul 2 wikitext text/x-wiki {| class="sortable" border="1" ! Wiki !! Main topic !! Page count (lower bound) !! Creation month !! State of development !! Content being added? !! Error log !! Pageview count (2013) (estimate) |- | [[Groupprops:Main Page|Groupprops]] || Group theory || 7200 || December 2006 || beta || Occasionally || [[Groupprops:Groupprops:Error log|here]] || 675,000 |- | [[Topospaces:Main Page|Topospaces]] || Topology (point set, algebraic) || 600 || May 2007 || alpha || No || [[Topospaces:Topospaces:Error log|here]] || 85,000 |- | [[Commalg:Main Page|Commalg]] || Commutative algebra || 400 || January 2007 || alpha || No || Not yet || 14,000 |- | [[Diffgeom:Main Page|Diffgeom]] || Differential geometry || 400 || February 2007 || alpha || No || Not yet || 5,000 |- | [[Number:Main Page|Number]] || Number theory || 200 || March 2009 || alpha || No || Not yet || 7,000 |- | [[Market:Main Page|Market]] || Economic theory of markets, choices, and prices || 100 || January 2009 || alpha || Occasionally || [[Market:Market:Error log|here]] || 195,000 |- | [[Mech:Main Page|Mech]] || Classical mechanics || 50 || January 2009 || alpha || No || [[Mech:Mech:Error log|here]] || 35,000 |- | [[Calculus:Main Page|Calculus]] || Calculus || 50 || August 2011 || alpha || Occasionally || [[Calculus:Calculus:Error log|here]] || 150,000 |- | [[Learning:Main Page|Learning]] || Learning, teaching, and mastery || 25 || September 2013 || alpha || Occasionally || Not yet || NA |} '''NOTE''': The creation month need not coincide with the month of MediaWiki installation. In cases where the wiki was ported from elsewhere, the creation month may be earlier; in other cases, it may be later if the first edits happened long after installation. 'Pageview count note''': Estimates are based on Google Analytics data, ''not'' based on MediaWiki internal logs, which records about 2-3X the pageviews recorded by Google Analytics. 964546048f3317a556f55133af828b5b67ba9c3b 680 679 2013-12-18T04:12:38Z Vipul 2 wikitext text/x-wiki {| class="sortable" border="1" ! Wiki !! Main topic !! Page count (lower bound) !! Creation month !! State of development !! Content being added? !! Error log !! Pageview count (2013) (estimate) |- | [[Groupprops:Main Page|Groupprops]] || Group theory || 7200 || December 2006 || beta || Occasionally || [[Groupprops:Groupprops:Error log|here]] || 675,000 |- | [[Topospaces:Main Page|Topospaces]] || Topology (point set, algebraic) || 600 || May 2007 || alpha || No || [[Topospaces:Topospaces:Error log|here]] || 85,000 |- | [[Commalg:Main Page|Commalg]] || Commutative algebra || 400 || January 2007 || alpha || No || Not yet || 14,000 |- | [[Diffgeom:Main Page|Diffgeom]] || Differential geometry || 400 || February 2007 || alpha || No || Not yet || 5,000 |- | [[Number:Main Page|Number]] || Number theory || 200 || March 2009 || alpha || No || Not yet || 7,000 |- | [[Market:Main Page|Market]] || Economic theory of markets, choices, and prices || 100 || January 2009 || alpha || Occasionally || [[Market:Market:Error log|here]] || 195,000 |- | [[Mech:Main Page|Mech]] || Classical mechanics || 50 || January 2009 || alpha || No || [[Mech:Mech:Error log|here]] || 35,000 |- | [[Calculus:Main Page|Calculus]] || Calculus || 50 || August 2011 || alpha || Occasionally || [[Calculus:Calculus:Error log|here]] || 150,000 |- | [[Learning:Main Page|Learning]] || Learning, teaching, and mastery || 25 || September 2013 || alpha || Occasionally || Not yet || NA |} '''NOTE''': The creation month need not coincide with the month of MediaWiki installation. In cases where the wiki was ported from elsewhere, the creation month may be earlier; in other cases, it may be later if the first edits happened long after installation. '''Pageview count note''': Estimates are based on Google Analytics data, ''not'' based on MediaWiki internal logs, which records about 2-3X the pageviews recorded by Google Analytics. 8e1b225cb0795332e4564fe0f435ae5d65c48c81 682 680 2013-12-19T00:04:14Z Vipul 2 wikitext text/x-wiki {| class="sortable" border="1" ! Wiki !! Main topic !! Page count (lower bound) !! Creation month !! State of development !! Content being added? !! Error log !! Pageview count (2013) (estimate) |- | [[Groupprops:Main Page|Groupprops]] || Group theory || [[Groupprops:Special:Statistics|7200]] || December 2006 || beta || Occasionally || [[Groupprops:Groupprops:Error log|here]] || 675,000 |- | [[Topospaces:Main Page|Topospaces]] || Topology (point set, algebraic) || [[Topospaces:Special:Statistics|600]] || May 2007 || alpha || No || [[Topospaces:Topospaces:Error log|here]] || 85,000 |- | [[Commalg:Main Page|Commalg]] || Commutative algebra || [[Commalg:Special:Statistics|400]] || January 2007 || alpha || No || Not yet || 14,000 |- | [[Diffgeom:Main Page|Diffgeom]] || Differential geometry || [[Diffgeom:Special:Statistics|400]] || February 2007 || alpha || No || Not yet || 5,000 |- | [[Number:Main Page|Number]] || Number theory || [[Number:Special:Statistics|200]] || March 2009 || alpha || No || Not yet || 7,000 |- | [[Market:Main Page|Market]] || Economic theory of markets, choices, and prices || [[Market:Special:Statistics|100]] || January 2009 || alpha || Occasionally || [[Market:Market:Error log|here]] || 195,000 |- | [[Mech:Main Page|Mech]] || Classical mechanics || [[Mech:Special:Statistics|50]] || January 2009 || alpha || No || [[Mech:Mech:Error log|here]] || 35,000 |- | [[Calculus:Main Page|Calculus]] || Calculus || [[Calculus:Special:Statistics|200]] || August 2011 || alpha || Occasionally || [[Calculus:Calculus:Error log|here]] || 150,000 |- | [[Learning:Main Page|Learning]] || Learning, teaching, and mastery || 25 || September 2013 || alpha || Occasionally || Not yet || NA |} '''NOTE''': The creation month need not coincide with the month of MediaWiki installation. In cases where the wiki was ported from elsewhere, the creation month may be earlier; in other cases, it may be later if the first edits happened long after installation. '''Pageview count note''': Estimates are based on Google Analytics data, ''not'' based on MediaWiki internal logs, which records about 2-3X the pageviews recorded by Google Analytics. e820bead80c0e830ae9d8dd1c13f5d06a1278036 684 682 2013-12-19T00:09:21Z Vipul 2 wikitext text/x-wiki {| class="sortable" border="1" ! Wiki !! Main topic !! Page count (lower bound) !! Creation month !! State of development !! Content being added? !! Error log !! Pageview count (2013) (estimate) |- | [[Groupprops:Main Page|Groupprops]] || Group theory || [[Groupprops:Special:Statistics|7200]] || December 2006 || beta || Occasionally || [[Groupprops:Groupprops:Error log|here]] || 675,000 |- | [[Topospaces:Main Page|Topospaces]] || Topology (point set, algebraic) || [[Topospaces:Special:Statistics|600]] || May 2007 || alpha || No || [[Topospaces:Topospaces:Error log|here]] || 85,000 |- | [[Commalg:Main Page|Commalg]] || Commutative algebra || [[Commalg:Special:Statistics|400]] || January 2007 || alpha || No || Not yet || 14,000 |- | [[Diffgeom:Main Page|Diffgeom]] || Differential geometry || [[Diffgeom:Special:Statistics|400]] || February 2007 || alpha || No || Not yet || 5,000 |- | [[Number:Main Page|Number]] || Number theory || [[Number:Special:Statistics|200]] || March 2009 || alpha || No || Not yet || 7,000 |- | [[Market:Main Page|Market]] || Economic theory of markets, choices, and prices || [[Market:Special:Statistics|100]] || January 2009 || alpha || Occasionally || [[Market:Market:Error log|here]] || 195,000 |- | [[Mech:Main Page|Mech]] || Classical mechanics || [[Mech:Special:Statistics|50]] || January 2009 || alpha || No || [[Mech:Mech:Error log|here]] || 35,000 |- | [[Calculus:Main Page|Calculus]] || Calculus || [[Calculus:Special:Statistics|200]] || August 2011 || alpha || Occasionally || [[Calculus:Calculus:Error log|here]] || 150,000 |- | [[Learning:Main Page|Learning]] || Learning, teaching, and mastery || [[Learning:Special:Statistics|25]] || September 2013 || alpha || Occasionally || Not yet || NA |} '''NOTE''': The creation month need not coincide with the month of MediaWiki installation. In cases where the wiki was ported from elsewhere, the creation month may be earlier; in other cases, it may be later if the first edits happened long after installation. '''Pageview count note''': Estimates are based on Google Analytics data, ''not'' based on MediaWiki internal logs, which records about 2-3X the pageviews recorded by Google Analytics. 0610fa48333a78d3b01e8b63a2c0cfe64ff356b1 685 684 2013-12-19T00:18:34Z Vipul 2 wikitext text/x-wiki {| class="sortable" border="1" ! Wiki !! Main topic !! Page count (lower bound) !! Creation month !! State of development !! Content being added? !! Error log !! Pageview count (2013) (estimate) |- | [[Groupprops:Main Page|Groupprops]] || Group theory || [[Groupprops:Special:Statistics|7200]] || December 2006 || beta || Occasionally || [[Groupprops:Groupprops:Error log|here]] || 675,000 |- | [[Topospaces:Main Page|Topospaces]] || Topology (point set, algebraic) || [[Topospaces:Special:Statistics|600]] || May 2007 || alpha || No || [[Topospaces:Topospaces:Error log|here]] || 85,000 |- | [[Commalg:Main Page|Commalg]] || Commutative algebra || [[Commalg:Special:Statistics|400]] || January 2007 || alpha || No || Not yet || 14,000 |- | [[Diffgeom:Main Page|Diffgeom]] || Differential geometry || [[Diffgeom:Special:Statistics|400]] || February 2007 || alpha || No || Not yet || 5,000 |- | [[Number:Main Page|Number]] || Number theory || [[Number:Special:Statistics|200]] || March 2009 || alpha || No || Not yet || 7,000 |- | [[Market:Main Page|Market]] || Economic theory of markets, choices, and prices || [[Market:Special:Statistics|100]] || January 2009 || alpha || Occasionally || [[Market:Market:Error log|here]] || 195,000 |- | [[Mech:Main Page|Mech]] || Classical mechanics || [[Mech:Special:Statistics|50]] || January 2009 || alpha || No || [[Mech:Mech:Error log|here]] || 35,000 |- | [[Calculus:Main Page|Calculus]] || Calculus || [[Calculus:Special:Statistics|200]] || August 2011 || alpha || Occasionally || [[Calculus:Calculus:Error log|here]] || 150,000 |- | [[Cellbio:Main Page|Cellbio]] || Cellbio || [[Cellbio:Special:Statistics|30]] || September 2011 || alpha || No || Not yet || ~0 |- | [[Learning:Main Page|Learning]] || Learning, teaching, and mastery || [[Learning:Special:Statistics|25]] || September 2013 || alpha || Occasionally || Not yet || NA |} '''NOTE''': The creation month need not coincide with the month of MediaWiki installation. In cases where the wiki was ported from elsewhere, the creation month may be earlier; in other cases, it may be later if the first edits happened long after installation. '''Pageview count note''': Estimates are based on Google Analytics data, ''not'' based on MediaWiki internal logs, which records about 2-3X the pageviews recorded by Google Analytics. 7ee8fa569930214d7335be4719953ce646f519e2 686 685 2013-12-19T00:29:41Z Vipul 2 wikitext text/x-wiki {| class="sortable" border="1" ! Wiki !! Main topic !! Page count (lower bound) !! Creation month !! State of development !! Content being added? !! Error log !! Pageview count (2013) (estimate) |- | [[Groupprops:Main Page|Groupprops]] || Group theory || [[Groupprops:Special:Statistics|7200]] || December 2006 || beta || Occasionally || [[Groupprops:Groupprops:Error log|here]] || 675,000 |- | [[Topospaces:Main Page|Topospaces]] || Topology (point set, algebraic) || [[Topospaces:Special:Statistics|600]] || May 2007 || alpha || No || [[Topospaces:Topospaces:Error log|here]] || 85,000 |- | [[Commalg:Main Page|Commalg]] || Commutative algebra || [[Commalg:Special:Statistics|400]] || January 2007 || alpha || No || Not yet || 14,000 |- | [[Diffgeom:Main Page|Diffgeom]] || Differential geometry || [[Diffgeom:Special:Statistics|400]] || February 2007 || alpha || No || Not yet || 5,000 |- | [[Number:Main Page|Number]] || Number theory || [[Number:Special:Statistics|200]] || March 2009 || alpha || No || Not yet || 7,000 |- | [[Market:Main Page|Market]] || Economic theory of markets, choices, and prices || [[Market:Special:Statistics|100]] || January 2009 || alpha || Occasionally || [[Market:Market:Error log|here]] || 195,000 |- | [[Mech:Main Page|Mech]] || Classical mechanics || [[Mech:Special:Statistics|50]] || January 2009 || alpha || No || [[Mech:Mech:Error log|here]] || 35,000 |- | [[Calculus:Main Page|Calculus]] || Calculus || [[Calculus:Special:Statistics|200]] || August 2011 || alpha || Occasionally || [[Calculus:Calculus:Error log|here]] || 150,000 |- | [[Cellbio:Main Page|Cellbio]] || Cellbio || [[Cellbio:Special:Statistics|30]] || September 2011 || alpha || No || Not yet || ~0 |- | [[Learning:Main Page|Learning]] || Learning, teaching, and mastery || [[Learning:Special:Statistics|25]] || September 2013 || alpha || Occasionally || Not yet || ~0 (tracking begun only in December 2013) |} '''NOTE''': The creation month need not coincide with the month of MediaWiki installation. In cases where the wiki was ported from elsewhere, the creation month may be earlier; in other cases, it may be later if the first edits happened long after installation. '''Pageview count note''': Estimates are based on Google Analytics data, ''not'' based on MediaWiki internal logs, which records about 2-3X the pageviews recorded by Google Analytics. fb6b897ec5effbb02623ed72787a42ca02fa9c85 687 686 2013-12-19T00:29:55Z Vipul 2 wikitext text/x-wiki {| class="sortable" border="1" ! Wiki !! Main topic !! Page count (lower bound) !! Creation month !! State of development !! Content being added? !! Error log !! Pageview count (2013) (estimate) |- | [[Groupprops:Main Page|Groupprops]] || Group theory || [[Groupprops:Special:Statistics|7200]] || December 2006 || beta || Occasionally || [[Groupprops:Groupprops:Error log|here]] || 675,000 |- | [[Topospaces:Main Page|Topospaces]] || Topology (point set, algebraic) || [[Topospaces:Special:Statistics|600]] || May 2007 || alpha || No || [[Topospaces:Topospaces:Error log|here]] || 85,000 |- | [[Commalg:Main Page|Commalg]] || Commutative algebra || [[Commalg:Special:Statistics|400]] || January 2007 || alpha || No || Not yet || 14,000 |- | [[Diffgeom:Main Page|Diffgeom]] || Differential geometry || [[Diffgeom:Special:Statistics|400]] || February 2007 || alpha || No || Not yet || 5,000 |- | [[Number:Main Page|Number]] || Number theory || [[Number:Special:Statistics|200]] || March 2009 || alpha || No || Not yet || 7,000 |- | [[Market:Main Page|Market]] || Economic theory of markets, choices, and prices || [[Market:Special:Statistics|100]] || January 2009 || alpha || Occasionally || [[Market:Market:Error log|here]] || 195,000 |- | [[Mech:Main Page|Mech]] || Classical mechanics || [[Mech:Special:Statistics|50]] || January 2009 || alpha || No || [[Mech:Mech:Error log|here]] || 35,000 |- | [[Calculus:Main Page|Calculus]] || Calculus || [[Calculus:Special:Statistics|200]] || August 2011 || alpha || Occasionally || [[Calculus:Calculus:Error log|here]] || 150,000 |- | [[Cellbio:Main Page|Cellbio]] || Cellbio || [[Cellbio:Special:Statistics|30]] || September 2011 || alpha || No || Not yet || ~0 |- | [[Learning:Main Page|Learning]] || Learning, teaching, and mastery || [[Learning:Special:Statistics|25]] || September 2013 || alpha || Occasionally || Not yet || ~0 |} '''NOTE''': The creation month need not coincide with the month of MediaWiki installation. In cases where the wiki was ported from elsewhere, the creation month may be earlier; in other cases, it may be later if the first edits happened long after installation. '''Pageview count note''': Estimates are based on Google Analytics data, ''not'' based on MediaWiki internal logs, which records about 2-3X the pageviews recorded by Google Analytics. edd9bf3cfed57497b7cea050f46705009a530bf0 688 687 2013-12-19T00:33:42Z Vipul 2 Vipul moved page [[Ref:Subwiki list]] to [[Ref:Subwiki:List of subject wikis]] wikitext text/x-wiki {| class="sortable" border="1" ! Wiki !! Main topic !! Page count (lower bound) !! Creation month !! State of development !! Content being added? !! Error log !! Pageview count (2013) (estimate) |- | [[Groupprops:Main Page|Groupprops]] || Group theory || [[Groupprops:Special:Statistics|7200]] || December 2006 || beta || Occasionally || [[Groupprops:Groupprops:Error log|here]] || 675,000 |- | [[Topospaces:Main Page|Topospaces]] || Topology (point set, algebraic) || [[Topospaces:Special:Statistics|600]] || May 2007 || alpha || No || [[Topospaces:Topospaces:Error log|here]] || 85,000 |- | [[Commalg:Main Page|Commalg]] || Commutative algebra || [[Commalg:Special:Statistics|400]] || January 2007 || alpha || No || Not yet || 14,000 |- | [[Diffgeom:Main Page|Diffgeom]] || Differential geometry || [[Diffgeom:Special:Statistics|400]] || February 2007 || alpha || No || Not yet || 5,000 |- | [[Number:Main Page|Number]] || Number theory || [[Number:Special:Statistics|200]] || March 2009 || alpha || No || Not yet || 7,000 |- | [[Market:Main Page|Market]] || Economic theory of markets, choices, and prices || [[Market:Special:Statistics|100]] || January 2009 || alpha || Occasionally || [[Market:Market:Error log|here]] || 195,000 |- | [[Mech:Main Page|Mech]] || Classical mechanics || [[Mech:Special:Statistics|50]] || January 2009 || alpha || No || [[Mech:Mech:Error log|here]] || 35,000 |- | [[Calculus:Main Page|Calculus]] || Calculus || [[Calculus:Special:Statistics|200]] || August 2011 || alpha || Occasionally || [[Calculus:Calculus:Error log|here]] || 150,000 |- | [[Cellbio:Main Page|Cellbio]] || Cellbio || [[Cellbio:Special:Statistics|30]] || September 2011 || alpha || No || Not yet || ~0 |- | [[Learning:Main Page|Learning]] || Learning, teaching, and mastery || [[Learning:Special:Statistics|25]] || September 2013 || alpha || Occasionally || Not yet || ~0 |} '''NOTE''': The creation month need not coincide with the month of MediaWiki installation. In cases where the wiki was ported from elsewhere, the creation month may be earlier; in other cases, it may be later if the first edits happened long after installation. '''Pageview count note''': Estimates are based on Google Analytics data, ''not'' based on MediaWiki internal logs, which records about 2-3X the pageviews recorded by Google Analytics. edd9bf3cfed57497b7cea050f46705009a530bf0 690 688 2013-12-19T00:34:04Z Vipul 2 Vipul moved page [[Ref:Subwiki:List of subject wikis]] to [[Subwiki:List of subject wikis]] wikitext text/x-wiki {| class="sortable" border="1" ! Wiki !! Main topic !! Page count (lower bound) !! Creation month !! State of development !! Content being added? !! Error log !! Pageview count (2013) (estimate) |- | [[Groupprops:Main Page|Groupprops]] || Group theory || [[Groupprops:Special:Statistics|7200]] || December 2006 || beta || Occasionally || [[Groupprops:Groupprops:Error log|here]] || 675,000 |- | [[Topospaces:Main Page|Topospaces]] || Topology (point set, algebraic) || [[Topospaces:Special:Statistics|600]] || May 2007 || alpha || No || [[Topospaces:Topospaces:Error log|here]] || 85,000 |- | [[Commalg:Main Page|Commalg]] || Commutative algebra || [[Commalg:Special:Statistics|400]] || January 2007 || alpha || No || Not yet || 14,000 |- | [[Diffgeom:Main Page|Diffgeom]] || Differential geometry || [[Diffgeom:Special:Statistics|400]] || February 2007 || alpha || No || Not yet || 5,000 |- | [[Number:Main Page|Number]] || Number theory || [[Number:Special:Statistics|200]] || March 2009 || alpha || No || Not yet || 7,000 |- | [[Market:Main Page|Market]] || Economic theory of markets, choices, and prices || [[Market:Special:Statistics|100]] || January 2009 || alpha || Occasionally || [[Market:Market:Error log|here]] || 195,000 |- | [[Mech:Main Page|Mech]] || Classical mechanics || [[Mech:Special:Statistics|50]] || January 2009 || alpha || No || [[Mech:Mech:Error log|here]] || 35,000 |- | [[Calculus:Main Page|Calculus]] || Calculus || [[Calculus:Special:Statistics|200]] || August 2011 || alpha || Occasionally || [[Calculus:Calculus:Error log|here]] || 150,000 |- | [[Cellbio:Main Page|Cellbio]] || Cellbio || [[Cellbio:Special:Statistics|30]] || September 2011 || alpha || No || Not yet || ~0 |- | [[Learning:Main Page|Learning]] || Learning, teaching, and mastery || [[Learning:Special:Statistics|25]] || September 2013 || alpha || Occasionally || Not yet || ~0 |} '''NOTE''': The creation month need not coincide with the month of MediaWiki installation. In cases where the wiki was ported from elsewhere, the creation month may be earlier; in other cases, it may be later if the first edits happened long after installation. '''Pageview count note''': Estimates are based on Google Analytics data, ''not'' based on MediaWiki internal logs, which records about 2-3X the pageviews recorded by Google Analytics. edd9bf3cfed57497b7cea050f46705009a530bf0 695 690 2013-12-30T04:21:52Z Vipul 2 wikitext text/x-wiki {| class="sortable" border="1" ! Wiki !! Main topic !! Page count (lower bound) !! Creation month !! State of development !! Content being added? !! Error log !! Pageview count (2013) (estimate) |- | [[Groupprops:Main Page|Groupprops]] || Group theory || [[Groupprops:Special:Statistics|7200]] || December 2006 || beta || Occasionally || [[Groupprops:Groupprops:Error log|here]] || 675,000 |- | [[Topospaces:Main Page|Topospaces]] || Topology (point set, algebraic) || [[Topospaces:Special:Statistics|600]] || May 2007 || alpha || No || [[Topospaces:Topospaces:Error log|here]] || 85,000 |- | [[Commalg:Main Page|Commalg]] || Commutative algebra || [[Commalg:Special:Statistics|400]] || January 2007 || alpha || No || Not yet || 14,000 |- | [[Diffgeom:Main Page|Diffgeom]] || Differential geometry || [[Diffgeom:Special:Statistics|400]] || February 2007 || alpha || No || Not yet || 5,000 |- | [[Number:Main Page|Number]] || Number theory || [[Number:Special:Statistics|200]] || March 2009 || alpha || No || Not yet || 7,000 |- | [[Market:Main Page|Market]] || Economic theory of markets, choices, and prices || [[Market:Special:Statistics|100]] || January 2009 || alpha || Occasionally || [[Market:Market:Error log|here]] || 195,000 |- | [[Mech:Main Page|Mech]] || Classical mechanics || [[Mech:Special:Statistics|50]] || January 2009 || alpha || No || [[Mech:Mech:Error log|here]] || 35,000 |- | [[Calculus:Main Page|Calculus]] || Calculus || [[Calculus:Special:Statistics|200]] || August 2011 || alpha || Occasionally || [[Calculus:Calculus:Error log|here]] || 150,000 |- | [[Cellbio:Main Page|Cellbio]] || Cellbio || [[Cellbio:Special:Statistics|30]] || September 2011 || alpha || No || Not yet || ~0 |- | [[Learning:Main Page|Learning]] || Learning, teaching, and mastery || [[Learning:Special:Statistics|25]] || September 2013 || alpha || Occasionally || Not yet || ~0 |} '''NOTE''': The creation month need not coincide with the month of MediaWiki installation. In cases where the wiki was ported from elsewhere, the creation month may be earlier; in other cases, it may be later if the first edits happened long after installation. '''Pageview count note''': Estimates are based on Google Analytics data, ''not'' based on MediaWiki internal logs, which record about 2-3X the pageviews recorded by Google Analytics. e2b218bbe8960c5a2d400b75bb7cdb5f971fb0c1 696 695 2015-01-01T23:41:23Z Vipul 2 wikitext text/x-wiki {| class="sortable" border="1" ! Wiki !! Main topic !! Page count (lower bound) !! Creation month !! State of development !! Content being added? !! Error log !! Pageview count (2014) (Google Analytics) |- | [[Groupprops:Main Page|Groupprops]] || Group theory || [[Groupprops:Special:Statistics|7200]] || December 2006 || beta || Occasionally || [[Groupprops:Groupprops:Error log|here]] || 740,227 |- | [[Topospaces:Main Page|Topospaces]] || Topology (point set, algebraic) || [[Topospaces:Special:Statistics|600]] || May 2007 || alpha || No || [[Topospaces:Topospaces:Error log|here]] || 86,678 |- | [[Commalg:Main Page|Commalg]] || Commutative algebra || [[Commalg:Special:Statistics|400]] || January 2007 || alpha || No || Not yet || 11,181 |- | [[Diffgeom:Main Page|Diffgeom]] || Differential geometry || [[Diffgeom:Special:Statistics|400]] || February 2007 || alpha || No || Not yet || 4,034 |- | [[Number:Main Page|Number]] || Number theory || [[Number:Special:Statistics|200]] || March 2009 || alpha || No || Not yet || 5,271 |- | [[Market:Main Page|Market]] || Economic theory of markets, choices, and prices || [[Market:Special:Statistics|100]] || January 2009 || alpha || Occasionally || [[Market:Market:Error log|here]] || 224,132 |- | [[Mech:Main Page|Mech]] || Classical mechanics || [[Mech:Special:Statistics|50]] || January 2009 || alpha || No || [[Mech:Mech:Error log|here]] || 43,735 |- | [[Calculus:Main Page|Calculus]] || Calculus || [[Calculus:Special:Statistics|200]] || August 2011 || alpha || Occasionally || [[Calculus:Calculus:Error log|here]] || 182,397 |- | [[Cellbio:Main Page|Cellbio]] || Cellbio || [[Cellbio:Special:Statistics|30]] || September 2011 || alpha || No || Not yet || 59 |- | [[Learning:Main Page|Learning]] || Learning, teaching, and mastery || [[Learning:Special:Statistics|25]] || September 2013 || alpha || Occasionally || Not yet || 873 |} '''NOTE''': The creation month need not coincide with the month of MediaWiki installation. In cases where the wiki was ported from elsewhere, the creation month may be earlier; in other cases, it may be later if the first edits happened long after installation. '''Pageview count note''': Estimates are based on Google Analytics data, ''not'' based on MediaWiki internal logs, which record about 2-3X the pageviews recorded by Google Analytics. 18c21c3d5da07b144fe09612b8482638c6773bb9 8 93 683 223 2013-12-19T00:06:41Z Vipul 2 wikitext text/x-wiki * SEARCH * navigation ** mainpage|mainpage ** recentchanges-url|recentchanges ** randompage-url|randompage * subject wikis ** Groupprops:Main Page|Groupprops ** Topospaces:Main Page|Topospaces ** Commalg:Main Page|Commalg ** Diffgeom:Main Page|Diffgeom ** Number:Main Page|Number ** Market:Main Page|Market ** Mech:Main Page|Mech ** Calculus:Main Page|Calculus ** Learning:Main Page|Learning 213624b8dacb5e26017f4f7207649a52c9af8db4 4 366 689 2013-12-19T00:33:42Z Vipul 2 Vipul moved page [[Ref:Subwiki list]] to [[Ref:Subwiki:List of subject wikis]] wikitext text/x-wiki #REDIRECT [[Ref:Subwiki:List of subject wikis]] cdb8890475878e67a15ddae3f7ea8892e730c266 4 367 691 2013-12-19T00:34:04Z Vipul 2 Vipul moved page [[Ref:Subwiki:List of subject wikis]] to [[Subwiki:List of subject wikis]] wikitext text/x-wiki #REDIRECT [[Subwiki:List of subject wikis]] ef8f4a70c6d25d7cc7a4df254d8025ac44768fe5 0 335 697 557 2015-10-28T15:26:02Z Jon Awbrey 3 /* Preliminaries */ add a bit wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. In [[logic]] and [[mathematics]], '''relation composition''', or the composition of [[relation (mathematics)|relations]], is the generalization of [[function composition]], or the composition of [[function (mathematics)|functions]]. ==Preliminaries== There are several ways to formalize the subject matter of relations. Relations and their combinations may be described in the logic of relative terms, in set theories of various kinds, and through a broadening of category theory from functions to relations in general. The first order of business is to define the operation on [[relation (mathematics)|relations]] that is variously known as the ''composition of relations'', ''relational composition'', or ''relative multiplication''. In approaching the more general constructions, it pays to begin with the composition of dyadic and triadic relations. As an incidental observation on usage, there are many different conventions of syntax for denoting the application and composition of relations, with perhaps even more options in general use than are common for the application and composition of functions. In this case there is little chance of standardization, since the convenience of conventions is relative to the context of use, and the same writers use different styles of syntax in different settings, depending on the ease of analysis and computation. {| align="center" cellpadding="4" width="90%" | <p>The first dimension of variation in syntax has to do with the correspondence between the order of operation and the linear order of terms on the page.</p> |- | <p>The second dimension of variation in syntax has to do with the automatic assumptions in place about the associations of terms in the absence of associations marked by parentheses. This becomes a significant factor with relations in general because the usual property of associativity is lost as both the complexities of compositions and the dimensions of relations increase.</p> |} These two factors together generate the following four styles of syntax: {| align="center" cellpadding="4" width="90%" | LALA = left application, left association. |- | LARA = left application, right association. |- | RALA = right application, left association. |- | RARA = right application, right association. |} ==Definition== A notion of relational composition is to be defined that generalizes the usual notion of functional composition: {| align="center" cellpadding="4" width="90%" | <p>Composing ''on the right'', <math>f : X \to Y</math> followed by <math>g : Y \to Z</math></p> <p>results in a ''composite function'' formulated as <math>fg : X \to Z.</math></p> |- | <p>Composing ''on the left'', <math>f : X \to Y</math> followed by <math>g : Y \to Z</math></p> <p>results in a ''composite function'' formulated as <math>gf : X \to Z.</math></p> |} Note on notation. The ordinary symbol for functional composition is the ''[[composition sign]]'', a small circle "<math>\circ</math>" written between the names of the functions being composed, as <math>f \circ g,</math> but the sign is often omitted if there is no risk of confusing the composition of functions with their algebraic product. In contexts where both compositions and products occur, either the composition is marked on each occasion or else the product is marked by means of a ''center dot'' "<math>\cdot</math>", as <math>f \cdot g.</math> Generalizing the paradigm along parallel lines, the ''composition'' of a pair of 2-adic relations is formulated in the following two ways: {| align="center" cellpadding="4" width="90%" | <p>Composing ''on the right'', <math>P \subseteq X \times Y</math> followed by <math>Q \subseteq Y \times Z</math></p> <p>results in a ''composite relation'' formulated as <math>PQ \subseteq X \times Z.</math></p> |- | <p>Composing ''on the left'', <math>P \subseteq X \times Y</math> followed by <math>Q \subseteq Y \times Z</math></p> <p>results in a ''composite relation'' formulated as <math>QP \subseteq X \times Z.</math></p> |} ==Geometric construction== There is a neat way of defining relational compositions in geometric terms, not only showing their relationship to the [[projection (set theory)|projection]] operations that come with any [[cartesian product]], but also suggesting natural directions for generalizing relational compositions beyond the 2-adic case, and even beyond relations that have any fixed [[arity]], in effect, to the general case of [[formal language]]s as generalized relations. This way of looking at relational compositions is sometimes referred to as [[Alfred Tarski|Tarski]]'s Trick (T²), on account of his having put it to especially good use in his work (Ulam and Bednarek, 1977). It supplies the imagination with a geometric way of visualizing the relational composition of a pair of 2-adic relations, doing this by attaching concrete imagery to the basic [[set theory|set-theoretic]] operations, namely, [[intersection (set theory)|intersection]]s, [[projection (set theory)|projection]]s, and a certain class of operations [[inverse relation|inverse]] to projections, here called ''[[tacit extension]]s''. The stage is set for Tarski's trick by highlighting the links between two topics that are likely to appear wholly unrelated at first, namely: :* The use of logical [[conjunction]], as denoted by the symbol "&and;" in expressions of the form ''F''(''x'',&nbsp;''y'',&nbsp;''z'') = ''G''(''x'',&nbsp;''y'')&nbsp;&and;&nbsp;''H''(''y'',&nbsp;''z''), to define a 3-adic relation ''F'' in terms of a pair of 2-adic relations ''G'' and ''H''. :* The concepts of 2-adic ''[[projection (set theory)|projection]]'' and ''projective determination'', that are invoked in the 'weak' notion of ''projective reducibility''. The relational composition ''G''&nbsp;&omicron;&nbsp;''H'' of a pair of 2-adic relations ''G'' and ''H'' will be constructed in three stages, first, by taking the tacit extensions of ''G'' and ''H'' to 3-adic relations that reside in the same space, next, by taking the intersection of these extensions, tantamount to the maximal 3-adic relation that is consistent with the ''prima facie'' 2-adic relation data, finally, by projecting this intersection on a suitable plane to form a third 2-adic relation, constituting in fact the relational composition ''G''&nbsp;&omicron;&nbsp;''H'' of the relations ''G'' and ''H''. The construction of a relational composition in a specifically mathematical setting normally begins with [[relation (mathematics)|mathematical relations]] at a higher level of abstraction than the corresponding objects in linguistic or logical settings. This is due to the fact that mathematical objects are typically specified only ''up to [[isomorphism]]'' as the conventional saying goes, that is, any objects that have the 'same form' are generally regarded as the being the same thing, for most all intents and mathematical purposes. Thus, the mathematical construction of a relational composition begins by default with a pair of 2-adic relations that reside, without loss of generality, in the same plane, say, ''G'', ''H''&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''Y'', as depicted in Figure 1. o-------------------------------------------------o | | | o o | | |\ |\ | | | \ | \ | | | \ | \ | | | \ | \ | | | \ | \ | | | \ | \ | | | * \ | * \ | | X * Y X * Y | | \ * | \ * | | | \ G | \ H | | | \ | \ | | | \ | \ | | | \ | \ | | | \ | \ | | | \| \| | | o o | | | o-------------------------------------------------o Figure 1. Dyadic Relations G, H c X x Y The 2-adic relations ''G'' and ''H'' cannot be composed at all at this point, not without additional information or further stipulation. In order for their relational composition to be possible, one of two types of cases has to happen: :* The first type of case occurs when ''X''&nbsp;=&nbsp;''Y''. In this case, both of the compositions ''G''&nbsp;&omicron;&nbsp;''H'' and ''H''&nbsp;&omicron;&nbsp;''G'' are defined. :* The second type of case occurs when ''X'' and ''Y'' are distinct, but when it nevertheless makes sense to speak of a 2-adic relation ''&#292;'' that is isomorphic to ''H'', but living in the plane ''YZ'', that is, in the space of the cartesian product ''Y''&nbsp;&times;&nbsp;''Z'', for some set ''Z''. Whether you view isomorphic things to be the same things or not, you still have to specify the exact isomorphisms that are needed to transform any given representation of a thing into a required representation of the same thing. Let us imagine that we have done this, and say how later: o-------------------------------------------------o | | | o o | | |\ /| | | | \ / | | | | \ / | | | | \ / | | | | \ / | | | | \ / | | | | * \ / * | | | X * Y Y * Z | | \ * | | * / | | \ G | | &#292; / | | \ | | / | | \ | | / | | \ | | / | | \ | | / | | \| |/ | | o o | | | o-------------------------------------------------o Figure 2. Dyadic Relations G c X x Y and &#292; c Y x Z With the required spaces carefully swept out, the stage is set for the presentation of Tarski's trick, and the invocation of the following symbolic formula, claimed to be a definition of the relational composition ''P''&nbsp;&omicron;&nbsp;''Q'' of a pair of 2-adic relations ''P'',&nbsp;''Q''&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''X''. : '''Definition.''' ''P''&nbsp;&omicron;&nbsp;''Q'' = ''proj''₁₃ (''P''&nbsp;&times;&nbsp;''X''&nbsp;&cap;&nbsp;''X''&nbsp;&times;&nbsp;''Q''). To get this drift of this definition, one needs to understand that it's written within a school of thought that holds that all 2-adic relations are, 'without loss of generality', covered well enough, 'for all practical purposes', under the aegis of subsets of a suitable cartesian square, and thus of the form ''L''&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''X''. So, if one has started out with a 2-adic relation of the shape ''L''&nbsp;&sube;&nbsp;''U''&nbsp;&times;&nbsp;''V'', one merely lets ''X''&nbsp;=&nbsp;''U''&nbsp;&cup;&nbsp;''V'', trading in the initial ''L'' for a new ''L''&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''X'' as need be. The projection ''proj''₁₃ is just the projection of the cartesian cube ''X''&nbsp;&times;&nbsp;''X''&nbsp;&times;&nbsp;''X'' on the space of shape ''X''&nbsp;&times;&nbsp;''X'' that is spanned by the first and the third domains, but since they now have the same names and the same contents it is necessary to distinguish them by numbering their relational places. Finally, the notation of the cartesian product sign "&times;" is extended to signify two other products with respect to a 2-adic relation ''L''&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''X'' and a subset ''W''&nbsp;&sube;&nbsp;''X'', as follows: : '''Definition.''' ''L'' &times; ''W'' = {(''x'', ''y'', ''z'') &isin; ''X''³ : (''x'', ''y'') &isin; ''L'' &and; ''z'' &isin; ''W''}. : '''Definition.''' ''W'' &times; ''L'' = {(''x'', ''y'', ''z'') &isin; ''X''³ : ''x'' &isin; ''W'' &and; (''y'', ''z'') &isin; ''L''}. Applying these definitions to the case ''P'',&nbsp;''Q''&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''X'', the two 2-adic relations whose relational composition ''P''&nbsp;&omicron;&nbsp;''Q''&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''X'' is about to be defined, one finds: : ''P'' &times; ''X'' = {(''x'', ''y'', ''z'') &isin; ''X''³ : (''x'', ''y'') &isin; ''P'' &and; ''z'' &isin; ''X''}, : ''X'' &times; ''Q'' = {(''x'', ''y'', ''z'') &isin; ''X''³ : ''x'' &isin; ''X'' &and; (''y'', ''z'') &isin; ''Q''}. These are just the appropriate special cases of the tacit extensions already defined. : ''P'' &times; ''X'' = ''te''₁₂³(''P''), : ''X'' &times; ''Q'' = ''te''₂₃¹(''Q''). In summary, then, the expression: : ''proj''₁₃(''P'' &times; ''X'' &cap; ''X'' &times; ''Q'') is equivalent to the expression: : ''proj''₁₃(''te''₁₂³(''P'') &cap; ''te''₂₃¹(''Q'')) and this form is generalized — although, relative to one's school of thought, perhaps inessentially so — by the form that was given above as follows: : '''Definition.''' ''P''&nbsp;&omicron;&nbsp;''Q'' = ''proj''_''XZ''(''te''_''XY''^''Z''(''P'')&nbsp;&cap;&nbsp;''te''_''YZ''^''X''(''Q'')). Figure 3 presents a geometric picture of what is involved in formulating a definition of the 3-adic relation ''F''&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''Y''&nbsp;&times;&nbsp;''Z'' by way of a conjunction of the 2-adic relation ''G''&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''Y'' and the 2-adic relation ''H''&nbsp;&sube;&nbsp;''Y''&nbsp;&times;&nbsp;''Z'', as done for example by means of an expression of the following form: :* ''F''(''x'', ''y'', ''z'') = ''G''(''x'', ''y'') &and; ''H''(''y'', ''z''). o-------------------------------------------------o | | | o | | /|\ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | o o o | | |\ / \ /| | | | \ / F \ / | | | | \ / * \ / | | | | \ /*\ / | | | | / \//*\\/ \ | | | | / /\/ \/\ \ | | | |/ ///\ /\\\ \| | | o X /// Y \\\ Z o | | |\ \/// | \\\/ /| | | | \ /// | \\\ / | | | | \ ///\ | /\\\ / | | | | \ /// \ | / \\\ / | | | | \/// \ | / \\\/ | | | | /\/ \ | / \/\ | | | | *//\ \|/ /\\* | | | X */ Y o Y \* Z | | \ * | | * / | | \ G | | H / | | \ | | / | | \ | | / | | \ | | / | | \ | | / | | \| |/ | | o o | | | o-------------------------------------------------o Figure 3. Projections of F onto G and H To interpret the Figure, visualize the 3-adic relation ''F''&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''Y''&nbsp;&times;&nbsp;''Z'' as a body in ''XYZ''-space, while ''G'' is a figure in ''XY''-space and ''H'' is a figure in ''YZ''-space. The 2-adic '''projections''' that accompany a 3-adic relation over ''X'', ''Y'', ''Z'' are defined as follows: :* ''proj''_''XY''(''L'') = {(''x'', ''y'') &isin; ''X'' &times; ''Y'' : (&exist; ''z'' &isin; ''Z'') (''x'', ''y'', ''z'') &isin; ''L''}, :* ''proj''_''XZ''(''L'') = {(''x'', ''z'') &isin; ''X'' &times; ''Z'' : (&exist; ''y'' &isin; ''Y'') (''x'', ''y'', ''z'') &isin; ''L''}, :* ''proj''_''YZ''(''L'') = {(''y'', ''z'') &isin; ''Y'' &times; ''Z'' : (&exist; ''x'' &isin; ''X'') (''x'', ''y'', ''z'') &isin; ''L''}. For many purposes it suffices to indicate the 2-adic projections of a 3-adic relation ''L'' by means of the briefer equivalents listed here: :* ''L''_''XY'' = ''proj''_''XY''(''L''), :* ''L''_''XZ'' = ''proj''_''XZ''(''L''), :* ''L''_''YZ'' = ''proj''_''YZ''(''L''). In light of these definitions, ''proj''_''XY'' is a mapping from the set <font face=signature>L</font>_''XYZ'' of 3-adic relations over ''X'', ''Y'', ''Z'' to the set <font face=signature>L</font>_''XY'' of 2-adic relations over ''X'' and ''Y'', with similar relationships holding for the other projections. To formalize these relationships in a concise but explicit manner, it serves to add a few more definitions. The set <font face=signature>L</font>_''XYZ'', whose membership is just the 3-adic relations over ''X'', ''Y'', ''Z'', can be recognized as the set of all subsets of the cartesian product ''X''&nbsp;&times;&nbsp;''Y''&nbsp;&times;&nbsp;''Z'', also known as the '''power set''' of ''X''&nbsp;&times;&nbsp;''Y''&nbsp;&times;&nbsp;''Z'', and notated here as ''Pow''(''X''&nbsp;&times;&nbsp;''Y''&nbsp;&times;&nbsp;''Z''). :* <font face=signature>L</font>_''XYZ'' = {''L''&nbsp;:&nbsp;''L''&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''Y''&nbsp;&times;&nbsp;''Z''} = ''Pow''(''X''&nbsp;&times;&nbsp;''Y''&nbsp;&times;&nbsp;''Z''). Likewise, the power sets of the pairwise cartesian products encompass all of the 2-adic relations on pairs of distinct domains that can be chosen from {''X'', ''Y'', ''Z''}. :* <font face=signature>L</font>_''XY'' = {''L'' : ''L'' &sube; ''X'' &times; ''Y''} = ''Pow''(''X'' &times; ''Y''), :* <font face=signature>L</font>_''XZ'' = {''L'' : ''L'' &sube; ''X'' &times; ''Z''} = ''Pow''(''X'' &times; ''Z''), :* <font face=signature>L</font>_''YZ'' = {''L'' : ''L'' &sube; ''Y'' &times; ''Z''} = ''Pow''(''Y'' &times; ''Z''). In mathematics, the inverse relation corresponding to a projection map is usually called an '''extension'''. To avoid confusion with other senses of the word, however, it is probably best for the sake of this discussion to stick with the more specific term '''tacit extension'''. The ''tacit extensions'' ''te''_''XY''^''Z'', ''te''_''XZ''^''Y'', ''te''_''YZ''^''X'', of the 2-adic relations ''U''&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''Y'', ''V''&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''Z'', ''W''&nbsp;&sube;&nbsp;''Y''&nbsp;&times; ''Z'', respectively, are defined in the following way: :* ''te''_''XY''^''Z''(''U'') = {(''x'', ''y'', ''z'') : (''x'', ''y'') &isin; ''U''} :* ''te''_''XZ''^''Y''(''V'') = {(''x'', ''y'', ''z'') : (''x'', ''z'') &isin; ''V''} :* ''te''_''YZ''^''X''(''W'') = {(''x'', ''y'', ''z'') : (''y'', ''z'') &isin; ''W''} So long as the intended indices attaching to the tacit extensions can be gathered from context, it is usually clear enough to use the abbreviated forms, ''te''(''U''), ''te''(''V''), ''te''(''W''). The definition and illustration of relational composition presently under way makes use of the tacit extension of ''G''&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''Y'' to ''te''(''G'')&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''Y''&nbsp;&times;&nbsp;''Z'' and the tacit extension of ''H''&nbsp;&sube;&nbsp;''Y''&nbsp;&times;&nbsp;''Z'' to ''te''(''H'')&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''Y''&nbsp;&times; ''Z'', only. Geometric illustrations of ''te''(''G'') and ''te''(''H'') are afforded by Figures 4 and 5, respectively. o-------------------------------------------------o | | | o | | /|\ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | / | * \ | | o o ** o | | |\ / \*** /| | | | \ / *** / | | | | \ / ***\ / | | | | \ *** / | | | | / \*** / \ | | | | / *** / \ | | | |/ ***\ / \| | | o X /** Y Z o | | |\ \//* | / /| | | | \ /// | / / | | | | \ ///\ | / / | | | | \ /// \ | / / | | | | \/// \ | / / | | | | /\/ \ | / / | | | | *//\ \|/ / * | | | X */ Y o Y * Z | | \ * | | * / | | \ G | | H / | | \ | | / | | \ | | / | | \ | | / | | \ | | / | | \| |/ | | o o | | | o-------------------------------------------------o Figure 4. Tacit Extension of G to X x Y x Z <br> o-------------------------------------------------o | | | o | | /|\ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | / * | \ | | o ** o o | | |\ ***/ \ /| | | | \ *** \ / | | | | \ /*** \ / | | | | \ *** / | | | | / \ ***/ \ | | | | / \ *** \ | | | |/ \ /*** \| | | o X Y **\ Z o | | |\ \ | *\\/ /| | | | \ \ | \\\ / | | | | \ \ | /\\\ / | | | | \ \ | / \\\ / | | | | \ \ | / \\\/ | | | | \ \ | / \/\ | | | | * \ \|/ /\\* | | | X * Y o Y \* Z | | \ * | | * / | | \ G | | H / | | \ | | / | | \ | | / | | \ | | / | | \ | | / | | \| |/ | | o o | | | o-------------------------------------------------o Figure 5. Tacit Extension of H to X x Y x Z A geometric interpretation can now be given that fleshes out in graphic form the meaning of a formula like the following: :* ''F''(''x'', ''y'', ''z'') = ''G''(''x'', ''y'') &and; ''H''(''y'', ''z''). The conjunction that is indicated by "&and;" corresponds as usual to an intersection of two sets, however, in this case it is the intersection of the tacit extensions ''te''(''G'') and ''te''(''H''). o-------------------------------------------------o | | | o | | /|\ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | o o o | | |\ / \ /| | | | \ / F \ / | | | | \ / * \ / | | | | \ /*\ / | | | | / \//*\\/ \ | | | | / /\/ \/\ \ | | | |/ ///\ /\\\ \| | | o X /// Y \\\ Z o | | |\ \/// | \\\/ /| | | | \ /// | \\\ / | | | | \ ///\ | /\\\ / | | | | \ /// \ | / \\\ / | | | | \/// \ | / \\\/ | | | | /\/ \ | / \/\ | | | | *//\ \|/ /\\* | | | X */ Y o Y \* Z | | \ * | | * / | | \ G | | H / | | \ | | / | | \ | | / | | \ | | / | | \ | | / | | \| |/ | | o o | | | o-------------------------------------------------o Figure 6. F as the Intersection of te(G) and te(H) ==Algebraic construction== The transition from a geometric picture of relation composition to an algebraic formulation is accomplished through the introduction of coordinates, in other words, identifiable names for the objects that are related through the various forms of relations, 2-adic and 3-adic in the present case. Adding coordinates to the running Example produces the following Figure: o-------------------------------------------------o | | | o | | /|\ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | o o o | | |\ / \ /| | | | \ / F \ / | | | | \ / * \ / | | | | \ /*\ / | | | | / \//*\\/ \ | | | | / /\/ \/\ \ | | | |/ ///\ /\\\ \| | | o X /// Y \\\ Z o | | |\ 7\/// | \\\/7 /| | | | \ 6// | \\6 / | | | | \ //5\ | /5\\ / | | | | \ /// 4\ | /4 \\\ / | | | | \/// 3\ | /3 \\\/ | | | | G/\/ 2\ | /2 \/\H | | | | *//\ 1\|/1 /\\* | | | X *\ Y o Y /* Z | | 7\ *\\ |7 7| //* /7 | | 6\ |\\\|6 6|///| /6 | | 5\| \\@5 5@// |/5 | | 4@ \@4 4@/ @4 | | 3\ @3 3@ /3 | | 2\ |2 2| /2 | | 1\|1 1|/1 | | o o | | | o-------------------------------------------------o Figure 7. F as the Intersection of te(G) and te(H) Thinking of relations in operational terms is facilitated by using a variant notation for tuples and sets of tuples, namely, the ordered pair (''x'',&nbsp;''y'') is written ''x'':''y'', the ordered triple (''x'',&nbsp;''y'',&nbsp;''z'') is written ''x'':''y'':''z'', and so on, and a set of tuples is conceived as a logical-algebraic sum, which can be written out in the smaller finite cases in forms like ''a'':''b''&nbsp;+&nbsp;''b'':''c''&nbsp;+&nbsp;''c'':''d'' and so on. For example, translating the relations ''F''&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''Y''&nbsp;&times;&nbsp;''Z'', ''G''&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''Y'', ''H''&nbsp;&sube;&nbsp;''Y''&nbsp;&times;&nbsp;''Z'' into this notation produces the following summary of the data: {| cellpadding=8 style="text-align:center" | &nbsp; || ''F'' || = || 4:3:4 || + || 4:4:4 || + || 4:5:4 |- | &nbsp; || ''G'' || = || 4:3 || + || 4:4 || + || 4:5 |- | &nbsp; || ''H'' || = || 3:4 || + || 4:4 || + || 5:4 |} As often happens with abstract notations for functions and relations, the ''type information'', in this case, the fact that ''G'' and ''H'' live in different spaces, is left implicit in the context of use. Let us now verify that all of the proposed definitions, formulas, and other relationships check out against the concrete data of the current composition example. The ultimate goal is to develop a clearer picture of what is going on in the formula that expresses the relational composition of a couple of 2-adic relations in terms of the medial projection of the intersection of their tacit extensions: : ''G'' &omicron; ''H'' = ''proj''_''XZ''(''te''_''XY''^''Z''(''G'') &cap; ''te''_''YZ''^''X''(''H'')). Here is the big picture, with all of the pieces in place: o-------------------------------------------------o | | | o | | / \ | | / \ | | / \ | | / \ | | / \ | | / \ | | / G o H \ | | X * Z | | 7\ /|\ /7 | | 6\ / | \ /6 | | 5\ / | \ /5 | | 4@ | @4 | | 3\ | /3 | | 2\ | /2 | | 1\|/1 | | | | | | | | | | | /|\ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | o | o | | |\ /|\ /| | | | \ / F \ / | | | | \ / * \ / | | | | \ /*\ / | | | | / \//*\\/ \ | | | | / /\/ \/\ \ | | | |/ ///\ /\\\ \| | | o X /// Y \\\ Z o | | |\ \/// | \\\/ /| | | | \ /// | \\\ / | | | | \ ///\ | /\\\ / | | | | \ /// \ | / \\\ / | | | | \/// \ | / \\\/ | | | | G/\/ \ | / \/\H | | | | *//\ \|/ /\\* | | | X *\ Y o Y /* Z | | 7\ *\\ |7 7| //* /7 | | 6\ |\\\|6 6|///| /6 | | 5\| \\@5 5@// |/5 | | 4@ \@4 4@/ @4 | | 3\ @3 3@ /3 | | 2\ |2 2| /2 | | 1\|1 1|/1 | | o o | | | o-------------------------------------------------o Figure 8. G o H = proj_XZ (te(G) |^| te(H)) All that remains is to check the following collection of data and derivations against the situation represented in Figure 8. {| cellpadding=8 style="text-align:center" | &nbsp; || ''F'' || = || 4:3:4 || + || 4:4:4 || + || 4:5:4 |- | &nbsp; || ''G'' || = || 4:3 || + || 4:4 || + || 4:5 |- | &nbsp; || ''H'' || = || 3:4 || + || 4:4 || + || 5:4 |} {| cellpadding=8 style="text-align:center" | &nbsp; || ''G'' &omicron; ''H'' || = || (4:3 + 4:4 + 4:5)(3:4 + 4:4 + 5:4) |- | &nbsp; || &nbsp; || = || 4:4 |} {| cellpadding=8 style="text-align:center" | &nbsp; || ''te''(''G'') || = || ''te''_''XY''^''Z''(''G'') |- | &nbsp; || &nbsp; || = || &sum;_''z''=1..7 (4:3:''z'' + 4:4:''z'' + 4:5:''z'') |} {| cellpadding=8 style="text-align:center" | &nbsp; || ''te''(''G'') || = || 4:3:1 || + || 4:4:1 || + || 4:5:1 || + |- | &nbsp; || &nbsp; || &nbsp; || 4:3:2 || + || 4:4:2 || + || 4:5:2 || + |- | &nbsp; || &nbsp; || &nbsp; || 4:3:3 || + || 4:4:3 || + || 4:5:3 || + |- | &nbsp; || &nbsp; || &nbsp; || 4:3:4 || + || 4:4:4 || + || 4:5:4 || + |- | &nbsp; || &nbsp; || &nbsp; || 4:3:5 || + || 4:4:5 || + || 4:5:5 || + |- | &nbsp; || &nbsp; || &nbsp; || 4:3:6 || + || 4:4:6 || + || 4:5:6 || + |- | &nbsp; || &nbsp; || &nbsp; || 4:3:7 || + || 4:4:7 || + || 4:5:7 |} {| cellpadding=8 style="text-align:center" | &nbsp; || ''te''(''H'') || = || ''te''_''YZ''^''X''(''H'') |- | &nbsp; || &nbsp; || = || &sum;_''x''=1..7 (''x'':3:4 + ''x'':4:4 + ''x'':5:4) |} {| cellpadding=8 style="text-align:center" | &nbsp; || ''te''(''H'') || = || 1:3:4 || + || 1:4:4 || + || 1:5:4 || + |- | &nbsp; || &nbsp; || &nbsp; || 2:3:4 || + || 2:4:4 || + || 2:5:4 || + |- | &nbsp; || &nbsp; || &nbsp; || 3:3:4 || + || 3:4:4 || + || 3:5:4 || + |- | &nbsp; || &nbsp; || &nbsp; || 4:3:4 || + || 4:4:4 || + || 4:5:4 || + |- | &nbsp; || &nbsp; || &nbsp; || 5:3:4 || + || 5:4:4 || + || 5:5:4 || + |- | &nbsp; || &nbsp; || &nbsp; || 6:3:4 || + || 6:4:4 || + || 6:5:4 || + |- | &nbsp; || &nbsp; || &nbsp; || 7:3:4 || + || 7:4:4 || + || 7:5:4 |} {| |- | align="center" | ''te''(''G'') &cap; ''te''(''H'') | = | 4:3:4 + 4:4:4 + 4:5:4 |- | align="center" | ''G'' &omicron; ''H'' | = | ''proj''_''XZ''(''te''(''G'') &cap; ''te''(''H'')) |- | &nbsp; | = | ''proj''_''XZ''(4:3:4 + 4:4:4 + 4:5:4) |- | &nbsp; | = | 4:4 |} ==Matrix representation== We have it within our reach to pick up another way of representing 2-adic relations, namely, the representation as [[logical matrix|logical matrices]], and also to grasp the analogy between relational composition and ordinary [[matrix multiplication]] as it appears in [[linear algebra]]. First of all, while we still have the data of a very simple concrete case in mind, let us reflect on what we did in our last Example in order to find the composition ''G''&nbsp;&omicron;&nbsp;''H'' of the 2-adic relations ''G''&nbsp;and&nbsp;''H''. Here is the setup that we had before: {| cellpadding=8 style="text-align:center" | &nbsp; || ''X'' || = || {1, 2, 3, 4, 5, 6, 7} |} {| cellpadding=8 style="text-align:center" | &nbsp; || ''G'' || = || 4:3 || + || 4:4 || + || 4:5 || &sube; || ''X'' &times; ''X'' |- | &nbsp; || ''H'' || = || 3:4 || + || 4:4 || + || 5:4 || &sube; || ''X'' &times; ''X '' |} Let us recall the rule for finding the relational composition of a pair of 2-adic relations. Given the 2-adic relations ''P''&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''Y'', ''Q''&nbsp;&sube;&nbsp;''Y''&nbsp;&times;&nbsp;''Z'', the relational composition of ''P'' and ''Q'', in that order, is written as ''P''&nbsp;&omicron;&nbsp;''Q'' or more simply as ''PQ'' and obtained as follows: To compute ''PQ'', in general, where ''P'' and ''Q'' are 2-adic relations, simply multiply out the two sums in the ordinary distributive algebraic way, but subject to the following rule for finding the product of two elementary relations of shapes ''a'':''b'' and ''c'':''d''. {| cellpadding=8 style="text-align:center" | &nbsp; || (a:b)(c:d) || = || (a:d) || if b = c |- | &nbsp; || (a:b)(c:d) || = || 0 || otherwise |} To find the relational composition ''G'' &omicron; ''H'', one may begin by writing it as a quasi-algebraic product: {| cellpadding=8 style="text-align:center" | &nbsp; || ''G'' &omicron; ''H'' || = || (4:3 + 4:4 + 4:5)(3:4 + 4:4 + 5:4) |} Multiplying this out in accord with the applicable form of distributive law one obtains the following expansion: {| cellpadding=8 style="text-align:center" | &nbsp; || ''G'' &omicron; ''H'' || = || (4:3)(3:4) || + || (4:3)(4:4) || + || (4:3)(5:4) || + |- | &nbsp; || &nbsp; || &nbsp; || (4:4)(3:4) || + || (4:4)(4:4) || + || (4:4)(5:4) || + |- | &nbsp; || &nbsp; || &nbsp; ||(4:5)(3:4) || + || (4:5)(4:4) || + || (4:5)(5:4) |} Applying the rule that determines the product of elementary relations produces the following array: {| cellpadding=8 style="text-align:center" | &nbsp; || ''G'' &omicron; ''H'' || = || (4:4) || + || 0 || + || 0 || + |- | &nbsp; || &nbsp; || &nbsp; || 0 || + || (4:4) || + || 0 || + |- | &nbsp; || &nbsp; || &nbsp; || 0 || + || 0 || + || (4:4) |} Since the plus sign in this context represents an operation of logical disjunction or set-theoretic aggregation, all of the positive multiplicites count as one, and this gives the ultimate result: {| cellpadding=8 style="text-align:center" | &nbsp; || ''G'' &omicron; ''H'' || = || (4:4) |} With an eye toward extracting a general formula for relation composition, viewed here on analogy to algebraic multiplication, let us examine what we did in multiplying the 2-adic relations ''G'' and ''H'' together to obtain their relational composite ''G'' o ''H''. Given the space ''X'' = {1, 2, 3, 4, 5, 6, 7}, whose cardinality |''X''| is 7, there are |''X'' &times; ''X''| = |''X''| &middot; |''X''| = 7 &middot; 7 = 49 elementary relations of the form ''i'':''j'', where ''i'' and ''j'' range over the space ''X''. Although they might be organized in many different ways, it is convenient to regard the collection of elementary relations as being arranged in a lexicographic block of the following form: {| cellpadding=8 style="text-align:center" | &nbsp; || 1:1 || 1:2 || 1:3 || 1:4 || 1:5 || 1:6 || 1:7 |- | &nbsp; || 2:1 || 2:2 || 2:3 || 2:4 || 2:5 || 2:6 || 2:7 |- | &nbsp; || 3:1 || 3:2 || 3:3 || 3:4 || 3:5 || 3:6 || 3:7 |- | &nbsp; || 4:1 || 4:2 || 4:3 || 4:4 || 4:5 || 4:6 || 4:7 |- | &nbsp; || 5:1 || 5:2 || 5:3 || 5:4 || 5:5 || 5:6 || 5:7 |- | &nbsp; || 6:1 || 6:2 || 6:3 || 6:4 || 6:5 || 6:6 || 6:7 |- | &nbsp; || 7:1 || 7:2 || 7:3 || 7:4 || 7:5 || 7:6 || 7:7 |} The relations ''G'' and ''H'' may then be regarded as logical sums of the following forms: {| cellpadding=8 style="text-align:center" | &nbsp; || ''G'' || = || &sum;_''ij'' ''G''_''ij''(''i'':''j'') |- | &nbsp; || ''H'' || = || &sum;_''ij'' ''H''_''ij''(''i'':''j'') |} The notation &sum;_''ij'' indicates a logical sum over the collection of elementary relations ''i'':''j'', while the factors ''G''_''ij'' and ''H''_''ij'' are values in the boolean domain '''B''' = {0,&nbsp;1} that are known as the ''coefficients'' of the relations ''G'' and ''H'', respectively, with regard to the corresponding elementary relations ''i'':''j''. In general, for a 2-adic relation ''L'', the coefficient ''L''_''ij'' of the elementary relation ''i'':''j'' in the relation ''L'' will be 0 or 1, respectively, as ''i'':''j'' is excluded from or included in ''L''. With these conventions in place, the expansions of ''G'' and ''H'' may be written out as follows: {| cellpadding=6 style="text-align:center" | &nbsp; || ''G'' || = || 4:3 || + || 4:4 || + || 4:5 || = |} {| | style="width:20px" | &nbsp; | 0(1:1) + | 0(1:2) + | 0(1:3) + | 0(1:4) + | 0(1:5) + | 0(1:6) + | 0(1:7) + |- | &nbsp; | 0(2:1) + | 0(2:2) + | 0(2:3) + | 0(2:4) + | 0(2:5) + | 0(2:6) + | 0(2:7) + |- | &nbsp; | 0(3:1) + | 0(3:2) + | 0(3:3) + | 0(3:4) + | 0(3:5) + | 0(3:6) + | 0(3:7) + |- | &nbsp; | 0(4:1) + | 0(4:2) + | '''1'''(4:3) + | '''1'''(4:4) + | '''1'''(4:5) + | 0(4:6) + | 0(4:7) + |- | &nbsp; | 0(5:1) + | 0(5:2) + | 0(5:3) + | 0(5:4) + | 0(5:5) + | 0(5:6) + | 0(5:7) + |- | &nbsp; | 0(6:1) + | 0(6:2) + | 0(6:3) + | 0(6:4) + | 0(6:5) + | 0(6:6) + | 0(6:7) + |- | &nbsp; | 0(7:1) + | 0(7:2) + | 0(7:3) + | 0(7:4) + | 0(7:5) + | 0(7:6) + | 0(7:7) |} <br> {| cellpadding=6 style="text-align:center" | &nbsp; || ''H'' || = || 3:4 || + || 4:4 || + || 5:4 || = |} {| | style="width:20px" | &nbsp; | 0(1:1) + | 0(1:2) + | 0(1:3) + | 0(1:4) + | 0(1:5) + | 0(1:6) + | 0(1:7) + |- | &nbsp; | 0(2:1) + | 0(2:2) + | 0(2:3) + | 0(2:4) + | 0(2:5) + | 0(2:6) + | 0(2:7) + |- | &nbsp; | 0(3:1) + | 0(3:2) + | 0(3:3) + | '''1'''(3:4) + | 0(3:5) + | 0(3:6) + | 0(3:7) + |- | &nbsp; | 0(4:1) + | 0(4:2) + | 0(4:3) + | '''1'''(4:4) + | 0(4:5) + | 0(4:6) + | 0(4:7) + |- | &nbsp; | 0(5:1) + | 0(5:2) + | 0(5:3) + | '''1'''(5:4) + | 0(5:5) + | 0(5:6) + | 0(5:7) + |- | &nbsp; | 0(6:1) + | 0(6:2) + | 0(6:3) + | 0(6:4) + | 0(6:5) + | 0(6:6) + | 0(6:7) + |- | &nbsp; | 0(7:1) + | 0(7:2) + | 0(7:3) + | 0(7:4) + | 0(7:5) + | 0(7:6) + | 0(7:7) |} Stripping down to the bare essentials, one obtains the following matrices of coefficients for the relations ''G''&nbsp;and&nbsp;''H''. {| style="text-align:center; width=30%" | style="width:20px" | &nbsp; || ''G'' || = |- | &nbsp; || 0 || 0 || 0 || 0 || 0 || 0 || 0 |- | &nbsp; || 0 || 0 || 0 || 0 || 0 || 0 || 0 |- | &nbsp; || 0 || 0 || 0 || 0 || 0 || 0 || 0 |- | &nbsp; || 0 || 0 || 1 || 1 || 1 || 0 || 0 |- | &nbsp; || 0 || 0 || 0 || 0 || 0 || 0 || 0 |- | &nbsp; || 0 || 0 || 0 || 0 || 0 || 0 || 0 |- | &nbsp; || 0 || 0 || 0 || 0 || 0 || 0 || 0 |} <br> {| style="text-align:center; width=30%" | style="width:20px" | &nbsp; || ''H'' || = |- | &nbsp; || 0 || 0 || 0 || 0 || 0 || 0 || 0 |- | &nbsp; || 0 || 0 || 0 || 0 || 0 || 0 || 0 |- | &nbsp; || 0 || 0 || 0 || 1 || 0 || 0 || 0 |- | &nbsp; || 0 || 0 || 0 || 1 || 0 || 0 || 0 |- | &nbsp; || 0 || 0 || 0 || 1 || 0 || 0 || 0 |- | &nbsp; || 0 || 0 || 0 || 0 || 0 || 0 || 0 |- | &nbsp; || 0 || 0 || 0 || 0 || 0 || 0 || 0 |} These are the logical matrix representations of the 2-adic relations ''G''&nbsp;and&nbsp;''H''. If the 2-adic relations ''G'' and ''H'' are viewed as logical sums, then their relational composition ''G''&nbsp;&omicron;&nbsp;''H'' can be regarded as a product of sums, a fact that can be indicated as follows: : ''G''&nbsp;&omicron;&nbsp;''H'' = (&sum;_''ij'' ''G''_''ij''(''i'':''j''))(&sum;_''ij'' ''H''_''ij''(''i'':''j'')). The composite relation ''G''&nbsp;&omicron;&nbsp;''H'' is itself a 2-adic relation over the same space ''X'', in other words, ''G''&nbsp;&omicron;&nbsp;''H''&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''X'', and this means that ''G''&nbsp;&omicron;&nbsp;''H'' must be amenable to being written as a logical sum of the following form: : ''G''&nbsp;&omicron;&nbsp;''H'' = &sum;_''ij'' (''G''&nbsp;&omicron;&nbsp;''H'')_''ij''(''i'':''j''). In this formula, (''G''&nbsp;&omicron;&nbsp;''H'')_''ij'' is the coefficient of ''G''&nbsp;&omicron;&nbsp;''H'' with respect to the elementary relation ''i'':''j''. One of the best ways to reason out what ''G''&nbsp;&omicron;&nbsp;''H'' should be is to ask oneself what its coefficient (''G''&nbsp;&omicron;&nbsp;''H'')_''ij'' should be for each of the elementary relations ''i'':''j'' in turn. So let us pose the question: : (''G''&nbsp;&omicron;&nbsp;''H'')_''ij'' = ? In order to answer this question, it helps to realize that the indicated product given above can be written in the following equivalent form: : ''G''&nbsp;&omicron;&nbsp;''H'' = (&sum;_''ik'' ''G''_''ik''(''i'':''k''))(&sum;_''kj'' ''H''_''kj''(''k'':''j'')). A moment's thought will tell us that (''G''&nbsp;&omicron;&nbsp;''H'')_''ij''&nbsp;=&nbsp;1 if and only if there is an element ''k'' in ''X'' such that ''G''_''ik''&nbsp;=&nbsp;1 and ''H''_''kj''&nbsp;=&nbsp;1. Consequently, we have the result: : (''G''&nbsp;&omicron;&nbsp;''H'')_''ij'' = &sum;_k ''G''_''ik''''H''_''kj''. This follows from the properties of boolean arithmetic, specifically, from the fact that the product ''G''_''ik''''H''_''kj'' is 1 if and only if both ''G''_''ik'' and ''H''_''kj'' are 1, and from the fact that &sum;_''k'' ''F''_''k'' is equal to 1 just in case some ''F''_''k'' is 1. All that remains in order to obtain a computational formula for the relational composite ''G''&nbsp;&omicron;&nbsp;''H'' of the 2-adic relations ''G'' and ''H'' is to collect the coefficients (''G''&nbsp;&omicron;&nbsp;''H'')_''ij'' over the appropriate basis of elementary relations ''i'':''j'', as ''i'' and ''j'' range through ''X''. : ''G''&nbsp;&omicron;&nbsp;''H'' = &sum;_''ij'' (''G''&nbsp;&omicron;&nbsp;''H'')_''ij''(''i'':''j'') = &sum;_''ij''(&sum;_''k'' ''G''_''ik''''H''_''kj'')(''i'':''j''). This is the logical analogue of matrix multiplication in linear algebra, the difference in the logical setting being that all of the operations performed on coefficients take place in a system of boolean arithmetic where summation corresponds to logical disjunction and multiplication corresponds to logical conjunction. By way of disentangling this formula, one may notice that the form &sum;_''k'' (''G''_''ik''''H''_''kj'') is what is usually called a "scalar product". In this case it is the scalar product of the ''i''^th row of ''G'' with the ''j''^th column of ''H''. To make this statement more concrete, let us go back to the particular examples of ''G'' and ''H'' that we came in with: {| style="text-align:center; width=30%" | style="width:20px" | &nbsp; || ''G'' || = |- | &nbsp; || 0 || 0 || 0 || 0 || 0 || 0 || 0 |- | &nbsp; || 0 || 0 || 0 || 0 || 0 || 0 || 0 |- | &nbsp; || 0 || 0 || 0 || 0 || 0 || 0 || 0 |- | &nbsp; || 0 || 0 || 1 || 1 || 1 || 0 || 0 |- | &nbsp; || 0 || 0 || 0 || 0 || 0 || 0 || 0 |- | &nbsp; || 0 || 0 || 0 || 0 || 0 || 0 || 0 |- | &nbsp; || 0 || 0 || 0 || 0 || 0 || 0 || 0 |} <br> {| style="text-align:center; width=30%" | style="width:20px" | &nbsp; || ''H'' || = |- | &nbsp; || 0 || 0 || 0 || 0 || 0 || 0 || 0 |- | &nbsp; || 0 || 0 || 0 || 0 || 0 || 0 || 0 |- | &nbsp; || 0 || 0 || 0 || 1 || 0 || 0 || 0 |- | &nbsp; || 0 || 0 || 0 || 1 || 0 || 0 || 0 |- | &nbsp; || 0 || 0 || 0 || 1 || 0 || 0 || 0 |- | &nbsp; || 0 || 0 || 0 || 0 || 0 || 0 || 0 |- | &nbsp; || 0 || 0 || 0 || 0 || 0 || 0 || 0 |} The formula for computing ''G''&nbsp;&omicron;&nbsp;''H'' says the following: {| cellpadding="2" |- | style="width:20px" | &nbsp; | align="center" | (''G''&nbsp;&omicron;&nbsp;''H'')_''ij'' | &nbsp; |- | &nbsp; | align="center" | = | the ''ij''^th entry in the matrix representation for ''G''&nbsp;&omicron;&nbsp;''H'' |- | &nbsp; | align="center" | = | the entry in the ''i''^th row and the ''j''^th column of ''G''&nbsp;&omicron;&nbsp;''H'' |- | &nbsp; | align="center" | = | the scalar product of the ''i''^th row of ''G'' with the ''j''^th column of ''H'' |- | &nbsp; | align="center" | = | &sum;_''k'' ''G''_''ik''''H''_''kj'' |} As it happens, it is possible to make exceedingly light work of this example, since there is only one row of ''G'' and one column of ''H'' that are not all zeroes. Taking the scalar product, in a logical way, of the fourth row of ''G'' with the fourth column of ''H'' produces the sole non-zero entry for the matrix of ''G''&nbsp;&omicron;&nbsp;''H''. {| cellpadding="2px" style="text-align:center" | style="width:20px" | &nbsp; | ''G''&nbsp;&omicron;&nbsp;''H'' | = |} {| style="text-align:center; width=30%" | style="width:20px" | &nbsp; | 0 || 0 || 0 || 0 || 0 || 0 || 0 |- | &nbsp; | 0 || 0 || 0 || 0 || 0 || 0 || 0 |- | &nbsp; | 0 || 0 || 0 || 0 || 0 || 0 || 0 |- | &nbsp; | 0 || 0 || 0 || 1 || 0 || 0 || 0 |- | &nbsp; | 0 || 0 || 0 || 0 || 0 || 0 || 0 |- | &nbsp; | 0 || 0 || 0 || 0 || 0 || 0 || 0 |- | &nbsp; | 0 || 0 || 0 || 0 || 0 || 0 || 0 |} ==Graph-theoretic picture== There is another form of representation for 2-adic relations that is useful to keep in mind, especially for its ability to render the logic of many complex formulas almost instantly understandable to the mind's eye. This is the representation in terms of ''[[bipartite graph]]s'', or ''bigraphs'' for short. Here is what ''G'' and ''H'' look like in the bigraph picture: o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | /|\ | | / | \ G | | / | \ | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 9. G = 4:3 + 4:4 + 4:5 o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | \ | / | | \ | / H | | \|/ | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 10. H = 3:4 + 4:4 + 5:4 These graphs may be read to say: :* ''G'' puts 4 in relation to 3, 4, 5. :* ''H'' puts 3, 4, 5 in relation to 4. To form the composite relation ''G''&nbsp;&omicron;&nbsp;''H'', one simply follows the bigraph for ''G'' by the bigraph for ''H'', here arranging the bigraphs in order down the page, and then treats any non-empty set of paths of length two between two nodes as being equivalent to a single directed edge between those nodes in the composite bigraph for ''G''&nbsp;&omicron;&nbsp;''H''. Here's how it looks in pictures: o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | /|\ | | / | \ G | | / | \ | | o o o o o o o X | | \ | / | | \ | / H | | \|/ | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 11. G Followed By H o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | | | | | G o H | | | | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 12. G Composed With H Once again we find that ''G''&nbsp;&omicron;&nbsp;''H'' = 4:4. We have now seen three different representations of 2-adic relations. If one has a strong preference for letters, or numbers, or pictures, then one may be tempted to take one or another of these as being canonical, but each of them will be found to have its peculiar advantages and disadvantages in any given application, and the maximum advantage is therefore approached by keeping all three of them in mind. To see the promised utility of the bigraph picture of 2-adic relations, let us devise a slightly more complex example of a composition problem, and use it to illustrate the logic of the matrix multiplication formula. Keeping to the same space ''X'' = {1, 2, 3, 4, 5, 6, 7}, define the 2-adic relations ''M'',&nbsp;''N''&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''X'' as follows: {| cellpadding="2px" style="text-align:center" | style="width:20px" | &nbsp; | ''M'' | &nbsp; | = | &nbsp; | 2:1 | + | 2:2 | + | 2:3 | + | 4:3 | + | 4:4 | + | 4:5 | + | 6:5 | + | 6:6 | + | 6:7 |- | &nbsp; | ''N'' | &nbsp; | = | &nbsp; | 1:1 | + | 2:1 | + | 3:3 | + | 4:3 | &nbsp; | + | &nbsp; | 4:5 | + | 5:5 | + | 6:7 | + | 7:7 |} Here are the bigraph pictures: o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | /|\ /|\ /|\ | | / | \ / | \ / | \ M | | / | \ / | \ / | \ | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 13. Dyadic Relation M o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | | / | / \ | \ | | | | / | / \ | \ | N | | |/ |/ \| \| | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 14. Dyadic Relation N To form the composite relation ''M''&nbsp;&omicron;&nbsp;''N'', one simply follows the bigraph for ''M'' by the bigraph for ''N'', here arranging the bigraphs in order down the page, and then counts any non-empty set of paths of length two between two nodes as being equivalent to a single directed edge between those two nodes in the composite bigraph for ''M''&nbsp;&omicron;&nbsp;''N''. Here's how it looks in pictures: o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | /|\ /|\ /|\ | | / | \ / | \ / | \ M | | / | \ / | \ / | \ | | o o o o o o o X | | | / | / \ | \ | | | | / | / \ | \ | N | | |/ |/ \| \| | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 15. M Followed By N o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | / \ / \ / \ | | / \ / \ / \ M o N | | / \ / \ / \ | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 16. M Composed With N Let us hark back to that mysterious matrix multiplication formula, and see how it appears in the light of the bigraph representation. The coefficient of the composition ''M''&nbsp;&omicron;&nbsp;''N'' between ''i'' and ''j'' in ''X'' is given as follows: : (''M''&nbsp;&omicron;&nbsp;''N'')_''ij'' = &sum;_''k''(''M''_''ik''''N''_''kj'') Graphically interpreted, this is a ''sum over paths''. Starting at the node ''i'', ''M''_''ik'' being 1 indicates that there is an edge in the bigraph of ''M'' from node ''i'' to node ''k'', and ''N''_''kj'' being 1 indicates that there is an edge in the bigraph of ''N'' from node ''k'' to node ''j''. So the &sum;_''k'' ranges over all possible intermediaries ''k'', ascending from 0 to 1 just as soon as there happens to be some path of length two between nodes ''i'' and ''j''. It is instructive at this point to compute the other possible composition that can be formed from ''M'' and ''N'', namely, the composition ''N''&nbsp;&omicron;&nbsp;''M'', that takes ''M'' and ''N'' in the opposite order. Here is the graphic computation: o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | | / | / \ | \ | | | | / | / \ | \ | N | | |/ |/ \| \| | | o o o o o o o X | | /|\ /|\ /|\ | | / | \ / | \ / | \ M | | / | \ / | \ / | \ | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 17. N Followed By M o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | | | N o M | | | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 18. N Composed With M In sum, ''N''&nbsp;&omicron;&nbsp;''M'' = 0. This example affords sufficient evidence that relational composition, just like its kindred, matrix multiplication, is a ''[[non-commutative]]'' algebraic operation. ==References== * [[Stanislaw Marcin Ulam|Ulam, S.M.]] and [[Al Bednarek|Bednarek, A.R.]], "On the Theory of Relational Structures and Schemata for Parallel Computation" (1977), pp. 477-508 in A.R. Bednarek and Françoise Ulam (eds.), ''Analogies Between Analogies: The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators'', University of California Press, Berkeley, CA, 1990. ==Bibliography== * [[Mathematical Society of Japan]], ''Encyclopedic Dictionary of Mathematics'', 2nd edition, 2 vols., Kiyosi Itô (ed.), MIT Press, Cambridge, MA, 1993. * Mili, A., Desharnais, J., Mili, F., with Frappier, M., ''Computer Program Construction'', Oxford University Press, New York, NY, 1994. — Introduction to Tarskian relation theory and its applications within the relational programming paradigm. * [[Stanislaw Marcin Ulam|Ulam, S.M.]], ''Analogies Between Analogies: The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators'', A.R. Bednarek and Françoise Ulam (eds.), University of California Press, Berkeley, CA, 1990. ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://mywikibiz.com/Relation_composition Relation Composition @ MyWikiBiz] * [http://mathweb.org/wiki/Relation_composition Relation Composition @ MathWeb Wiki] * [http://netknowledge.org/wiki/Relation_composition Relation Composition @ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Relation_composition Relation Composition @ OER Commons] {{col-break}} * [http://p2pfoundation.net/Relation_Composition Relation Composition @ P2P Foundation] * [http://semanticweb.org/wiki/Relation_composition Relation Composition @ SemanticWeb] * [http://ref.subwiki.org/wiki/Relation_composition Relation Composition @ Subject Wikis] * [http://beta.wikiversity.org/wiki/Relation_composition Relation Composition @ Wikiversity Beta] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;] * [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;] * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;] ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://mywikibiz.com/Relation_composition Relation Composition], [http://mywikibiz.com/ MyWikiBiz] * [http://mathweb.org/wiki/Relation_composition Relation Composition], [http://mathweb.org/wiki/ MathWeb Wiki] * [http://netknowledge.org/wiki/Relation_composition Relation Composition], [http://netknowledge.org/ NetKnowledge] * [http://wiki.oercommons.org/mediawiki/index.php/Relation_composition Relation Composition], [http://wiki.oercommons.org/ OER Commons] * [http://p2pfoundation.net/Relation_Composition Relation Composition], [http://p2pfoundation.net/ P2P Foundation] {{col-break}} * [http://semanticweb.org/wiki/Relation_composition Relation Composition], [http://semanticweb.org/ Semantic Web] * [http://planetmath.org/encyclopedia/RelationComposition2.html Relation Composition], [http://planetmath.org/ PlanetMath] * [http://getwiki.net/-Relational_Composition Relation Composition], [http://getwiki.net/ GetWiki] * [http://wikinfo.org/index.php/Relation_composition Relation Composition], [http://wikinfo.org/ Wikinfo] * [http://en.wikipedia.org/w/index.php?title=Relation_composition&oldid=43467878 Relation Composition], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Database Theory]] [[Category:Discrete Mathematics]] [[Category:Formal Sciences]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Relation Theory]] [[Category:Set Theory]] ceba6137212487fa34813e9df040a73c86821f1f 2 163 698 650 2015-10-28T15:28:44Z Jon Awbrey 3 update wikitext text/x-wiki <br> <p><font face="lucida calligraphy" size="7">Jon Awbrey</font></p> <br> __NOTOC__ ==Presently &hellip;== <center> [https://oeis.org/wiki/Riffs_and_Rotes Riffs and Rotes] [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] [http://intersci.ss.uci.edu/wiki/index.php/Theme_One_Program Theme One Program] [http://intersci.ss.uci.edu/wiki/index.php/Propositions_As_Types Propositions As Types] [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems] [http://intersci.ss.uci.edu/wiki/index.php/Peirce%27s_1870_Logic_Of_Relatives Peirce's Logic Of Relatives] Logical Graphs : [http://inquiryintoinquiry.com/2008/07/29/logical-graphs-1/ One] [http://inquiryintoinquiry.com/2008/09/19/logical-graphs-2/ Two] [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] [http://intersci.ss.uci.edu/wiki/index.php/Differential_Analytic_Turing_Automata Differential Analytic Turing Automata] [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to 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operator]] [[Multigrade operator]] [[Normative science]] [[Null graph]] [[On a New List of Categories]] [[Parametric operator]] [[Peirce's law]] [[Philosophy of mathematics]] [[Pragmatic information]] [[Pragmatic maxim]] [[Pragmatic theory of truth]] [[Pragmaticism]] [[Pragmatism]] [[Prescisive abstraction]] [[Propositional calculus]] [[Relation (mathematics)]] [[Relation composition]] [[Relation construction]] [[Relation reduction]] [[Relation theory]] [[Relation type]] [[Relative term]] [[Semeiotic]] [[Semiotic information]] [[Semiotics]] [[Sign relation]] [[Sign relational complex]] [[Sole sufficient operator]] [[Tacit extension]] [[The Simplest Mathematics]] [[Triadic relation]] [[Truth table]] [[Truth theory]] [[Universe of discourse]] [[What we've got here is (a) failure to communicate]] [[Zeroth order logic]] ===Papers=== ====Functional Logic==== * [[Functional Logic : Inquiry and Analogy]] * [[Functional Logic : Higher Order Propositions]] * [[Functional Logic : Quantification Theory]] ====Differential Logic==== * [[Differential Logic : Introduction]] * [[Differential Propositional Calculus]] * [[Differential Logic and Dynamic Systems 2.0|Differential Logic and Dynamic Systems]] ====Logic and Semiotics==== * [[Futures Of Logical Graphs]] * [[Peirce's 1870 Logic Of Relatives]] * [[Peirce's Logic Of Information]] * [[Propositional Equation Reasoning Systems]] * [[Semiotic Information]] ====Inquiry Driven Systems==== * [[Prospects for Inquiry Driven Systems]] * [[Introduction to Inquiry Driven Systems]] * [[Inquiry Driven Systems : Fields Of Inquiry]] * [[Inquiry Driven Systems|Inquiry Driven Systems : Inquiry Into Inquiry]] ===Projects=== * [[Cactus Language]] * [[User:Jon Awbrey/Differential Logic|Differential Logic]] * [[Theme One Program]] ===Poetry=== * [[Past All Reckoning]] * [[Poems Of Emediate Moment]] * [[Questionable Verses]] ==Presentations and Publications== * Awbrey, S.M., and Awbrey, J.L. (May 2001), &ldquo;Conceptual Barriers to Creating Integrative Universities&rdquo;, ''Organization : The Interdisciplinary Journal of Organization, Theory, and Society'' 8(2), Sage Publications, London, UK, pp. 269&ndash;284. [http://org.sagepub.com/content/8/2/269.abstract Abstract]. * Awbrey, S.M., and Awbrey, J.L. (September 1999), &ldquo;Organizations of Learning or Learning Organizations : The Challenge of Creating Integrative Universities for the Next Century&rdquo;, ''Second International Conference of the Journal &lsquo;Organization&rsquo;'', ''Re-Organizing Knowledge, Trans-Forming Institutions : Knowing, Knowledge, and the University in the 21st Century'', University of Massachusetts, Amherst, MA. [http://cspeirce.com/menu/library/aboutcsp/awbrey/integrat.htm Online]. * Awbrey, J.L., and Awbrey, S.M. (Autumn 1995), &ldquo;Interpretation as Action : The Risk of Inquiry&rdquo;, ''Inquiry : Critical Thinking Across the Disciplines'' 15(1), pp. 40&ndash;52. [https://web.archive.org/web/19970626071826/http://chss.montclair.edu/inquiry/fall95/awbrey.html Archive]. [https://www.pdcnet.org/inquiryct/content/inquiryct_1995_0015_0001_0040_0052 Journal]. [https://www.academia.edu/1266493/Interpretation_as_Action_The_Risk_of_Inquiry Online]. * Awbrey, J.L., and Awbrey, S.M. (June 1992), &ldquo;Interpretation as Action : The Risk of Inquiry&rdquo;, ''The Eleventh International Human Science Research Conference'', Oakland University, Rochester, Michigan. * Awbrey, S.M., and Awbrey, J.L. (May 1991), &ldquo;An Architecture for Inquiry : Building Computer Platforms for Discovery&rdquo;, ''Proceedings of the Eighth International Conference on Technology and Education'', Toronto, Canada, pp. 874&ndash;875. [http://www.academia.edu/1270327/An_Architecture_for_Inquiry_Building_Computer_Platforms_for_Discovery Online]. * Awbrey, J.L., and Awbrey, S.M. (January 1991), &ldquo;Exploring Research Data Interactively : Developing a Computer Architecture for Inquiry&rdquo;, Poster presented at the ''Annual Sigma Xi Research Forum'', University of Texas Medical Branch, Galveston, TX. * Awbrey, J.L., and Awbrey, S.M. (August 1990), &ldquo;Exploring Research Data Interactively. Theme One : A Program of Inquiry&rdquo;, ''Proceedings of the Sixth Annual Conference on Applications of Artificial Intelligence and CD-ROM in Education and Training'', Society for Applied Learning Technology, Washington, DC, pp. 9&ndash;15. [http://academia.edu/1272839/Exploring_Research_Data_Interactively._Theme_One_A_Program_of_Inquiry Online]. ==Education== * 1993&ndash;2003. [http://web.archive.org/web/20120202222443/http://www2.oakland.edu/secs/dispprofile.asp?Fname=Fatma&Lname=Mili Graduate Study], [http://web.archive.org/web/20120203004703/http://www2.oakland.edu/secs/dispprofile.asp?Fname=Mohamed&Lname=Zohdy Systems Engineering], [http://meadowbrookhall.org/explore/history/meadowbrookhall Oakland University]. * '''1989. [http://web.archive.org/web/20001206101800/http://www.msu.edu/dig/msumap/psychology.html M.A. Psychology]''', [http://web.archive.org/web/20000902103838/http://www.msu.edu/dig/msumap/beaumont.html Michigan State University]. * 1985&ndash;1986. [http://quod.lib.umich.edu/cgi/i/image/image-idx?id=S-BHL-X-BL001808%5DBL001808 Graduate Study, Mathematics], [http://www.umich.edu/ University of Michigan]. * 1985. [http://web.archive.org/web/20061230174652/http://www.uiuc.edu/navigation/buildings/altgeld.top.html Graduate Study, Mathematics], [http://www.uiuc.edu/ University of Illinois at Urbana&ndash;Champaign]. * 1984. [http://www.psych.uiuc.edu/graduate/ Graduate Study, Psychology], [http://www.uiuc.edu/ University of Illinois at Champaign&ndash;Urbana]. * '''1980. [http://www.mth.msu.edu/images/wells_medium.jpg M.A. Mathematics]''', [http://www.msu.edu/~hvac/survey/BeaumontTower.html Michigan State University]. * '''1976. [http://web.archive.org/web/20001206050600/http://www.msu.edu/dig/msumap/phillips.html B.A. Mathematical and Philosophical Method]''', <br> [http://www.enolagaia.com/JMC.html Justin Morrill College], [http://www.msu.edu/ Michigan State University]. a396f9a23f1c3d143b70fbdecd2da75c13eebbc6 0 312 700 649 2015-10-29T15:18:18Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. This article treats relations from the perspective of combinatorics, in other words, as a subject matter in discrete mathematics, with special attention to finite structures and concrete set-theoretic constructions, many of which arise quite naturally in applications. This approach to relation theory, or the theory of relations, is distinguished from, though closely related to, its study from the perspectives of abstract algebra on the one hand and formal logic on the other. ==Preliminaries== Two definitions of the relation concept are common in the literature. Although it is usually clear in context which definition is being used at a given time, it tends to become less clear as contexts collide, or as discussion moves from one context to another. The same sort of ambiguity arose in the development of the function concept and it may save some effort to follow the pattern of resolution that worked itself out there. When we speak of a function <math>f : X \to Y\!</math> we are thinking of a mathematical object whose articulation requires three pieces of data, specifying the set <math>X,\!</math> the set <math>Y,\!</math> and a particular subset of their cartesian product <math>{X \times Y}.\!</math> So far so good. Let us write <math>f = (\mathrm{obj_1}f, \mathrm{obj_2}f, \mathrm{obj_{12}}f)\!</math> to express what has been said so far. When it comes to parsing the notation <math>{}^{\backprime\backprime} f : X \to Y {}^{\prime\prime},\!</math> everyone takes the part <math>{}^{\backprime\backprime} X \to Y {}^{\prime\prime}\!</math> to specify the ''type'' of the function, that is, the pair <math>(\mathrm{obj_1}f, \mathrm{obj_2}f),\!</math> but <math>{}^{\backprime\backprime} f {}^{\prime\prime}\!</math> is used equivocally to denote both the triple and the subset <math>\mathrm{obj_{12}}f\!</math> that forms one part of it. One way to resolve the ambiguity is to formalize a distinction between a function and its ''graph'', letting <math>\mathrm{graph}(f) := \mathrm{obj_{12}}f.\!</math> Another tactic treats the whole notation <math>{}^{\backprime\backprime} f : X \to Y {}^{\prime\prime}\!</math> as sufficient denotation for the triple, letting <math>{}^{\backprime\backprime} f {}^{\prime\prime}\!</math> denote <math>\mathrm{graph}(f).\!</math> In categorical and computational contexts, at least initially, the type is regarded as an essential attribute or an integral part of the function itself. In other contexts it may be desirable to use a more abstract concept of function, treating a function as a mathematical object that appears in connection with many different types. Following the pattern of the functional case, let the notation <math>{}^{\backprime\backprime} L \subseteq X \times Y {}^{\prime\prime}\!</math> bring to mind a mathematical object that is specified by three pieces of data, the set <math>X,\!</math> the set <math>Y,\!</math> and a particular subset of their cartesian product <math>{X \times Y}.\!</math> As before we have two choices, either let <math>L = (X, Y, \mathrm{graph}(L))\!</math> or let <math>{}^{\backprime\backprime} L {}^{\prime\prime}\!</math> denote <math>\mathrm{graph}(L)\!</math> and choose another name for the triple. ==Definition== It is convenient to begin with the definition of a ''<math>k\!</math>-place relation'', where <math>k\!</math> is a positive integer. '''Definition.''' A ''<math>k\!</math>-place relation'' <math>L \subseteq X_1 \times \ldots \times X_k\!</math> over the nonempty sets <math>X_1, \ldots, X_k\!</math> is a <math>(k+1)\!</math>-tuple <math>(X_1, \ldots, X_k, L)\!</math> where <math>L\!</math> is a subset of the cartesian product <math>X_1 \times \ldots \times X_k.\!</math> ==Remarks== Though usage varies as usage will, there are several bits of optional language that are frequently useful in discussing relations. The sets <math>X_1, \ldots, X_k\!</math> are called the ''domains'' of the relation <math>L \subseteq X_1 \times \ldots \times X_k,\!</math> with <math>{X_j}\!</math> being the <math>j^\text{th}\!</math> domain. If all of the <math>{X_j}\!</math> are the same set <math>X\!</math> then <math>L \subseteq X_1 \times \ldots \times X_k\!</math> is more simply described as a <math>k\!</math>-place relation over <math>X.\!</math> The set <math>L\!</math> is called the ''graph'' of the relation <math>L \subseteq X_1 \times \ldots \times X_k,\!</math> on analogy with the graph of a function. If the sequence of sets <math>X_1, \ldots, X_k\!</math> is constant throughout a given discussion or is otherwise determinate in context, then the relation <math>L \subseteq X_1 \times \ldots \times X_k\!</math> is determined by its graph <math>L,\!</math> making it acceptable to denote the relation by referring to its graph. Other synonyms for the adjective ''<math>k\!</math>-place'' are ''<math>k\!</math>-adic'' and ''<math>k\!</math>-ary'', all of which leads to the integer <math>k\!</math> being called the ''dimension'', ''adicity'', or ''arity'' of the relation <math>L.\!</math> ==Local incidence properties== A ''local incidence property'' (LIP) of a relation <math>L\!</math> is a property that depends in turn on the properties of special subsets of <math>L\!</math> that are known as its ''local flags''. The local flags of a relation are defined in the following way: Let <math>L\!</math> be a <math>k\!</math>-place relation <math>L \subseteq X_1 \times \ldots \times X_k.\!</math> Select a relational domain <math>{X_j}\!</math> and one of its elements <math>x.\!</math> Then <math>L_{x \operatorname{at} j}\!</math> is a subset of <math>L\!</math> that is referred to as the ''flag'' of <math>L\!</math> with <math>x\!</math> at <math>{j},\!</math> or the ''<math>x \operatorname{at} j\!</math>-flag'' of <math>L,\!</math> an object that has the following definition: {| align="center" cellpadding="8" style="text-align:center" | <math>L_{x \operatorname{at} j} ~=~ \{ (x_1, \ldots, x_j, \ldots, x_k) \in L ~:~ x_j = x \}.\!</math> |} Any property <math>C\!</math> of the local flag <math>L_{x \operatorname{at} j} \subseteq L\!</math> is said to be a ''local incidence property'' of <math>L\!</math> with respect to the ''locus'' <math>x \operatorname{at} j.\!</math> A <math>k\!</math>-adic relation <math>L \subseteq X_1 \times \ldots \times X_k\!</math> is said to be ''<math>C\!</math>-regular'' at <math>j\!</math> if and only if every flag of <math>L\!</math> with <math>x\!</math> at <math>j\!</math> has the property <math>C,\!</math> where <math>x\!</math> is taken to vary over the ''theme'' of the fixed domain <math>X_j.\!</math> Expressed in symbols, <math>L\!</math> is ''<math>C\!</math>-regular'' at <math>j\!</math> if and only if <math>C(L_{x \operatorname{at} j})\!</math> is true for all <math>x\!</math> in <math>X_j.\!</math> ==Regional incidence properties== The definition of a local flag can be broadened from a point <math>x\!</math> in <math>{X_j}\!</math> to a subset <math>M\!</math> of <math>X_j,\!</math> arriving at the definition of a ''regional flag'' in the following way: Suppose that <math>L \subseteq X_1 \times \ldots \times X_k,\!</math> and choose a subset <math>M \subseteq X_j.\!</math> Then <math>L_{M \operatorname{at} j}\!</math> is a subset of <math>L\!</math> that is said to be the ''flag'' of <math>L\!</math> with <math>M\!</math> at <math>{j},\!</math> or the ''<math>M \operatorname{at} j\!</math>-flag'' of <math>L,\!</math> an object which has the following definition: {| align="center" cellpadding="8" style="text-align:center" | <math>L_{M \operatorname{at} j} ~=~ \{ (x_1, \ldots, x_j, \ldots, x_k) \in L ~:~ x_j \in M \}.\!</math> |} ==Numerical incidence properties== A ''numerical incidence property'' (NIP) of a relation is a local incidence property that depends on the cardinalities of its local flags. For example, <math>L\!</math> is said to be ''<math>c\!</math>-regular at <math>j\!</math>'' if and only if the cardinality of the local flag <math>L_{x \operatorname{at} j}\!</math> is <math>c\!</math> for all <math>x\!</math> in <math>{X_j},\!</math> or, to write it in symbols, if and only if <math>|L_{x \operatorname{at} j}| = c\!</math> for all <math>{x \in X_j}.\!</math> In a similar fashion, one can define the NIPs, ''<math>(<\!c)\!</math>-regular at <math>{j},\!</math>'' ''<math>(>\!c)\!</math>-regular at <math>{j},\!</math>'' and so on. For ease of reference, a few of these definitions are recorded here: {| align="center" cellpadding="8" style="text-align:center" | <math>\begin{matrix} L & \text{is} & c\textit{-regular} & \text{at}\ j & \text{if and only if} & |L_{x \operatorname{at} j}| & = & c & \text{for all}\ x \in X_j. \\[4pt] L & \text{is} & (<\!c)\textit{-regular} & \text{at}\ j & \text{if and only if} & |L_{x \operatorname{at} j}| & < & c & \text{for all}\ x \in X_j. \\[4pt] L & \text{is} & (>\!c)\textit{-regular} & \text{at}\ j & \text{if and only if} & |L_{x \operatorname{at} j}| & > & c & \text{for all}\ x \in X_j. \end{matrix}</math> |} ==Species of 2-adic relations== Returning to 2-adic relations, it is useful to describe some familiar classes of objects in terms of their local and numerical incidence properties. Let <math>L \subseteq S \times T\!</math> be an arbitrary 2-adic relation. The following properties of <math>L\!</math> can be defined: {| align="center" cellpadding="8" style="text-align:center" | <math>\begin{matrix} L & \text{is} & \textit{total} & \text{at}~ S & \text{if and only if} & L & \text{is} & (\ge 1)\text{-regular} & \text{at}~ S. \\[4pt] L & \text{is} & \textit{total} & \text{at}~ T & \text{if and only if} & L & \text{is} & (\ge 1)\text{-regular} & \text{at}~ T. \\[4pt] L & \text{is} & \textit{tubular} & \text{at}~ S & \text{if and only if} & L & \text{is} & (\le 1)\text{-regular} & \text{at}\ S. \\[4pt] L & \text{is} & \textit{tubular} & \text{at}~ T & \text{if and only if} & L & \text{is} & (\le 1)\text{-regular} & \text{at}~ T. \end{matrix}</math> |} If <math>L \subseteq S \times T\!</math> is tubular at <math>S\!</math> then <math>L\!</math> is called a ''partial function'' or a ''prefunction'' from <math>S\!</math> to <math>T.\!</math> This is sometimes indicated by giving <math>L\!</math> an alternate name, say, <math>{}^{\backprime\backprime} p {}^{\prime\prime},~\!</math> and writing <math>L = p : S \rightharpoonup T.\!</math> Just by way of formalizing the definition: {| align="center" cellpadding="8" style="text-align:center" | <math>\begin{matrix} L & = & p : S \rightharpoonup T & \text{if and only if} & L & \text{is} & \text{tubular} & \text{at}~ S. \end{matrix}</math> |} If <math>L\!</math> is a prefunction <math>p : S \rightharpoonup T\!</math> that happens to be total at <math>S,\!</math> then <math>L\!</math> is called a ''function'' from <math>S\!</math> to <math>T,\!</math> indicated by writing <math>L = f : S \to T.\!</math> To say that a relation <math>L \subseteq S \times T\!</math> is ''totally tubular'' at <math>S\!</math> is to say that it is <math>1\!</math>-regular at <math>S.\!</math> Thus, we may formalize the following definition: {| align="center" cellpadding="8" style="text-align:center" | <math>\begin{matrix} L & = & f : S \to T & \text{if and only if} & L & \text{is} & 1\text{-regular} & \text{at}~ S. \end{matrix}</math> |} In the case of a function <math>f : S \to T,\!</math> one has the following additional definitions: {| align="center" cellpadding="8" style="text-align:center" | <math>\begin{matrix} f & \text{is} & \textit{surjective} & \text{if and only if} & f & \text{is} & \text{total} & \text{at}~ T. \\[4pt] f & \text{is} & \textit{injective} & \text{if and only if} & f & \text{is} & \text{tubular} & \text{at}~ T. \\[4pt] f & \text{is} & \textit{bijective} & \text{if and only if} & f & \text{is} & 1\text{-regular} & \text{at}~ T. \end{matrix}</math> |} ==Variations== Because the concept of a relation has been developed quite literally from the beginnings of logic and mathematics, and because it has incorporated contributions from a diversity of thinkers from many different times and intellectual climes, there is a wide variety of terminology that the reader may run across in connection with the subject. One dimension of variation is reflected in the names that are given to <math>k\!</math>-place relations, for <math>k = 0, 1, 2, 3, \ldots,\!</math> with some writers using the Greek forms, ''medadic'', ''monadic'', ''dyadic'', ''triadic'', ''<math>k\!</math>-adic'', and other writers using the Latin forms, ''nullary'', ''unary'', ''binary'', ''ternary'', ''<math>k\!</math>-ary''. The number of relational domains may be referred to as the ''adicity'', ''arity'', or ''dimension'' of the relation. Accordingly, one finds a relation on a finite number of domains described as a ''polyadic'' relation or a ''finitary'' relation, but others count infinitary relations among the polyadic. If the number of domains is finite, say equal to <math>k,\!</math> then the relation may be described as a ''<math>k\!</math>-adic'' relation, a ''<math>k\!</math>-ary'' relation, or a ''<math>k\!</math>-dimensional'' relation, respectively. A more conceptual than nominal variation depends on whether one uses terms like ''predicate'', ''relation'', and even ''term'' to refer to the formal object proper or else to the allied syntactic items that are used to denote them. Compounded with this variation is still another, frequently associated with philosophical differences over the status in reality accorded formal objects. Among those who speak of numbers, functions, properties, relations, and sets as being real, that is to say, as having objective properties, there are divergences as to whether some things are more real than others, especially whether particulars or properties are equally real or else one is derivative in relationship to the other. Historically speaking, just about every combination of modalities has been used by one school of thought or another, but it suffices here merely to indicate how the options are generated. ==Examples== : ''See the articles on [[relation (mathematics)|relations]], [[relation composition]], [[relation reduction]], [[sign relation]]s, and [[triadic relation]]s for concrete examples of relations.'' Many relations of the greatest interest in mathematics are triadic relations, but this fact is somewhat disguised by the circumstance that many of them are referred to as ''binary operations'', and because the most familiar of these have very specific properties that are dictated by their axioms. This makes it practical to study these operations for quite some time by focusing on their dyadic aspects before being forced to consider their proper characters as triadic relations. ==References== * Peirce, C.S., &ldquo;Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic&rdquo;, ''Memoirs of the American Academy of Arts and Sciences'', 9, 317&ndash;378, 1870. Reprinted, ''Collected Papers'' CP 3.45&ndash;149, ''Chronological Edition'' CE 2, 359&ndash;429. * Ulam, S.M. and Bednarek, A.R., &ldquo;On the Theory of Relational Structures and Schemata for Parallel Computation&rdquo;, pp. 477&ndash;508 in A.R. Bednarek and Françoise Ulam (eds.), ''Analogies Between Analogies : The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators'', University of California Press, Berkeley, CA, 1990. ==Bibliography== * Barr, Michael, and Wells, Charles (1990), ''Category Theory for Computing Science'', Prentice Hall, Hemel Hempstead, UK. * Bourbaki, Nicolas (1994), ''Elements of the History of Mathematics'', John Meldrum (trans.), Springer-Verlag, Berlin, Germany. * Carnap, Rudolf (1958), ''Introduction to Symbolic Logic with Applications'', Dover Publications, New York, NY. * Chang, C.C., and Keisler, H.J. (1973), ''Model Theory'', North-Holland, Amsterdam, Netherlands. * van Dalen, Dirk (1980), ''Logic and Structure'', 2nd edition, Springer-Verlag, Berlin, Germany. * Devlin, Keith J. (1993), ''The Joy of Sets : Fundamentals of Contemporary Set Theory'', 2nd edition, Springer-Verlag, New York, NY. * Halmos, Paul Richard (1960), ''Naive Set Theory'', D. Van Nostrand Company, Princeton, NJ. * van Heijenoort, Jean (1967/1977), ''From Frege to Gödel : A Source Book in Mathematical Logic, 1879&ndash;1931'', Harvard University Press, Cambridge, MA, 1967. Reprinted with corrections, 1977. * Kelley, John L. (1955), ''General Topology'', Van Nostrand Reinhold, New York, NY. * Kneale, William; and Kneale, Martha (1962/1975), ''The Development of Logic'', Oxford University Press, Oxford, UK, 1962. Reprinted with corrections, 1975. * Lawvere, Francis William; and Rosebrugh, Robert (2003), ''Sets for Mathematics'', Cambridge University Press, Cambridge, UK. * Lawvere, Francis William; and Schanuel, Stephen H. (1997/2000), ''Conceptual Mathematics : A First Introduction to Categories'', Cambridge University Press, Cambridge, UK, 1997. Reprinted with corrections, 2000. * Manin, Yu. I. (1977), ''A Course in Mathematical Logic'', Neal Koblitz (trans.), Springer-Verlag, New York, NY. * Mathematical Society of Japan (1993), ''Encyclopedic Dictionary of Mathematics'', 2nd edition, 2 vols., Kiyosi Itô (ed.), MIT Press, Cambridge, MA. * Mili, A., Desharnais, J., Mili, F., with Frappier, M. (1994), ''Computer Program Construction'', Oxford University Press, New York, NY. (Introduction to Tarskian relation theory and relational programming.) * Mitchell, John C. (1996), ''Foundations for Programming Languages'', MIT Press, Cambridge, MA. * Peirce, Charles Sanders (1870), ``Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic", ''Memoirs of the American Academy of Arts and Sciences'' 9 (1870), 317&ndash;378. Reprinted (CP 3.45&ndash;149), (CE 2, 359&ndash;429). * Peirce, Charles Sanders (1931&ndash;1935, 1958), ''Collected Papers of Charles Sanders Peirce'', vols. 1&ndash;6, Charles Hartshorne and Paul Weiss (eds.), vols. 7&ndash;8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA. Cited as (CP volume.paragraph). * Peirce, Charles Sanders (1981&ndash;), ''Writings of Charles S. Peirce : A Chronological Edition'', Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN. Cited as (CE volume, page). * Poizat, Bruno (2000), ''A Course in Model Theory : An Introduction to Contemporary Mathematical Logic'', Moses Klein (trans.), Springer-Verlag, New York, NY. * Quine, Willard Van Orman (1940/1981), ''Mathematical Logic'', 1940. Revised edition, Harvard University Press, Cambridge, MA, 1951. New preface, 1981. * Royce, Josiah (1961), ''The Principles of Logic'', Philosophical Library, New York, NY. * Runes, Dagobert D. (ed., 1962), ''Dictionary of Philosophy'', Littlefield, Adams, and Company, Totowa, NJ. * Shoenfield, Joseph R. (1967), ''Mathematical Logic'', Addison-Wesley, Reading, MA. * Styazhkin, N.I. (1969), ''History of Mathematical Logic from Leibniz to Peano'', MIT Press, Cambridge, MA. * Suppes, Patrick (1957/1999), ''Introduction to Logic'', 1st published 1957. Reprinted, Dover Publications, New York, NY, 1999. * Suppes, Patrick (1960/1972), ''Axiomatic Set Theory'', 1st published 1960. Reprinted, Dover Publications, New York, NY, 1972. * Tarski, Alfred (1956/1983), ''Logic, Semantics, Metamathematics : Papers from 1923 to 1938'', J.H. Woodger (trans.), Oxford University Press, 1956. 2nd edition, J. Corcoran (ed.), Hackett Publishing, Indianapolis, IN, 1983. * Ulam, Stanislaw Marcin; and Bednarek, A.R. (1977), &ldquo;On the Theory of Relational Structures and Schemata for Parallel Computation&rdquo;, pp. 477&ndash;508 in A.R. Bednarek and Françoise Ulam (eds.), ''Analogies Between Analogies : The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators'', University of California Press, Berkeley, CA, 1990. * Ulam, Stanislaw Marcin (1990), ''Analogies Between Analogies : The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators'', A.R. Bednarek and Françoise Ulam (eds.), University of California Press, Berkeley, CA. * Ullman, Jeffrey D. (1980), ''Principles of Database Systems'', Computer Science Press, Rockville, MD. * Venetus, Paulus (1472/1984), ''Logica Parva : Translation of the 1472 Edition with Introduction and Notes'', Alan R. Perreiah (trans.), Philosophia Verlag, Munich, Germany. ==Syllabus== ===Focal nodes=== {{col-begin}} {{col-break}} * [[Inquiry Live]] {{col-break}} * [[Logic Live]] {{col-end}} ===Peer nodes=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Relation_theory Relation Theory @ InterSciWiki] * [http://mywikibiz.com/Relation_theory Relation Theory @ MyWikiBiz] {{col-break}} * [http://beta.wikiversity.org/wiki/Relation_theory Relation Theory @ Wikiversity Beta] * [http://ref.subwiki.org/wiki/Relation_theory Relation Theory @ Subject Wikis] {{col-end}} ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * 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Theory], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Charles Sanders Peirce]] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Database Theory]] [[Category:Discrete Mathematics]] [[Category:Formal Sciences]] [[Category:Inquiry]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Relation Theory]] [[Category:Set Theory]] 8f18b489a3118304ec934cf2b026f3dc36808933 0 181 701 644 2015-10-29T20:54:29Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page serves as a '''focal node''' for a collection of related resources. ==Participants== * Interested parties may add their names on [[Logic Live/Participants|this page]]. ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Logic_Live Logic Live @ InterSciWiki] * [http://mywikibiz.com/Logic_Live Logic Live @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Logic_Live Logic Live @ Subject Wikis] * [https://en.wikiversity.org/wiki/Logic_Live Logic Live @ Wikiversity] * [https://beta.wikiversity.org/wiki/Logic_Live Logic Live @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] a085e5c098a65be1e093f8598a490fcfb249ffb7 0 361 702 632 2015-10-30T02:34:03Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. '''Exclusive disjunction''', also known as '''logical inequality''' or '''symmetric difference''', is an operation on two logical values, typically the values of two propositions, that produces a value of ''true'' just in case exactly one of its operands is true. The [[truth table]] of <math>p ~\operatorname{XOR}~ q,</math> also written <math>p + q\!</math> or <math>p \ne q,\!</math> appears below: <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:45%" |+ style="height:30px" | <math>\text{Exclusive Disjunction}\!</math> |- style="height:40px; background:#f0f0ff" | style="width:33%" | <math>p\!</math> | style="width:33%" | <math>q\!</math> | style="width:33%" | <math>p ~\operatorname{XOR}~ q</math> |- | <math>\operatorname{F}</math> || <math>\operatorname{F}</math> || <math>\operatorname{F}</math> |- | <math>\operatorname{F}</math> || <math>\operatorname{T}</math> || <math>\operatorname{T}</math> |- | <math>\operatorname{T}</math> || <math>\operatorname{F}</math> || <math>\operatorname{T}</math> |- | <math>\operatorname{T}</math> || <math>\operatorname{T}</math> || <math>\operatorname{F}</math> |} <br> The following equivalents may then be deduced: {| align="center" cellspacing="10" width="90%" | <math>\begin{matrix} p + q & = & (p \land \lnot q) & \lor & (\lnot p \land q) \\[6pt] & = & (p \lor q) & \land & (\lnot p \lor \lnot q) \\[6pt] & = & (p \lor q) & \land & \lnot (p \land q) \end{matrix}</math> |} ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Exclusive_disjunction Exclusive Disjunction @ InterSciWiki] * [http://mywikibiz.com/Exclusive_disjunction Exclusive Disjunction @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Exclusive_disjunction Exclusive Disjunction @ Subject Wikis] * [https://en.wikiversity.org/wiki/Exclusive_disjunction Exclusive Disjunction @ Wikiversity] * [https://beta.wikiversity.org/wiki/Exclusive_disjunction Exclusive Disjunction @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Exclusive_disjunction Exclusive Disjunction], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Exclusive_disjunction Exclusive Disjunction], [http://mywikibiz.com/ MyWikiBiz] * [http://ref.subwiki.org/wiki/Exclusive_disjunction Exclusive Disjunction], [http://ref.subwiki.org/ Subject Wikis] {{col-break}} * [http://wikinfo.org/index.php/Exclusive_disjunction Exclusive Disjunction], [http://wikinfo.org/ Wikinfo] * [http://en.wikiversity.org/wiki/Exclusive_disjunction Exclusive Disjunction], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Exclusive_disjunction Exclusive Disjunction], [http://beta.wikiversity.org/ Wikiversity Beta] {{col-break}} * [http://en.wikipedia.org/w/index.php?title=Exclusive_disjunction&oldid=75153068 Exclusive Disjunction], [http://en.wikipedia.org/ Wikipedia] {{col-end}} [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Charles Sanders Peirce]] [[Category:Computer Science]] [[Category:Formal Languages]] [[Category:Formal Sciences]] [[Category:Formal Systems]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Philosophy]] [[Category:Semiotics]] b1750147301d79c13288e74af15b7f6fd376137e 705 702 2015-10-31T01:46:48Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. '''Exclusive disjunction''', also known as '''logical inequality''' or '''symmetric difference''', is an operation on two logical values, typically the values of two propositions, that produces a value of ''true'' just in case exactly one of its operands is true. The [[truth table]] of <math>p ~\operatorname{XOR}~ q,</math> also written <math>p + q\!</math> or <math>p \ne q,\!</math> appears below: <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:45%" |+ style="height:30px" | <math>\text{Exclusive Disjunction}\!</math> |- style="height:40px; background:#f0f0ff" | style="width:33%" | <math>p\!</math> | style="width:33%" | <math>q\!</math> | style="width:33%" | <math>p ~\operatorname{XOR}~ q</math> |- | <math>\operatorname{F}</math> || <math>\operatorname{F}</math> || <math>\operatorname{F}</math> |- | <math>\operatorname{F}</math> || <math>\operatorname{T}</math> || <math>\operatorname{T}</math> |- | <math>\operatorname{T}</math> || <math>\operatorname{F}</math> || <math>\operatorname{T}</math> |- | <math>\operatorname{T}</math> || <math>\operatorname{T}</math> || <math>\operatorname{F}</math> |} <br> The following equivalents may then be deduced: {| align="center" cellspacing="10" width="90%" | <math>\begin{matrix} p + q & = & (p \land \lnot q) & \lor & (\lnot p \land q) \\[6pt] & = & (p \lor q) & \land & (\lnot p \lor \lnot q) \\[6pt] & = & (p \lor q) & \land & \lnot (p \land q) \end{matrix}</math> |} ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Exclusive_disjunction Exclusive Disjunction @ InterSciWiki] * [http://mywikibiz.com/Exclusive_disjunction Exclusive Disjunction @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Exclusive_disjunction Exclusive Disjunction @ Subject Wikis] * [http://en.wikiversity.org/wiki/Exclusive_disjunction Exclusive Disjunction @ Wikiversity] * [http://beta.wikiversity.org/wiki/Exclusive_disjunction Exclusive Disjunction @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Exclusive_disjunction Exclusive Disjunction], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Exclusive_disjunction Exclusive Disjunction], [http://mywikibiz.com/ MyWikiBiz] * [http://ref.subwiki.org/wiki/Exclusive_disjunction Exclusive Disjunction], [http://ref.subwiki.org/ Subject Wikis] * [http://wikinfo.org/w/index.php/Exclusive_disjunction Exclusive Disjunction], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Exclusive_disjunction Exclusive Disjunction], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Exclusive_disjunction Exclusive Disjunction], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Exclusive_disjunction&oldid=75153068 Exclusive Disjunction], [http://en.wikipedia.org/ Wikipedia] [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Charles Sanders Peirce]] [[Category:Computer Science]] [[Category:Formal Languages]] [[Category:Formal Sciences]] [[Category:Formal Systems]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Philosophy]] [[Category:Semiotics]] 50688dcc736ea7bae4b9ae4d3280716c2fe74bf5 0 180 703 645 2015-10-30T13:14:33Z Jon Awbrey 3 wikitext text/x-wiki <font size="3">&#9758;</font> This page serves as a '''focal node''' for a collection of related resources. ==Participants== Interested parties may add their names on [[Inquiry Live/Participants|this page]]. ==Rudiments of organization== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] '''Focal nodes''' serve as hubs for collections of related resources, in particular, activity sites and article contents. ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Live Inquiry Live @ InterSciWiki] * [http://mywikibiz.com/Inquiry_Live Inquiry Live @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Inquiry_Live Inquiry Live @ Subject Wikis] * [http://en.wikiversity.org/wiki/Inquiry_Live Inquiry Live @ Wikiversity] * [http://beta.wikiversity.org/wiki/Inquiry_Live Inquiry Live @ Wikiversity Beta] '''Peer nodes''' are roughly parallel pages on different sites that are not necessarily identical in content &mdash; especially as they develop in time across different environments through interaction with diverse populations &mdash; but they should preserve enough information to reproduce each other, more or less, in case of damage or loss. [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] 007cd5b29865868fe0b5ce5ed6acb7d3815edab7 0 335 704 697 2015-10-31T01:29:24Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. In logic and mathematics, '''relation composition''', or the composition of [[relation (mathematics)|relations]], is the generalization of function composition, or the composition of functions. ==Preliminaries== There are several ways to formalize the subject matter of relations. Relations and their combinations may be described in the logic of relative terms, in set theories of various kinds, and through a broadening of category theory from functions to relations in general. The first order of business is to define the operation on [[relation (mathematics)|relations]] that is variously known as the ''composition of relations'', ''relational composition'', or ''relative multiplication''. In approaching the more general constructions, it pays to begin with the composition of dyadic and triadic relations. As an incidental observation on usage, there are many different conventions of syntax for denoting the application and composition of relations, with perhaps even more options in general use than are common for the application and composition of functions. In this case there is little chance of standardization, since the convenience of conventions is relative to the context of use, and the same writers use different styles of syntax in different settings, depending on the ease of analysis and computation. {| align="center" cellpadding="4" width="90%" | <p>The first dimension of variation in syntax has to do with the correspondence between the order of operation and the linear order of terms on the page.</p> |- | <p>The second dimension of variation in syntax has to do with the automatic assumptions in place about the associations of terms in the absence of associations marked by parentheses. This becomes a significant factor with relations in general because the usual property of associativity is lost as both the complexities of compositions and the dimensions of relations increase.</p> |} These two factors together generate the following four styles of syntax: {| align="center" cellpadding="4" width="90%" | LALA = left application, left association. |- | LARA = left application, right association. |- | RALA = right application, left association. |- | RARA = right application, right association. |} ==Definition== A notion of relational composition is to be defined that generalizes the usual notion of functional composition: {| align="center" cellpadding="4" width="90%" | <p>Composing ''on the right'', <math>f : X \to Y</math> followed by <math>g : Y \to Z</math></p> <p>results in a ''composite function'' formulated as <math>fg : X \to Z.</math></p> |- | <p>Composing ''on the left'', <math>f : X \to Y</math> followed by <math>g : Y \to Z</math></p> <p>results in a ''composite function'' formulated as <math>gf : X \to Z.</math></p> |} Note on notation. The ordinary symbol for functional composition is the ''composition sign'', a small circle "<math>\circ</math>" written between the names of the functions being composed, as <math>f \circ g,</math> but the sign is often omitted if there is no risk of confusing the composition of functions with their algebraic product. In contexts where both compositions and products occur, either the composition is marked on each occasion or else the product is marked by means of a ''center dot'' &ldquo;<math>\cdot</math>&rdquo;, as <math>f \cdot g.</math> Generalizing the paradigm along parallel lines, the ''composition'' of a pair of dyadic relations is formulated in the following two ways: {| align="center" cellpadding="4" width="90%" | <p>Composing ''on the right'', <math>P \subseteq X \times Y</math> followed by <math>Q \subseteq Y \times Z</math></p> <p>results in a ''composite relation'' formulated as <math>PQ \subseteq X \times Z.</math></p> |- | <p>Composing ''on the left'', <math>P \subseteq X \times Y</math> followed by <math>Q \subseteq Y \times Z</math></p> <p>results in a ''composite relation'' formulated as <math>QP \subseteq X \times Z.</math></p> |} ==Geometric construction== There is a neat way of defining relational compositions in geometric terms, not only showing their relationship to the projection operations that come with any cartesian product, but also suggesting natural directions for generalizing relational compositions beyond the dyadic case, and even beyond relations that have any fixed arity, in effect, to the general case of formal languages as generalized relations. This way of looking at relational compositions is sometimes referred to as Tarski's Trick, on account of his having put it to especially good use in his work (Ulam and Bednarek, 1977). It supplies the imagination with a geometric way of visualizing the relational composition of a pair of dyadic relations, doing this by attaching concrete imagery to the basic set-theoretic operations, namely, intersections, projections, and a certain class of operations inverse to projections, here called ''tacit extensions''. The stage is set for Tarski's trick by highlighting the links between two topics that are likely to appear wholly unrelated at first, namely: :* The use of [[logical conjunction]], as denoted by the symbol <math>\land,\!</math> in expressions of the form <math>F(x, y, z) = G(x, y) \land H(y, z),\!</math> to define a triadic relation <math>F\!</math> in terms of a pair of dyadic relations <math>G\!</math> and <math>H.\!</math> :* The concepts of dyadic ''projection'' and ''projective determination'', that are invoked in the &ldquo;weak&rdquo; notion of ''projective reducibility''. The relational composition <math>G \circ H\!</math> of a pair of dyadic relations <math>G\!</math> and <math>H\!</math> will be constructed in three stages, first, by taking the tacit extensions of <math>G\!</math> and <math>H\!</math> to triadic relations that reside in the same space, next, by taking the intersection of these extensions, tantamount to the maximal triadic relation that is consistent with the ''prima facie'' dyadic relation data, finally, by projecting this intersection on a suitable plane to form a third dyadic relation, constituting in fact the relational composition <math>G \circ H\!</math> of the relations <math>G\!</math> and <math>H.\!</math> The construction of a relational composition in a specifically mathematical setting normally begins with [[relation (mathematics)|mathematical relations]] at a higher level of abstraction than the corresponding objects in linguistic or logical settings. This is due to the fact that mathematical objects are typically specified only ''up to isomorphism'' as the conventional saying goes, that is, any objects that have the &ldquo;same form&rdquo; are generally regarded as the being the same thing, for most all intents and mathematical purposes. Thus the mathematical construction of a relational composition begins by default with a pair of dyadic relations that reside, without loss of generality, in the same plane, say, <math>G, H \subseteq X \times Y,\!</math> as shown in Figure&nbsp;1. {| align="center" border="0" cellpadding="10" | <pre> o-------------------------------------------------o | | | o o | | |\ |\ | | | \ | \ | | | \ | \ | | | \ | \ | | | \ | \ | | | \ | \ | | | * \ | * \ | | X * Y X * Y | | \ * | \ * | | | \ G | \ H | | | \ | \ | | | \ | \ | | | \ | \ | | | \ | \ | | | \| \| | | o o | | | o-------------------------------------------------o Figure 1. Dyadic Relations G, H c X x Y </pre> |} The dyadic relations <math>G\!</math> and <math>H\!</math> cannot be composed at all at this point, not without additional information or further stipulation. In order for their relational composition to be possible, one of two types of cases has to happen: :* The first type of case occurs when <math>X = Y.\!</math> In this case, both of the compositions <math>G \circ H\!</math> and <math>H \circ G\!</math> are defined. :* The second type of case occurs when <math>X\!</math> and <math>Y\!</math> are distinct, but when it nevertheless makes sense to speak of a dyadic relation <math>\hat{H}\!</math> that is isomorphic to <math>H,\!</math> but living in the plane <math>YZ,\!</math> that is, in the space of the cartesian product <math>Y \times Z,\!</math> for some set <math>Z.\!</math> Whether you view isomorphic things to be the same things or not, you still have to specify the exact isomorphisms that are needed to transform any given representation of a thing into a required representation of the same thing. Let us imagine that we have done this, and say how later: {| align="center" border="0" cellpadding="10" | <pre> o-------------------------------------------------o | | | o o | | |\ /| | | | \ / | | | | \ / | | | | \ / | | | | \ / | | | | \ / | | | | * \ / * | | | X * Y Y * Z | | \ * | | * / | | \ G | | &#292; / | | \ | | / | | \ | | / | | \ | | / | | \ | | / | | \| |/ | | o o | | | o-------------------------------------------------o Figure 2. Dyadic Relations G c X x Y and &#292; c Y x Z </pre> |} With the required spaces carefully swept out, the stage is set for the presentation of Tarski's trick, and the invocation of the following symbolic formula, claimed to be a definition of the relational composition <math>P \circ Q\!</math> of a pair of dyadic relations <math>P, Q \subseteq X \times X.\!</math> : '''Definition.''' <math>P \circ Q = \mathrm{proj}_{13} (P \times X ~\cap~ X \times Q).\!</math> To get this drift of this definition one needs to understand that it comes from a point of view that regards all dyadic relations as covered well enough by subsets of a suitable cartesian square and thus of the form <math>L \subseteq X \times X.\!</math> So, if one has started out with a dyadic relation of the shape <math>L \subseteq U \times V,\!</math> one merely lets <math>X = U \cup V,\!</math> trading in the initial <math>L\!</math> for a new <math>L \subseteq X \times X\!</math> as need be. The projection <math>\mathrm{proj}_{13}\!</math> is just the projection of the cartesian cube <math>X \times X \times X\!</math> on the space of shape <math>X \times X\!</math> that is spanned by the first and the third domains, but since they now have the same names and the same contents it is necessary to distinguish them by numbering their relational places. Finally, the notation of the cartesian product sign &ldquo;<math>\times\!</math>&rdquo; is extended to signify two other products with respect to a dyadic relation <math>L \subseteq X \times X\!</math> and a subset <math>W \subseteq X,\!</math> as follows: : '''Definition.''' <math>L \times W ~=~ \{ (x, y, z) \in X^3 ~:~ (x, y) \in L ~\mathrm{and}~ z \in W \}.\!</math> : '''Definition.''' <math>W \times L ~=~ \{ (x, y, z) \in X^3 ~:~ x \in W ~\mathrm{and}~ (y, z) \in L \}.\!</math> Applying these definitions to the case <math>P, Q \subseteq X \times X,\!</math> the two dyadic relations whose relational composition <math>P \circ Q \subseteq X \times X\!</math> is about to be defined, one finds: : <math>P \times X ~=~ \{ (x, y, z) \in X^3 ~:~ (x, y) \in P ~\mathrm{and}~ z \in X \},\!</math> : <math>X \times Q ~=~ \{ (x, y, z) \in X^3 ~:~ x \in X ~\mathrm{and}~ (y, z) \in Q \}.\!</math> These are just the appropriate special cases of the tacit extensions already defined. : <math>P \times X ~=~ \mathrm{te}_{12}^3 (P),~\!</math> : <math>X \times Q ~=~ \mathrm{te}_{23}^1 (Q).~\!</math> In summary, then, the expression: : <math>\mathrm{proj}_{13} (P \times X ~\cap~ X \times Q)\!</math> is equivalent to the expression: : <math>\mathrm{proj}_{13} (\mathrm{te}_{12}^3 (P) ~\cap~ \mathrm{te}_{23}^1 (Q))\!</math> and this form is generalized &mdash; although, relative to one's school of thought, perhaps inessentially so &mdash; by the form that was given above as follows: : '''Definition.''' <math>P \circ Q ~=~ \mathrm{proj}_{XZ} (\mathrm{te}_{XY}^Z (P) ~\cap~ \mathrm{te}_{YZ}^X (Q)).\!</math> Figure&nbsp;3 presents a geometric picture of what is involved in formulating a definition of the triadic relation <math>F \subseteq X \times Y \times Z\!</math> by way of a conjunction between the dyadic relation <math>G \subseteq X \times Y\!</math> and the dyadic relation <math>H \subseteq Y \times Z,\!</math> as done for example by means of an expression of the following form: :* <math>F(x, y, z) ~=~ G(x, y) \land H(y, z).\!</math> {| align="center" border="0" cellpadding="10" | <pre> o-------------------------------------------------o | | | o | | /|\ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | o o o | | |\ / \ /| | | | \ / F \ / | | | | \ / * \ / | | | | \ /*\ / | | | | / \//*\\/ \ | | | | / /\/ \/\ \ | | | |/ ///\ /\\\ \| | | o X /// Y \\\ Z o | | |\ \/// | \\\/ /| | | | \ /// | \\\ / | | | | \ ///\ | /\\\ / | | | | \ /// \ | / \\\ / | | | | \/// \ | / \\\/ | | | | /\/ \ | / \/\ | | | | *//\ \|/ /\\* | | | X */ Y o Y \* Z | | \ * | | * / | | \ G | | H / | | \ | | / | | \ | | / | | \ | | / | | \ | | / | | \| |/ | | o o | | | o-------------------------------------------------o Figure 3. Projections of F onto G and H </pre> |} To interpret the Figure, visualize the triadic relation <math>F \subseteq X \times Y \times Z\!</math> as a body in <math>XYZ\!</math>-space, while <math>G\!</math> is a figure in <math>XY\!</math>-space and <math>H\!</math> is a figure in <math>YZ\!</math>-space. The dyadic '''projections''' that accompany a triadic relation over <math>X, Y, Z\!</math> are defined as follows: :* <math>\mathrm{proj}_{XY} (L) ~=~ \{ (x, y) \in X \times Y : (x, y, z) \in L ~\text{for some}~ z \in Z) \},\!</math> :* <math>\mathrm{proj}_{XZ} (L) ~=~ \{ (x, z) \in X \times Z : (x, y, z) \in L ~\text{for some}~ y \in Y) \},\!</math> :* <math>\mathrm{proj}_{YZ} (L) ~=~ \{ (y, z) \in Y \times Z : (x, y, z) \in L ~\text{for some}~ x \in X) \}.\!</math> For many purposes it suffices to indicate the dyadic projections of a triadic relation <math>L\!</math> by means of the briefer equivalents listed next: :* <math>L_{XY} ~=~ \mathrm{proj}_{XY}(L),\!</math> :* <math>L_{XZ} ~=~ \mathrm{proj}_{XZ}(L),\!</math> :* <math>L_{YZ} ~=~ \mathrm{proj}_{YZ}(L).\!</math> In light of these definitions, <math>\mathrm{proj}_{XY}\!</math> is a mapping from the set <math>\mathcal{L}_{XYZ}\!</math> of triadic relations over the domains <math>X, Y, Z\!</math> to the set <math>\mathcal{L}_{XY}\!</math> of dyadic relations over the domains <math>X, Y,\!</math> with similar relationships holding for the other projections. To formalize these relationships in a concise but explicit manner, it serves to add a few more definitions. The set <math>\mathcal{L}_{XYZ},~\!</math> whose members are just the triadic relations over <math>X, Y, Z,\!</math> can be recognized as the set of all subsets of the cartesian product <math>X \times Y \times Z,\!</math> also known as the ''power set'' of <math>X \times Y \times Z,\!</math> and notated here as <math>\mathrm{Pow} (X \times Y \times Z).\!</math> :* <math>\mathcal{L}_{XYZ} ~=~ \{ L : L \subseteq X \times Y \times Z \} ~=~ \mathrm{Pow} (X \times Y \times Z).\!</math> Likewise, the power sets of the pairwise cartesian products encompass all the dyadic relations on pairs of distinct domains that can be chosen from <math>\{ X, Y, Z \}.\!</math> :* <math>\mathcal{L}_{XY} ~=~ \{L : L \subseteq X \times Y \} ~=~ \mathrm{Pow} (X \times Y),~\!</math> :* <math>\mathcal{L}_{XZ} ~=~ \{L : L \subseteq X \times Z \} ~=~ \mathrm{Pow} (X \times Z),~\!</math> :* <math>\mathcal{L}_{YZ} ~=~ \{L : L \subseteq Y \times Z \} ~=~ \mathrm{Pow} (Y \times Z).~\!</math> In mathematics, the inverse relation corresponding to a projection map is usually called an ''extension''. To avoid confusion with other senses of the word, however, it is probably best for the sake of this discussion to stick with the more specific term ''tacit extension''. Given three sets, <math>X, Y, Z,\!</math> and three dyadic relations, :* <math>U \subseteq X \times Y,~\!</math> :* <math>V \subseteq X \times Z,~\!</math> :* <math>W \subseteq Y \times Z,~\!</math> the ''tacit extensions'', <math>\mathrm{te}_{XY}^Z, \mathrm{te}_{XZ}^Y, \mathrm{te}_{YZ}^X,~\!</math> of <math>U, V, W,\!</math> respectively, are defined as follows: :* <math>\mathrm{te}_{XY}^Z (U) ~=~ \{ (x, y, z) : (x, y) \in U \},\!</math> :* <math>\mathrm{te}_{XZ}^Y (V) ~=~ \{ (x, y, z) : (x, z) \in V \},\!</math> :* <math>\mathrm{te}_{YZ}^X (W) ~=~ \{ (x, y, z) : (y, z) \in W \}.\!</math> So long as the intended indices attaching to the tacit extensions can be gathered from context, it is usually clear enough to use the abbreviated forms, <math>\mathrm{te}(U), \mathrm{te}(V), \mathrm{te}(W).\!</math> The definition and illustration of relational composition presently under way makes use of the tacit extension of <math>G \subseteq X \times Y\!</math> to <math>\mathrm{te}(G) \subseteq X \times Y \times Z\!</math> and the tacit extension of <math>H \subseteq Y \times Z\!</math> to <math>\mathrm{te}(H) \subseteq X \times Y \times Z,\!</math> only. Geometric illustrations of <math>\mathrm{te}(G)\!</math> and <math>\mathrm{te}(H)\!</math> are afforded by Figures&nbsp;4 and 5, respectively. {| align="center" border="0" cellpadding="10" | <pre> o-------------------------------------------------o | | | o | | /|\ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | / | * \ | | o o ** o | | |\ / \*** /| | | | \ / *** / | | | | \ / ***\ / | | | | \ *** / | | | | / \*** / \ | | | | / *** / \ | | | |/ ***\ / \| | | o X /** Y Z o | | |\ \//* | / /| | | | \ /// | / / | | | | \ ///\ | / / | | | | \ /// \ | / / | | | | \/// \ | / / | | | | /\/ \ | / / | | | | *//\ \|/ / * | | | X */ Y o Y * Z | | \ * | | * / | | \ G | | H / | | \ | | / | | \ | | / | | \ | | / | | \ | | / | | \| |/ | | o o | | | o-------------------------------------------------o Figure 4. Tacit Extension of G to X x Y x Z </pre> |} {| align="center" border="0" cellpadding="10" | <pre> o-------------------------------------------------o | | | o | | /|\ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | / * | \ | | o ** o o | | |\ ***/ \ /| | | | \ *** \ / | | | | \ /*** \ / | | | | \ *** / | | | | / \ ***/ \ | | | | / \ *** \ | | | |/ \ /*** \| | | o X Y **\ Z o | | |\ \ | *\\/ /| | | | \ \ | \\\ / | | | | \ \ | /\\\ / | | | | \ \ | / \\\ / | | | | \ \ | / \\\/ | | | | \ \ | / \/\ | | | | * \ \|/ /\\* | | | X * Y o Y \* Z | | \ * | | * / | | \ G | | H / | | \ | | / | | \ | | / | | \ | | / | | \ | | / | | \| |/ | | o o | | | o-------------------------------------------------o Figure 5. Tacit Extension of H to X x Y x Z </pre> |} A geometric interpretation can now be given that fleshes out in graphic form the meaning of a formula like the following: :* <math>F(x, y, z) ~=~ G(x, y) \land H(y, z).\!</math> The conjunction that is indicated by &ldquo;<math>\land\!</math>&rdquo; corresponds as usual to an intersection of two sets, however, in this case it is the intersection of the tacit extensions <math>\mathrm{te}(G)\!</math> and <math>\mathrm{te}(H).\!</math> {| align="center" border="0" cellpadding="10" | <pre> o-------------------------------------------------o | | | o | | /|\ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | o o o | | |\ / \ /| | | | \ / F \ / | | | | \ / * \ / | | | | \ /*\ / | | | | / \//*\\/ \ | | | | / /\/ \/\ \ | | | |/ ///\ /\\\ \| | | o X /// Y \\\ Z o | | |\ \/// | \\\/ /| | | | \ /// | \\\ / | | | | \ ///\ | /\\\ / | | | | \ /// \ | / \\\ / | | | | \/// \ | / \\\/ | | | | /\/ \ | / \/\ | | | | *//\ \|/ /\\* | | | X */ Y o Y \* Z | | \ * | | * / | | \ G | | H / | | \ | | / | | \ | | / | | \ | | / | | \ | | / | | \| |/ | | o o | | | o-------------------------------------------------o Figure 6. F as the Intersection of te(G) and te(H) </pre> |} ==Algebraic construction== The transition from a geometric picture of relation composition to an algebraic formulation is accomplished through the introduction of coordinates, in other words, identifiable names for the objects that are related through the various forms of relations, dyadic and triadic in the present case. Adding coordinates to the running Example produces the following Figure: {| align="center" border="0" cellpadding="10" | <pre> o-------------------------------------------------o | | | o | | /|\ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | o o o | | |\ / \ /| | | | \ / F \ / | | | | \ / * \ / | | | | \ /*\ / | | | | / \//*\\/ \ | | | | / /\/ \/\ \ | | | |/ ///\ /\\\ \| | | o X /// Y \\\ Z o | | |\ 7\/// | \\\/7 /| | | | \ 6// | \\6 / | | | | \ //5\ | /5\\ / | | | | \ /// 4\ | /4 \\\ / | | | | \/// 3\ | /3 \\\/ | | | | G/\/ 2\ | /2 \/\H | | | | *//\ 1\|/1 /\\* | | | X *\ Y o Y /* Z | | 7\ *\\ |7 7| //* /7 | | 6\ |\\\|6 6|///| /6 | | 5\| \\@5 5@// |/5 | | 4@ \@4 4@/ @4 | | 3\ @3 3@ /3 | | 2\ |2 2| /2 | | 1\|1 1|/1 | | o o | | | o-------------------------------------------------o Figure 7. F as the Intersection of te(G) and te(H) </pre> |} Thinking of relations in operational terms is facilitated by using variant notations for ordered tuples and sets of ordered tuples, namely, the ordered pair <math>(x, y)\!</math> is written <math>x\!:\!y,\!</math> the ordered triple <math>(x, y, z)\!</math> is written <math>x\!:\!y\!:\!z,\!</math> and so on, and a set of tuples is conceived as a logical-algebraic sum, which can be written out in the smaller finite cases in forms like <math>a\!:\!b ~+~ b\!:\!c ~+~ c\!:\!d\!</math> and so on. For example, translating the relations <math>F \subseteq X \times Y \times Z, ~ G \subseteq X \times Y, ~ H \subseteq Y \times Z\!</math> into this notation produces the following summary of the data: {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix} F & = & 4:3:4 & + & 4:4:4 & + & 4:5:4 \\ G & = & 4:3 & + & 4:4 & + & 4:5 \\ H & = & 3:4 & + & 4:4 & + & 5:4 \end{matrix}</math> |} As often happens with abstract notations for functions and relations, the ''type information'', in this case, the fact that <math>G\!</math> and <math>H\!</math> live in different spaces, is left implicit in the context of use. Let us now verify that all of the proposed definitions, formulas, and other relationships check out against the concrete data of the current composition example. The ultimate goal is to develop a clearer picture of what is going on in the formula that expresses the relational composition of a couple of dyadic relations in terms of the medial projection of the intersection of their tacit extensions: {| align="center" cellpadding="8" width="90%" | <math>G \circ H ~=~ \mathrm{proj}_{XZ} (\mathrm{te}_{XY}^Z (G) ~\cap~ \mathrm{te}_{YZ}^X (H)).\!</math> |} Here is the big picture, with all the pieces in place: {| align="center" border="0" cellpadding="10" | <pre> o-------------------------------------------------o | | | o | | / \ | | / \ | | / \ | | / \ | | / \ | | / \ | | / G o H \ | | X * Z | | 7\ /|\ /7 | | 6\ / | \ /6 | | 5\ / | \ /5 | | 4@ | @4 | | 3\ | /3 | | 2\ | /2 | | 1\|/1 | | | | | | | | | | | /|\ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | o | o | | |\ /|\ /| | | | \ / F \ / | | | | \ / * \ / | | | | \ /*\ / | | | | / \//*\\/ \ | | | | / /\/ \/\ \ | | | |/ ///\ /\\\ \| | | o X /// Y \\\ Z o | | |\ \/// | \\\/ /| | | | \ /// | \\\ / | | | | \ ///\ | /\\\ / | | | | \ /// \ | / \\\ / | | | | \/// \ | / \\\/ | | | | G/\/ \ | / \/\H | | | | *//\ \|/ /\\* | | | X *\ Y o Y /* Z | | 7\ *\\ |7 7| //* /7 | | 6\ |\\\|6 6|///| /6 | | 5\| \\@5 5@// |/5 | | 4@ \@4 4@/ @4 | | 3\ @3 3@ /3 | | 2\ |2 2| /2 | | 1\|1 1|/1 | | o o | | | o-------------------------------------------------o Figure 8. G o H = proj_XZ (te(G) |^| te(H)) </pre> |} All that remains is to check the following collection of data and derivations against the situation represented in Figure&nbsp;8. {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix} F & = & 4:3:4 & + & 4:4:4 & + & 4:5:4 \\ G & = & 4:3 & + & 4:4 & + & 4:5 \\ H & = & 3:4 & + & 4:4 & + & 5:4 \end{matrix}</math> |} {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix} G \circ H & = & (4\!:\!3 ~+~ 4\!:\!4 ~+~ 4\!:\!5)(3\!:\!4 ~+~ 4\!:\!4 ~+~ 5\!:\!4) \\[6pt] & = & 4:4 \end{matrix}</math> |} {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix} \mathrm{te}(G) & = & \mathrm{te}_{XY}^Z (G) \\[4pt] & = & \displaystyle\sum_{z=1}^7 (4\!:\!3\!:\!z ~+~ 4\!:\!4\!:\!z ~+~ 4\!:\!5\!:\!z) \end{matrix}</math> |} {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix} \mathrm{te}(G) & = & 4:3:1 & + & 4:4:1 & + & 4:5:1 & + \\ & & 4:3:2 & + & 4:4:2 & + & 4:5:2 & + \\ & & 4:3:3 & + & 4:4:3 & + & 4:5:3 & + \\ & & 4:3:4 & + & 4:4:4 & + & 4:5:4 & + \\ & & 4:3:5 & + & 4:4:5 & + & 4:5:5 & + \\ & & 4:3:6 & + & 4:4:6 & + & 4:5:6 & + \\ & & 4:3:7 & + & 4:4:7 & + & 4:5:7 \end{matrix}</math> |} {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix} \mathrm{te}(H) & = & \mathrm{te}_{YZ}^X (H) \\[4pt] & = & \displaystyle\sum_{x=1}^7 (x\!:\!3\!:\!4 ~+~ x\!:\!4\!:\!4 ~+~ x\!:\!5\!:\!4) \end{matrix}</math> |} {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix} \mathrm{te}(H) & = & 1:3:4 & + & 1:4:4 & + & 1:5:4 & + \\ & & 2:3:4 & + & 2:4:4 & + & 2:5:4 & + \\ & & 3:3:4 & + & 3:4:4 & + & 3:5:4 & + \\ & & 4:3:4 & + & 4:4:4 & + & 4:5:4 & + \\ & & 5:3:4 & + & 5:4:4 & + & 5:5:4 & + \\ & & 6:3:4 & + & 6:4:4 & + & 6:5:4 & + \\ & & 7:3:4 & + & 7:4:4 & + & 7:5:4 \end{matrix}</math> |} {| align="center" cellpadding="8" width="90%" | <math>\begin{array}{ccl} \mathrm{te}(G) \cap \mathrm{te}(H) & = & 4\!:\!3:\!4 ~+~ 4\!:\!4\!:\!4 ~+~ 4\!:\!5\!:\!4 \\[4pt] G \circ H & = & \mathrm{proj}_{XZ} (\mathrm{te}(G) \cap \mathrm{te}(H)) \\[4pt] & = & \mathrm{proj}_{XZ} (4\!:\!3:\!4 ~+~ 4\!:\!4\!:\!4 ~+~ 4\!:\!5\!:\!4) \\[4pt] & = & 4:4 \end{array}</math> |} ==Matrix representation== We have it within our reach to pick up another way of representing dyadic relations, namely, the representation as ''[[logical matrix|logical matrices]]'', and also to grasp the analogy between relational composition and ordinary [[matrix multiplication]] as it appears in linear algebra. First of all, while we still have the data of a very simple concrete case in mind, let us reflect on what we did in our last Example in order to find the composition <math>G \circ H\!</math> of the dyadic relations <math>G\!</math> and <math>H.\!</math> Here is the setup that we had before: {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix}X & = & \{ 1, 2, 3, 4, 5, 6, 7 \}\end{matrix}</math> |} {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix} G & = & 4:3 & + & 4:4 & + & 4:5 & \subseteq & X \times X \\ H & = & 3:4 & + & 4:4 & + & 5:4 & \subseteq & X \times X \end{matrix}</math> |} Let us recall the rule for finding the relational composition of a pair of dyadic relations. Given the dyadic relations <math>P \subseteq X \times Y\!</math> and <math>Q \subseteq Y \times Z,\!</math> the composition of <math>P ~\text{on}~ Q\!</math> is written as <math>P \circ Q,\!</math> or more simply as <math>PQ,\!</math> and obtained as follows: To compute <math>PQ,\!</math> in general, where <math>P\!</math> and <math>Q\!</math> are dyadic relations, simply multiply out the two sums in the ordinary distributive algebraic way, but subject to the following rule for finding the product of two elementary relations of shapes <math>a:b\!</math> and <math>c:d.\!</math> {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix} (a:b)(c:d) & = & (a:d) & \text{if}~ b = c \\ (a:b)(c:d) & = & 0 & \text{otherwise} \end{matrix}</math> |} To find the relational composition <math>G \circ H,\!</math> one may begin by writing it as a quasi-algebraic product: {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix} G \circ H & = & (4\!:\!3 ~+~ 4\!:\!4 ~+~ 4\!:\!5)(3\!:\!4 ~+~ 4\!:\!4 ~+~ 5\!:\!4) \end{matrix}</math> |} Multiplying this out in accord with the applicable form of distributive law one obtains the following expansion: {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix} G \circ H & = & (4:3)(3:4) & + & (4:3)(4:4) & + & (4:3)(5:4) & + \\ & & (4:4)(3:4) & + & (4:4)(4:4) & + & (4:4)(5:4) & + \\ & & (4:5)(3:4) & + & (4:5)(4:4) & + & (4:5)(5:4) \end{matrix}</math> |} Applying the rule that determines the product of elementary relations produces the following array: {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix} G \circ H & = & 4:4 & + & 0 & + & 0 & + \\ & & 0 & + & 4:4 & + & 0 & + \\ & & 0 & + & 0 & + & 4:4 \end{matrix}</math> |} Since the plus sign in this context represents an operation of logical disjunction or set-theoretic aggregation, all of the positive multiplicites count as one, and this gives the ultimate result: {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix}G \circ H & = & 4:4\end{matrix}</math> |} With an eye toward extracting a general formula for relation composition, viewed here on analogy to algebraic multiplication, let us examine what we did in multiplying the dyadic relations <math>G\!</math> and <math>H\!</math> together to obtain their relational composite <math>G \circ H.\!</math> Given the space <math>X = \{ 1, 2, 3, 4, 5, 6, 7 \},\!</math> whose cardinality <math>|X|\!</math> is <math>7,\!</math> there are <math>|X \times X| = |X| \cdot |X|\!</math> <math>=\!</math> <math>7 \cdot 7 = 49\!</math> elementary relations of the form <math>i:j,\!</math> where <math>i\!</math> and <math>j\!</math> range over the space <math>X.\!</math> Although they might be organized in many different ways, it is convenient to regard the collection of elementary relations as arranged in a lexicographic block of the following form: {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix} 1\!:\!1 & 1\!:\!2 & 1\!:\!3 & 1\!:\!4 & 1\!:\!5 & 1\!:\!6 & 1\!:\!7 \\ 2\!:\!1 & 2\!:\!2 & 2\!:\!3 & 2\!:\!4 & 2\!:\!5 & 2\!:\!6 & 2\!:\!7 \\ 3\!:\!1 & 3\!:\!2 & 3\!:\!3 & 3\!:\!4 & 3\!:\!5 & 3\!:\!6 & 3\!:\!7 \\ 4\!:\!1 & 4\!:\!2 & 4\!:\!3 & 4\!:\!4 & 4\!:\!5 & 4\!:\!6 & 4\!:\!7 \\ 5\!:\!1 & 5\!:\!2 & 5\!:\!3 & 5\!:\!4 & 5\!:\!5 & 5\!:\!6 & 5\!:\!7 \\ 6\!:\!1 & 6\!:\!2 & 6\!:\!3 & 6\!:\!4 & 6\!:\!5 & 6\!:\!6 & 6\!:\!7 \\ 7\!:\!1 & 7\!:\!2 & 7\!:\!3 & 7\!:\!4 & 7\!:\!5 & 7\!:\!6 & 7\!:\!7 \end{matrix}</math> |} The relations <math>G\!</math> and <math>H\!</math> may then be regarded as logical sums of the following forms: {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix} G & = & \displaystyle\sum_{ij} G_{ij} (i\!:\!j) \\[20pt] H & = & \displaystyle\sum_{ij} H_{ij} (i\!:\!j) \end{matrix}\!</math> |} The notation <math>\textstyle\sum_{ij}\!</math> indicates a logical sum over the collection of elementary relations <math>i\!:\!j\!</math> while the factors <math>G_{ij}\!</math> and <math>H_{ij}\!</math> are values in the ''[[boolean domain]]'' <math>\mathbb{B} = \{ 0, 1 \}~\!</math> that are called the ''coefficients'' of the relations <math>G\!</math> and <math>H,\!</math> respectively, with regard to the corresponding elementary relations <math>i\!:\!j.\!</math> In general, for a dyadic relation <math>L,\!</math> the coefficient <math>L_{ij}\!</math> of the elementary relation <math>i\!:\!j\!</math> in the relation <math>L\!</math> will be <math>0\!</math> or <math>1,\!</math> respectively, as <math>i\!:\!j\!</math> is excluded from or included in <math>L.\!</math> With these conventions in place, the expansions of <math>G\!</math> and <math>H\!</math> may be written out as follows: {| align="center" cellpadding="4" width="90%" | <math>\begin{matrix}G & = & 4:3 & + & 4:4 & + & 4:5 & =\end{matrix}</math> |- | <math>\begin{smallmatrix} 0 \cdot (1:1) & + & 0 \cdot (1:2) & + & 0 \cdot (1:3) & + & 0 \cdot (1:4) & + & 0 \cdot (1:5) & + & 0 \cdot (1:6) & + & 0 \cdot (1:7) & + \\ 0 \cdot (2:1) & + & 0 \cdot (2:2) & + & 0 \cdot (2:3) & + & 0 \cdot (2:4) & + & 0 \cdot (2:5) & + & 0 \cdot (2:6) & + & 0 \cdot (2:7) & + \\ 0 \cdot (3:1) & + & 0 \cdot (3:2) & + & 0 \cdot (3:3) & + & 0 \cdot (3:4) & + & 0 \cdot (3:5) & + & 0 \cdot (3:6) & + & 0 \cdot (3:7) & + \\ 0 \cdot (4:1) & + & 0 \cdot (4:2) & + & \mathbf{1} \cdot (4:3) & + & \mathbf{1} \cdot (4:4) & + & \mathbf{1} \cdot (4:5) & + & 0 \cdot (4:6) & + & 0 \cdot (4:7) & + \\ 0 \cdot (5:1) & + & 0 \cdot (5:2) & + & 0 \cdot (5:3) & + & 0 \cdot (5:4) & + & 0 \cdot (5:5) & + & 0 \cdot (5:6) & + & 0 \cdot (5:7) & + \\ 0 \cdot (6:1) & + & 0 \cdot (6:2) & + & 0 \cdot (6:3) & + & 0 \cdot (6:4) & + & 0 \cdot (6:5) & + & 0 \cdot (6:6) & + & 0 \cdot (6:7) & + \\ 0 \cdot (7:1) & + & 0 \cdot (7:2) & + & 0 \cdot (7:3) & + & 0 \cdot (7:4) & + & 0 \cdot (7:5) & + & 0 \cdot (7:6) & + & 0 \cdot (7:7) \end{smallmatrix}</math> |} {| align="center" cellpadding="4" width="90%" | <math>\begin{matrix}H & = & 3:4 & + & 4:4 & + & 5:4 & =\end{matrix}</math> |- | <math>\begin{smallmatrix} 0 \cdot (1:1) & + & 0 \cdot (1:2) & + & 0 \cdot (1:3) & + & 0 \cdot (1:4) & + & 0 \cdot (1:5) & + & 0 \cdot (1:6) & + & 0 \cdot (1:7) & + \\ 0 \cdot (2:1) & + & 0 \cdot (2:2) & + & 0 \cdot (2:3) & + & 0 \cdot (2:4) & + & 0 \cdot (2:5) & + & 0 \cdot (2:6) & + & 0 \cdot (2:7) & + \\ 0 \cdot (3:1) & + & 0 \cdot (3:2) & + & 0 \cdot (3:3) & + & \mathbf{1} \cdot (3:4) & + & 0 \cdot (3:5) & + & 0 \cdot (3:6) & + & 0 \cdot (3:7) & + \\ 0 \cdot (4:1) & + & 0 \cdot (4:2) & + & 0 \cdot (4:3) & + & \mathbf{1} \cdot (4:4) & + & 0 \cdot (4:5) & + & 0 \cdot (4:6) & + & 0 \cdot (4:7) & + \\ 0 \cdot (5:1) & + & 0 \cdot (5:2) & + & 0 \cdot (5:3) & + & \mathbf{1} \cdot (5:4) & + & 0 \cdot (5:5) & + & 0 \cdot (5:6) & + & 0 \cdot (5:7) & + \\ 0 \cdot (6:1) & + & 0 \cdot (6:2) & + & 0 \cdot (6:3) & + & 0 \cdot (6:4) & + & 0 \cdot (6:5) & + & 0 \cdot (6:6) & + & 0 \cdot (6:7) & + \\ 0 \cdot (7:1) & + & 0 \cdot (7:2) & + & 0 \cdot (7:3) & + & 0 \cdot (7:4) & + & 0 \cdot (7:5) & + & 0 \cdot (7:6) & + & 0 \cdot (7:7) \end{smallmatrix}</math> |} Stripping down to the bare essentials, one obtains the following matrices of coefficients for the relations <math>G\!</math> and <math>H.\!</math> {| align="center" cellpadding="8" width="90%" | <math>G ~=~ \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}</math> |} {| align="center" cellpadding="8" width="90%" | <math>H ~=~ \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}</math> |} These are the logical matrix representations of the dyadic relations <math>G\!</math> and <math>H.\!</math> If the dyadic relations <math>G\!</math> and <math>H\!</math> are viewed as logical sums then their relational composition <math>G \circ H\!</math> can be regarded as a product of sums, a fact that can be indicated as follows: {| align="center" cellpadding="8" width="90%" | <math>G \circ H ~=~ (\sum_{ij} G_{ij} (i\!:\!j))(\sum_{ij} H_{ij} (i\!:\!j)).\!</math> |} The composite relation <math>G \circ H\!</math> is itself a dyadic relation over the same space <math>X,\!</math> in other words, <math>G \circ H \subseteq X \times X,\!</math> and this means that <math>G \circ H\!</math> must be amenable to being written as a logical sum of the following form: {| align="center" cellpadding="8" width="90%" | <math>G \circ H ~=~ \sum_{ij} (G \circ H)_{ij} (i\!:\!j).\!</math> |} In this formula, <math>(G \circ H)_{ij}\!</math> is the coefficient of <math>G \circ H\!</math> with respect to the elementary relation <math>i\!:\!j.\!</math> One of the best ways to reason out what <math>G \circ H\!</math> should be is to ask oneself what its coefficient <math>(G \circ H)_{ij}\!</math> should be for each of the elementary relations <math>i\!:\!j\!</math> in turn. So let us pose the question: {| align="center" cellpadding="8" width="90%" | <math>(G \circ H)_{ij} ~=~ ?\!</math> |} In order to answer this question, it helps to realize that the indicated product given above can be written in the following equivalent form: {| align="center" cellpadding="8" width="90%" | <math>G \circ H ~=~ (\sum_{ik} G_{ik} (i\!:\!k))(\sum_{kj} H_{kj} (k\!:\!j)).\!</math> |} A moment's thought will tell us that <math>(G \circ H)_{ij} = 1\!</math> if and only if there is an element <math>k\!</math> in <math>X\!</math> such that <math>G_{ik} = 1\!</math> and <math>H_{kj} = 1.\!</math> Consequently, we have the result: {| align="center" cellpadding="8" width="90%" | <math>(G \circ H)_{ij} ~=~ \sum_{k} G_{ik} H_{kj}.\!</math> |} This follows from the properties of boolean arithmetic, specifically, from the fact that the product <math>G_{ik} H_{kj}\!</math> is <math>1\!</math> if and only if both <math>G_{ik}\!</math> and <math>H_{kj}\!</math> are <math>1\!</math> and from the fact that <math>\textstyle\sum_{k} F_{k}\!</math> is equal to <math>1\!</math> just in case some <math>F_{k}\!</math> is <math>1.\!</math> All that remains in order to obtain a computational formula for the relational composite <math>G \circ H\!</math> of the dyadic relations <math>G\!</math> and <math>H\!</math> is to collect the coefficients <math>(G \circ H)_{ij}\!</math> as <math>i\!</math> and <math>j\!</math> range over <math>X.\!</math> {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix}G \circ H & = & \displaystyle \sum_{ij} (G \circ H)_{ij} (i\!:\!j) & = & \displaystyle \sum_{ij} (\sum_{k} G_{ik} H_{kj}) (i\!:\!j). \end{matrix}</math> |} This is the logical analogue of matrix multiplication in linear algebra, the difference in the logical setting being that all of the operations performed on coefficients take place in a system of boolean arithmetic where summation corresponds to logical disjunction and multiplication corresponds to logical conjunction. By way of disentangling this formula, one may notice that the form <math>\textstyle \sum_{k} G_{ik} H_{kj}\!</math> is what is usually called a ''scalar product''. In this case it is the scalar product of the <math>i^\text{th}\!</math> row of <math>G\!</math> with the <math>j^\text{th}\!</math> column of <math>H.\!</math> To make this statement more concrete, let us go back to the examples of <math>G\!</math> and <math>H\!</math> we came in with: {| align="center" cellpadding="8" width="90%" | <math>G ~=~ \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}</math> |} {| align="center" cellpadding="8" width="90%" | <math>H ~=~ \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}</math> |} The formula for computing <math>G \circ H\!</math> says the following: {| align="center" cellpadding="8" width="90%" | <math>\begin{array}{ccl} (G \circ H)_{ij} & = & \text{the}~ {ij}^\text{th} ~\text{entry in the matrix representation for}~ G \circ H \\[2pt] & = & \text{the entry in the}~ {i}^\text{th} ~\text{row and the}~ {j}^\text{th} ~\text{column of}~ G \circ H \\[2pt] & = & \text{the scalar product of the}~ {i}^\text{th} ~\text{row of}~ G ~\text{with the}~ {j}^\text{th} ~\text{column of}~ H \\[2pt] & = & \sum_{k} G_{ik} H_{kj} \end{array}</math> |} As it happens, it is possible to make exceedingly light work of this example, since there is only one row of <math>G\!</math> and one column of <math>H\!</math> that are not all zeroes. Taking the scalar product, in a logical way, of the fourth row of <math>G\!</math> with the fourth column of <math>H\!</math> produces the sole non-zero entry for the matrix of <math>G \circ H.\!</math> {| align="center" cellpadding="8" width="90%" | <math>G \circ H ~=~ \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}</math> |} ==Graph-theoretic picture== There is another form of representation for dyadic relations that is useful to keep in mind, especially for its ability to render the logic of many complex formulas almost instantly understandable to the mind's eye. This is the representation in terms of ''bipartite graphs'', or ''bigraphs'' for short. Here is what <math>G\!</math> and <math>H\!</math> look like in the bigraph picture: {| align="center" border="0" cellpadding="10" | <pre> o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | /|\ | | / | \ G | | / | \ | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 9. G = 4:3 + 4:4 + 4:5 </pre> |- | <pre> o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | \ | / | | \ | / H | | \|/ | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 10. H = 3:4 + 4:4 + 5:4 </pre> |} These graphs may be read to say: {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix} G ~\text{puts}~ 4 ~\text{in relation to}~ 3, 4, 5. \\[2pt] H ~\text{puts}~ 3, 4, 5 ~\text{in relation to}~ 4. \end{matrix}</math> |} To form the composite relation <math>G \circ H,\!</math> one simply follows the bigraph for <math>G\!</math> by the bigraph for <math>H,\!</math> here arranging the bigraphs in order down the page, and then treats any non-empty set of paths of length two between two nodes as being equivalent to a single directed edge between those nodes in the composite bigraph for <math>G \circ H.\!</math> Here's how it looks in pictures: {| align="center" border="0" cellpadding="10" | <pre> o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | /|\ | | / | \ G | | / | \ | | o o o o o o o X | | \ | / | | \ | / H | | \|/ | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 11. G Followed By H </pre> |- | <pre> o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | | | | | G o H | | | | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 12. G Composed With H </pre> |} Once again we find that <math>G \circ H = 4:4.\!</math> We have now seen three different representations of dyadic relations. If one has a strong preference for letters, or numbers, or pictures, then one may be tempted to take one or another of these as being canonical, but each of them will be found to have its peculiar advantages and disadvantages in any given application, and the maximum advantage is therefore approached by keeping all three of them in mind. To see the promised utility of the bigraph picture of dyadic relations, let us devise a slightly more complex example of a composition problem, and use it to illustrate the logic of the matrix multiplication formula. Keeping to the same space <math>X = \{ 1, 2, 3, 4, 5, 6, 7 \},\!</math> define the dyadic relations <math>M, N \subseteq X \times X\!</math> as follows: {| align="center" cellpadding="8" width="90%" | <math>\begin{array}{*{19}{c}} M & = & 2\!:\!1 & + & 2\!:\!2 & + & 2\!:\!3 & + & 4\!:\!3 & + & 4\!:\!4 & + & 4\!:\!5 & + & 6\!:\!5 & + & 6\!:\!6 & + & 6\!:\!7 \\[2pt] N & = & 1\!:\!1 & + & 2\!:\!1 & + & 3\!:\!3 & + & 4\!:\!3 & ~ & + & ~ & 4\!:\!5 & + & 5\!:\!5 & + & 6\!:\!7 & + & 7\!:\!7 \end{array}</math> |} Here are the bigraph pictures: {| align="center" border="0" cellpadding="10" | <pre> o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | /|\ /|\ /|\ | | / | \ / | \ / | \ M | | / | \ / | \ / | \ | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 13. Dyadic Relation M </pre> |- | <pre> o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | | / | / \ | \ | | | | / | / \ | \ | N | | |/ |/ \| \| | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 14. Dyadic Relation N </pre> |} To form the composite relation <math>M \circ N,\!</math> one simply follows the bigraph for <math>M\!</math> by the bigraph for <math>N,\!</math> arranging the bigraphs in order down the page, and then counts any non-empty set of paths of length two between two nodes as being equivalent to a single directed edge between those two nodes in the composite bigraph for <math>M \circ N.\!</math> Here's how it looks in pictures: {| align="center" border="0" cellpadding="10" | <pre> o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | /|\ /|\ /|\ | | / | \ / | \ / | \ M | | / | \ / | \ / | \ | | o o o o o o o X | | | / | / \ | \ | | | | / | / \ | \ | N | | |/ |/ \| \| | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 15. M Followed By N </pre> |- | <pre> o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | / \ / \ / \ | | / \ / \ / \ M o N | | / \ / \ / \ | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 16. M Composed With N </pre> |} Let us hark back to that mysterious matrix multiplication formula, and see how it appears in the light of the bigraph representation. The coefficient of the composition <math>M \circ N\!</math> between <math>i\!</math> and <math>j\!</math> in <math>X\!</math> is given as follows: {| align="center" cellpadding="8" width="90%" | <math>(M \circ N)_{ij} ~=~ \sum_{k} M_{ik} N_{kj}\!</math> |} Graphically interpreted, this is a ''sum over paths''. Starting at the node <math>i,\!</math> <math>M_{ik}\!</math> being <math>1\!</math> indicates that there is an edge in the bigraph of <math>M\!</math> from node <math>i\!</math> to node <math>k\!</math> and <math>N_{kj}\!</math> being <math>1\!</math> indicates that there is an edge in the bigraph of <math>N\!</math> from node <math>k\!</math> to node <math>j.\!</math> So the <math>\textstyle\sum_{k}\!</math> ranges over all possible intermediaries <math>k,\!</math> ascending from <math>0\!</math> to <math>1\!</math> just as soon as there happens to be a path of length two between nodes <math>i\!</math> and <math>j.\!</math> It is instructive at this point to compute the other possible composition that can be formed from <math>M\!</math> and <math>N,\!</math> namely, the composition <math>N \circ M,\!</math> that takes <math>M\!</math> and <math>N\!</math> in the opposite order. Here is the graphic computation: {| align="center" border="0" cellpadding="10" | <pre> o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | | / | / \ | \ | | | | / | / \ | \ | N | | |/ |/ \| \| | | o o o o o o o X | | /|\ /|\ /|\ | | / | \ / | \ / | \ M | | / | \ / | \ / | \ | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 17. N Followed By M </pre> |- | <pre> o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | | | N o M | | | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 18. N Composed With M </pre> |} In sum, <math>N \circ M = 0.\!</math> This example affords sufficient evidence that relational composition, just like its kindred, matrix multiplication, is a ''non-commutative'' algebraic operation. ==References== * Ulam, S.M., and Bednarek, A.R., &ldquo;On the Theory of Relational Structures and Schemata for Parallel Computation&rdquo; (1977), pp. 477&ndash;508 in A.R. Bednarek and Françoise Ulam (eds.), ''Analogies Between Analogies : The Mathematical Reports of S.M. Ulam and His Los&nbsp;Alamos Collaborators'', University of California Press, Berkeley, CA, 1990. ==Bibliography== * Mathematical Society of Japan, ''Encyclopedic Dictionary of Mathematics'', 2nd edition, 2&nbsp;volumes., Kiyosi Itô (ed.), MIT Press, Cambridge, MA, 1993. * Mili, A., Desharnais, J., Mili, F., with Frappier, M., ''Computer Program Construction'', Oxford University Press, New York, NY, 1994. * Ulam, S.M., ''Analogies Between Analogies : The Mathematical Reports of S.M. Ulam and His Los&nbsp;Alamos Collaborators'', A.R. Bednarek and Françoise Ulam (eds.), University of California Press, Berkeley, CA, 1990. ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Relation_composition Relation Composition @ InterSciWiki] * [http://mywikibiz.com/Relation_composition Relation Composition @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Relation_composition Relation Composition @ Subject Wikis] * [http://en.wikiversity.org/wiki/Relation_composition Relation Composition @ Wikiversity] * [http://beta.wikiversity.org/wiki/Relation_composition Relation Composition @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Relation_composition Relation Composition], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Relation_composition Relation Composition], [http://mywikibiz.com/ MyWikiBiz] * [http://semanticweb.org/wiki/Relation_composition Relation Composition], [http://semanticweb.org/ Semantic Web] * [http://ref.subwiki.org/wiki/Relation_composition Relation Composition], [http://ref.subwiki.org/ Subject Wikis] * [http://en.wikiversity.org/wiki/Relation_composition Relation Composition], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Relation_composition Relation Composition], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Relation_composition&oldid=43467878 Relation Composition], [http://en.wikipedia.org/ Wikipedia] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Database Theory]] [[Category:Discrete Mathematics]] [[Category:Formal Sciences]] [[Category:Inquiry]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Relation Theory]] [[Category:Set Theory]] 7abbec43919b7a4c4921d8a6bc2baddcbbf33700 724 704 2015-11-07T02:56:46Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. '''Relation composition''', or the composition of [[relation (mathematics)|relations]], is the generalization of function composition, or the composition of functions.&nbsp; The following treatment of relation composition takes the &ldquo;strongly typed&rdquo; approach to relations that is outlined in the article on [[relation theory]]. ==Preliminaries== There are several ways to formalize the subject matter of relations. Relations and their combinations may be described in the logic of relative terms, in set theories of various kinds, and through a broadening of category theory from functions to relations in general. The first order of business is to define the operation on [[relation (mathematics)|relations]] that is variously known as the ''composition of relations'', ''relational composition'', or ''relative multiplication''. In approaching the more general constructions, it pays to begin with the composition of dyadic and triadic relations. As an incidental observation on usage, there are many different conventions of syntax for denoting the application and composition of relations, with perhaps even more options in general use than are common for the application and composition of functions. In this case there is little chance of standardization, since the convenience of conventions is relative to the context of use, and the same writers use different styles of syntax in different settings, depending on the ease of analysis and computation. {| align="center" cellpadding="4" width="90%" | <p>The first dimension of variation in syntax has to do with the correspondence between the order of operation and the linear order of terms on the page.</p> |- | <p>The second dimension of variation in syntax has to do with the automatic assumptions in place about the associations of terms in the absence of associations marked by parentheses. This becomes a significant factor with relations in general because the usual property of associativity is lost as both the complexities of compositions and the dimensions of relations increase.</p> |} These two factors together generate the following four styles of syntax: {| align="center" cellpadding="4" width="90%" | LALA = left application, left association. |- | LARA = left application, right association. |- | RALA = right application, left association. |- | RARA = right application, right association. |} ==Definition== A notion of relational composition is to be defined that generalizes the usual notion of functional composition: {| align="center" cellpadding="4" width="90%" | <p>Composing ''on the right'', <math>f : X \to Y</math> followed by <math>g : Y \to Z</math></p> <p>results in a ''composite function'' formulated as <math>fg : X \to Z.</math></p> |- | <p>Composing ''on the left'', <math>f : X \to Y</math> followed by <math>g : Y \to Z</math></p> <p>results in a ''composite function'' formulated as <math>gf : X \to Z.</math></p> |} Note on notation. The ordinary symbol for functional composition is the ''composition sign'', a small circle "<math>\circ</math>" written between the names of the functions being composed, as <math>f \circ g,</math> but the sign is often omitted if there is no risk of confusing the composition of functions with their algebraic product. In contexts where both compositions and products occur, either the composition is marked on each occasion or else the product is marked by means of a ''center dot'' &ldquo;<math>\cdot</math>&rdquo;, as <math>f \cdot g.</math> Generalizing the paradigm along parallel lines, the ''composition'' of a pair of dyadic relations is formulated in the following two ways: {| align="center" cellpadding="4" width="90%" | <p>Composing ''on the right'', <math>P \subseteq X \times Y</math> followed by <math>Q \subseteq Y \times Z</math></p> <p>results in a ''composite relation'' formulated as <math>PQ \subseteq X \times Z.</math></p> |- | <p>Composing ''on the left'', <math>P \subseteq X \times Y</math> followed by <math>Q \subseteq Y \times Z</math></p> <p>results in a ''composite relation'' formulated as <math>QP \subseteq X \times Z.</math></p> |} ==Geometric construction== There is a neat way of defining relational compositions in geometric terms, not only showing their relationship to the projection operations that come with any cartesian product, but also suggesting natural directions for generalizing relational compositions beyond the dyadic case, and even beyond relations that have any fixed arity, in effect, to the general case of formal languages as generalized relations. This way of looking at relational compositions is sometimes referred to as Tarski's Trick, on account of his having put it to especially good use in his work (Ulam and Bednarek, 1977). It supplies the imagination with a geometric way of visualizing the relational composition of a pair of dyadic relations, doing this by attaching concrete imagery to the basic set-theoretic operations, namely, intersections, projections, and a certain class of operations inverse to projections, here called ''tacit extensions''. The stage is set for Tarski's trick by highlighting the links between two topics that are likely to appear wholly unrelated at first, namely: :* The use of [[logical conjunction]], as denoted by the symbol <math>\land,\!</math> in expressions of the form <math>F(x, y, z) = G(x, y) \land H(y, z),\!</math> to define a triadic relation <math>F\!</math> in terms of a pair of dyadic relations <math>G\!</math> and <math>H.\!</math> :* The concepts of dyadic ''projection'' and ''projective determination'', that are invoked in the &ldquo;weak&rdquo; notion of ''projective reducibility''. The relational composition <math>G \circ H\!</math> of a pair of dyadic relations <math>G\!</math> and <math>H\!</math> will be constructed in three stages, first, by taking the tacit extensions of <math>G\!</math> and <math>H\!</math> to triadic relations that reside in the same space, next, by taking the intersection of these extensions, tantamount to the maximal triadic relation that is consistent with the ''prima facie'' dyadic relation data, finally, by projecting this intersection on a suitable plane to form a third dyadic relation, constituting in fact the relational composition <math>G \circ H\!</math> of the relations <math>G\!</math> and <math>H.\!</math> The construction of a relational composition in a specifically mathematical setting normally begins with [[relation (mathematics)|mathematical relations]] at a higher level of abstraction than the corresponding objects in linguistic or logical settings. This is due to the fact that mathematical objects are typically specified only ''up to isomorphism'' as the conventional saying goes, that is, any objects that have the &ldquo;same form&rdquo; are generally regarded as the being the same thing, for most all intents and mathematical purposes. Thus the mathematical construction of a relational composition begins by default with a pair of dyadic relations that reside, without loss of generality, in the same plane, say, <math>G, H \subseteq X \times Y,\!</math> as shown in Figure&nbsp;1. {| align="center" border="0" cellpadding="10" | <pre> o-------------------------------------------------o | | | o o | | |\ |\ | | | \ | \ | | | \ | \ | | | \ | \ | | | \ | \ | | | \ | \ | | | * \ | * \ | | X * Y X * Y | | \ * | \ * | | | \ G | \ H | | | \ | \ | | | \ | \ | | | \ | \ | | | \ | \ | | | \| \| | | o o | | | o-------------------------------------------------o Figure 1. Dyadic Relations G, H c X x Y </pre> |} The dyadic relations <math>G\!</math> and <math>H\!</math> cannot be composed at all at this point, not without additional information or further stipulation. In order for their relational composition to be possible, one of two types of cases has to happen: :* The first type of case occurs when <math>X = Y.\!</math> In this case, both of the compositions <math>G \circ H\!</math> and <math>H \circ G\!</math> are defined. :* The second type of case occurs when <math>X\!</math> and <math>Y\!</math> are distinct, but when it nevertheless makes sense to speak of a dyadic relation <math>\hat{H}\!</math> that is isomorphic to <math>H,\!</math> but living in the plane <math>YZ,\!</math> that is, in the space of the cartesian product <math>Y \times Z,\!</math> for some set <math>Z.\!</math> Whether you view isomorphic things to be the same things or not, you still have to specify the exact isomorphisms that are needed to transform any given representation of a thing into a required representation of the same thing. Let us imagine that we have done this, and say how later: {| align="center" border="0" cellpadding="10" | <pre> o-------------------------------------------------o | | | o o | | |\ /| | | | \ / | | | | \ / | | | | \ / | | | | \ / | | | | \ / | | | | * \ / * | | | X * Y Y * Z | | \ * | | * / | | \ G | | &#292; / | | \ | | / | | \ | | / | | \ | | / | | \ | | / | | \| |/ | | o o | | | o-------------------------------------------------o Figure 2. Dyadic Relations G c X x Y and &#292; c Y x Z </pre> |} With the required spaces carefully swept out, the stage is set for the presentation of Tarski's trick, and the invocation of the following symbolic formula, claimed to be a definition of the relational composition <math>P \circ Q\!</math> of a pair of dyadic relations <math>P, Q \subseteq X \times X.\!</math> : '''Definition.''' <math>P \circ Q = \mathrm{proj}_{13} (P \times X ~\cap~ X \times Q).\!</math> To get this drift of this definition one needs to understand that it comes from a point of view that regards all dyadic relations as covered well enough by subsets of a suitable cartesian square and thus of the form <math>L \subseteq X \times X.\!</math> So, if one has started out with a dyadic relation of the shape <math>L \subseteq U \times V,\!</math> one merely lets <math>X = U \cup V,\!</math> trading in the initial <math>L\!</math> for a new <math>L \subseteq X \times X\!</math> as need be. The projection <math>\mathrm{proj}_{13}\!</math> is just the projection of the cartesian cube <math>X \times X \times X\!</math> on the space of shape <math>X \times X\!</math> that is spanned by the first and the third domains, but since they now have the same names and the same contents it is necessary to distinguish them by numbering their relational places. Finally, the notation of the cartesian product sign &ldquo;<math>\times\!</math>&rdquo; is extended to signify two other products with respect to a dyadic relation <math>L \subseteq X \times X\!</math> and a subset <math>W \subseteq X,\!</math> as follows: : '''Definition.''' <math>L \times W ~=~ \{ (x, y, z) \in X^3 ~:~ (x, y) \in L ~\mathrm{and}~ z \in W \}.\!</math> : '''Definition.''' <math>W \times L ~=~ \{ (x, y, z) \in X^3 ~:~ x \in W ~\mathrm{and}~ (y, z) \in L \}.\!</math> Applying these definitions to the case <math>P, Q \subseteq X \times X,\!</math> the two dyadic relations whose relational composition <math>P \circ Q \subseteq X \times X\!</math> is about to be defined, one finds: : <math>P \times X ~=~ \{ (x, y, z) \in X^3 ~:~ (x, y) \in P ~\mathrm{and}~ z \in X \},\!</math> : <math>X \times Q ~=~ \{ (x, y, z) \in X^3 ~:~ x \in X ~\mathrm{and}~ (y, z) \in Q \}.\!</math> These are just the appropriate special cases of the tacit extensions already defined. : <math>P \times X ~=~ \mathrm{te}_{12}^3 (P),~\!</math> : <math>X \times Q ~=~ \mathrm{te}_{23}^1 (Q).~\!</math> In summary, then, the expression: : <math>\mathrm{proj}_{13} (P \times X ~\cap~ X \times Q)\!</math> is equivalent to the expression: : <math>\mathrm{proj}_{13} (\mathrm{te}_{12}^3 (P) ~\cap~ \mathrm{te}_{23}^1 (Q))\!</math> and this form is generalized &mdash; although, relative to one's school of thought, perhaps inessentially so &mdash; by the form that was given above as follows: : '''Definition.''' <math>P \circ Q ~=~ \mathrm{proj}_{XZ} (\mathrm{te}_{XY}^Z (P) ~\cap~ \mathrm{te}_{YZ}^X (Q)).\!</math> Figure&nbsp;3 presents a geometric picture of what is involved in formulating a definition of the triadic relation <math>F \subseteq X \times Y \times Z\!</math> by way of a conjunction between the dyadic relation <math>G \subseteq X \times Y\!</math> and the dyadic relation <math>H \subseteq Y \times Z,\!</math> as done for example by means of an expression of the following form: :* <math>F(x, y, z) ~=~ G(x, y) \land H(y, z).\!</math> {| align="center" border="0" cellpadding="10" | <pre> o-------------------------------------------------o | | | o | | /|\ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | o o o | | |\ / \ /| | | | \ / F \ / | | | | \ / * \ / | | | | \ /*\ / | | | | / \//*\\/ \ | | | | / /\/ \/\ \ | | | |/ ///\ /\\\ \| | | o X /// Y \\\ Z o | | |\ \/// | \\\/ /| | | | \ /// | \\\ / | | | | \ ///\ | /\\\ / | | | | \ /// \ | / \\\ / | | | | \/// \ | / \\\/ | | | | /\/ \ | / \/\ | | | | *//\ \|/ /\\* | | | X */ Y o Y \* Z | | \ * | | * / | | \ G | | H / | | \ | | / | | \ | | / | | \ | | / | | \ | | / | | \| |/ | | o o | | | o-------------------------------------------------o Figure 3. Projections of F onto G and H </pre> |} To interpret the Figure, visualize the triadic relation <math>F \subseteq X \times Y \times Z\!</math> as a body in <math>XYZ\!</math>-space, while <math>G\!</math> is a figure in <math>XY\!</math>-space and <math>H\!</math> is a figure in <math>YZ\!</math>-space. The dyadic '''projections''' that accompany a triadic relation over <math>X, Y, Z\!</math> are defined as follows: :* <math>\mathrm{proj}_{XY} (L) ~=~ \{ (x, y) \in X \times Y : (x, y, z) \in L ~\text{for some}~ z \in Z) \},\!</math> :* <math>\mathrm{proj}_{XZ} (L) ~=~ \{ (x, z) \in X \times Z : (x, y, z) \in L ~\text{for some}~ y \in Y) \},\!</math> :* <math>\mathrm{proj}_{YZ} (L) ~=~ \{ (y, z) \in Y \times Z : (x, y, z) \in L ~\text{for some}~ x \in X) \}.\!</math> For many purposes it suffices to indicate the dyadic projections of a triadic relation <math>L\!</math> by means of the briefer equivalents listed next: :* <math>L_{XY} ~=~ \mathrm{proj}_{XY}(L),\!</math> :* <math>L_{XZ} ~=~ \mathrm{proj}_{XZ}(L),\!</math> :* <math>L_{YZ} ~=~ \mathrm{proj}_{YZ}(L).\!</math> In light of these definitions, <math>\mathrm{proj}_{XY}\!</math> is a mapping from the set <math>\mathcal{L}_{XYZ}\!</math> of triadic relations over the domains <math>X, Y, Z\!</math> to the set <math>\mathcal{L}_{XY}\!</math> of dyadic relations over the domains <math>X, Y,\!</math> with similar relationships holding for the other projections. To formalize these relationships in a concise but explicit manner, it serves to add a few more definitions. The set <math>\mathcal{L}_{XYZ},~\!</math> whose members are just the triadic relations over <math>X, Y, Z,\!</math> can be recognized as the set of all subsets of the cartesian product <math>X \times Y \times Z,\!</math> also known as the ''power set'' of <math>X \times Y \times Z,\!</math> and notated here as <math>\mathrm{Pow} (X \times Y \times Z).\!</math> :* <math>\mathcal{L}_{XYZ} ~=~ \{ L : L \subseteq X \times Y \times Z \} ~=~ \mathrm{Pow} (X \times Y \times Z).\!</math> Likewise, the power sets of the pairwise cartesian products encompass all the dyadic relations on pairs of distinct domains that can be chosen from <math>\{ X, Y, Z \}.\!</math> :* <math>\mathcal{L}_{XY} ~=~ \{L : L \subseteq X \times Y \} ~=~ \mathrm{Pow} (X \times Y),~\!</math> :* <math>\mathcal{L}_{XZ} ~=~ \{L : L \subseteq X \times Z \} ~=~ \mathrm{Pow} (X \times Z),~\!</math> :* <math>\mathcal{L}_{YZ} ~=~ \{L : L \subseteq Y \times Z \} ~=~ \mathrm{Pow} (Y \times Z).~\!</math> In mathematics, the inverse relation corresponding to a projection map is usually called an ''extension''. To avoid confusion with other senses of the word, however, it is probably best for the sake of this discussion to stick with the more specific term ''tacit extension''. Given three sets, <math>X, Y, Z,\!</math> and three dyadic relations, :* <math>U \subseteq X \times Y,~\!</math> :* <math>V \subseteq X \times Z,~\!</math> :* <math>W \subseteq Y \times Z,~\!</math> the ''tacit extensions'', <math>\mathrm{te}_{XY}^Z, \mathrm{te}_{XZ}^Y, \mathrm{te}_{YZ}^X,~\!</math> of <math>U, V, W,\!</math> respectively, are defined as follows: :* <math>\mathrm{te}_{XY}^Z (U) ~=~ \{ (x, y, z) : (x, y) \in U \},\!</math> :* <math>\mathrm{te}_{XZ}^Y (V) ~=~ \{ (x, y, z) : (x, z) \in V \},\!</math> :* <math>\mathrm{te}_{YZ}^X (W) ~=~ \{ (x, y, z) : (y, z) \in W \}.\!</math> So long as the intended indices attaching to the tacit extensions can be gathered from context, it is usually clear enough to use the abbreviated forms, <math>\mathrm{te}(U), \mathrm{te}(V), \mathrm{te}(W).\!</math> The definition and illustration of relational composition presently under way makes use of the tacit extension of <math>G \subseteq X \times Y\!</math> to <math>\mathrm{te}(G) \subseteq X \times Y \times Z\!</math> and the tacit extension of <math>H \subseteq Y \times Z\!</math> to <math>\mathrm{te}(H) \subseteq X \times Y \times Z,\!</math> only. Geometric illustrations of <math>\mathrm{te}(G)\!</math> and <math>\mathrm{te}(H)\!</math> are afforded by Figures&nbsp;4 and 5, respectively. {| align="center" border="0" cellpadding="10" | <pre> o-------------------------------------------------o | | | o | | /|\ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | / | * \ | | o o ** o | | |\ / \*** /| | | | \ / *** / | | | | \ / ***\ / | | | | \ *** / | | | | / \*** / \ | | | | / *** / \ | | | |/ ***\ / \| | | o X /** Y Z o | | |\ \//* | / /| | | | \ /// | / / | | | | \ ///\ | / / | | | | \ /// \ | / / | | | | \/// \ | / / | | | | /\/ \ | / / | | | | *//\ \|/ / * | | | X */ Y o Y * Z | | \ * | | * / | | \ G | | H / | | \ | | / | | \ | | / | | \ | | / | | \ | | / | | \| |/ | | o o | | | o-------------------------------------------------o Figure 4. Tacit Extension of G to X x Y x Z </pre> |} {| align="center" border="0" cellpadding="10" | <pre> o-------------------------------------------------o | | | o | | /|\ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | / * | \ | | o ** o o | | |\ ***/ \ /| | | | \ *** \ / | | | | \ /*** \ / | | | | \ *** / | | | | / \ ***/ \ | | | | / \ *** \ | | | |/ \ /*** \| | | o X Y **\ Z o | | |\ \ | *\\/ /| | | | \ \ | \\\ / | | | | \ \ | /\\\ / | | | | \ \ | / \\\ / | | | | \ \ | / \\\/ | | | | \ \ | / \/\ | | | | * \ \|/ /\\* | | | X * Y o Y \* Z | | \ * | | * / | | \ G | | H / | | \ | | / | | \ | | / | | \ | | / | | \ | | / | | \| |/ | | o o | | | o-------------------------------------------------o Figure 5. Tacit Extension of H to X x Y x Z </pre> |} A geometric interpretation can now be given that fleshes out in graphic form the meaning of a formula like the following: :* <math>F(x, y, z) ~=~ G(x, y) \land H(y, z).\!</math> The conjunction that is indicated by &ldquo;<math>\land\!</math>&rdquo; corresponds as usual to an intersection of two sets, however, in this case it is the intersection of the tacit extensions <math>\mathrm{te}(G)\!</math> and <math>\mathrm{te}(H).\!</math> {| align="center" border="0" cellpadding="10" | <pre> o-------------------------------------------------o | | | o | | /|\ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | o o o | | |\ / \ /| | | | \ / F \ / | | | | \ / * \ / | | | | \ /*\ / | | | | / \//*\\/ \ | | | | / /\/ \/\ \ | | | |/ ///\ /\\\ \| | | o X /// Y \\\ Z o | | |\ \/// | \\\/ /| | | | \ /// | \\\ / | | | | \ ///\ | /\\\ / | | | | \ /// \ | / \\\ / | | | | \/// \ | / \\\/ | | | | /\/ \ | / \/\ | | | | *//\ \|/ /\\* | | | X */ Y o Y \* Z | | \ * | | * / | | \ G | | H / | | \ | | / | | \ | | / | | \ | | / | | \ | | / | | \| |/ | | o o | | | o-------------------------------------------------o Figure 6. F as the Intersection of te(G) and te(H) </pre> |} ==Algebraic construction== The transition from a geometric picture of relation composition to an algebraic formulation is accomplished through the introduction of coordinates, in other words, identifiable names for the objects that are related through the various forms of relations, dyadic and triadic in the present case. Adding coordinates to the running Example produces the following Figure: {| align="center" border="0" cellpadding="10" | <pre> o-------------------------------------------------o | | | o | | /|\ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | o o o | | |\ / \ /| | | | \ / F \ / | | | | \ / * \ / | | | | \ /*\ / | | | | / \//*\\/ \ | | | | / /\/ \/\ \ | | | |/ ///\ /\\\ \| | | o X /// Y \\\ Z o | | |\ 7\/// | \\\/7 /| | | | \ 6// | \\6 / | | | | \ //5\ | /5\\ / | | | | \ /// 4\ | /4 \\\ / | | | | \/// 3\ | /3 \\\/ | | | | G/\/ 2\ | /2 \/\H | | | | *//\ 1\|/1 /\\* | | | X *\ Y o Y /* Z | | 7\ *\\ |7 7| //* /7 | | 6\ |\\\|6 6|///| /6 | | 5\| \\@5 5@// |/5 | | 4@ \@4 4@/ @4 | | 3\ @3 3@ /3 | | 2\ |2 2| /2 | | 1\|1 1|/1 | | o o | | | o-------------------------------------------------o Figure 7. F as the Intersection of te(G) and te(H) </pre> |} Thinking of relations in operational terms is facilitated by using variant notations for ordered tuples and sets of ordered tuples, namely, the ordered pair <math>(x, y)\!</math> is written <math>x\!:\!y,\!</math> the ordered triple <math>(x, y, z)\!</math> is written <math>x\!:\!y\!:\!z,\!</math> and so on, and a set of tuples is conceived as a logical-algebraic sum, which can be written out in the smaller finite cases in forms like <math>a\!:\!b ~+~ b\!:\!c ~+~ c\!:\!d\!</math> and so on. For example, translating the relations <math>F \subseteq X \times Y \times Z, ~ G \subseteq X \times Y, ~ H \subseteq Y \times Z\!</math> into this notation produces the following summary of the data: {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix} F & = & 4:3:4 & + & 4:4:4 & + & 4:5:4 \\ G & = & 4:3 & + & 4:4 & + & 4:5 \\ H & = & 3:4 & + & 4:4 & + & 5:4 \end{matrix}</math> |} As often happens with abstract notations for functions and relations, the ''type information'', in this case, the fact that <math>G\!</math> and <math>H\!</math> live in different spaces, is left implicit in the context of use. Let us now verify that all of the proposed definitions, formulas, and other relationships check out against the concrete data of the current composition example. The ultimate goal is to develop a clearer picture of what is going on in the formula that expresses the relational composition of a couple of dyadic relations in terms of the medial projection of the intersection of their tacit extensions: {| align="center" cellpadding="8" width="90%" | <math>G \circ H ~=~ \mathrm{proj}_{XZ} (\mathrm{te}_{XY}^Z (G) ~\cap~ \mathrm{te}_{YZ}^X (H)).\!</math> |} Here is the big picture, with all the pieces in place: {| align="center" border="0" cellpadding="10" | <pre> o-------------------------------------------------o | | | o | | / \ | | / \ | | / \ | | / \ | | / \ | | / \ | | / G o H \ | | X * Z | | 7\ /|\ /7 | | 6\ / | \ /6 | | 5\ / | \ /5 | | 4@ | @4 | | 3\ | /3 | | 2\ | /2 | | 1\|/1 | | | | | | | | | | | /|\ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | o | o | | |\ /|\ /| | | | \ / F \ / | | | | \ / * \ / | | | | \ /*\ / | | | | / \//*\\/ \ | | | | / /\/ \/\ \ | | | |/ ///\ /\\\ \| | | o X /// Y \\\ Z o | | |\ \/// | \\\/ /| | | | \ /// | \\\ / | | | | \ ///\ | /\\\ / | | | | \ /// \ | / \\\ / | | | | \/// \ | / \\\/ | | | | G/\/ \ | / \/\H | | | | *//\ \|/ /\\* | | | X *\ Y o Y /* Z | | 7\ *\\ |7 7| //* /7 | | 6\ |\\\|6 6|///| /6 | | 5\| \\@5 5@// |/5 | | 4@ \@4 4@/ @4 | | 3\ @3 3@ /3 | | 2\ |2 2| /2 | | 1\|1 1|/1 | | o o | | | o-------------------------------------------------o Figure 8. G o H = proj_XZ (te(G) |^| te(H)) </pre> |} All that remains is to check the following collection of data and derivations against the situation represented in Figure&nbsp;8. {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix} F & = & 4:3:4 & + & 4:4:4 & + & 4:5:4 \\ G & = & 4:3 & + & 4:4 & + & 4:5 \\ H & = & 3:4 & + & 4:4 & + & 5:4 \end{matrix}</math> |} {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix} G \circ H & = & (4\!:\!3 ~+~ 4\!:\!4 ~+~ 4\!:\!5)(3\!:\!4 ~+~ 4\!:\!4 ~+~ 5\!:\!4) \\[6pt] & = & 4:4 \end{matrix}</math> |} {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix} \mathrm{te}(G) & = & \mathrm{te}_{XY}^Z (G) \\[4pt] & = & \displaystyle\sum_{z=1}^7 (4\!:\!3\!:\!z ~+~ 4\!:\!4\!:\!z ~+~ 4\!:\!5\!:\!z) \end{matrix}</math> |} {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix} \mathrm{te}(G) & = & 4:3:1 & + & 4:4:1 & + & 4:5:1 & + \\ & & 4:3:2 & + & 4:4:2 & + & 4:5:2 & + \\ & & 4:3:3 & + & 4:4:3 & + & 4:5:3 & + \\ & & 4:3:4 & + & 4:4:4 & + & 4:5:4 & + \\ & & 4:3:5 & + & 4:4:5 & + & 4:5:5 & + \\ & & 4:3:6 & + & 4:4:6 & + & 4:5:6 & + \\ & & 4:3:7 & + & 4:4:7 & + & 4:5:7 \end{matrix}</math> |} {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix} \mathrm{te}(H) & = & \mathrm{te}_{YZ}^X (H) \\[4pt] & = & \displaystyle\sum_{x=1}^7 (x\!:\!3\!:\!4 ~+~ x\!:\!4\!:\!4 ~+~ x\!:\!5\!:\!4) \end{matrix}</math> |} {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix} \mathrm{te}(H) & = & 1:3:4 & + & 1:4:4 & + & 1:5:4 & + \\ & & 2:3:4 & + & 2:4:4 & + & 2:5:4 & + \\ & & 3:3:4 & + & 3:4:4 & + & 3:5:4 & + \\ & & 4:3:4 & + & 4:4:4 & + & 4:5:4 & + \\ & & 5:3:4 & + & 5:4:4 & + & 5:5:4 & + \\ & & 6:3:4 & + & 6:4:4 & + & 6:5:4 & + \\ & & 7:3:4 & + & 7:4:4 & + & 7:5:4 \end{matrix}</math> |} {| align="center" cellpadding="8" width="90%" | <math>\begin{array}{ccl} \mathrm{te}(G) \cap \mathrm{te}(H) & = & 4\!:\!3:\!4 ~+~ 4\!:\!4\!:\!4 ~+~ 4\!:\!5\!:\!4 \\[4pt] G \circ H & = & \mathrm{proj}_{XZ} (\mathrm{te}(G) \cap \mathrm{te}(H)) \\[4pt] & = & \mathrm{proj}_{XZ} (4\!:\!3:\!4 ~+~ 4\!:\!4\!:\!4 ~+~ 4\!:\!5\!:\!4) \\[4pt] & = & 4:4 \end{array}</math> |} ==Matrix representation== We have it within our reach to pick up another way of representing dyadic relations, namely, the representation as ''[[logical matrix|logical matrices]]'', and also to grasp the analogy between relational composition and ordinary [[matrix multiplication]] as it appears in linear algebra. First of all, while we still have the data of a very simple concrete case in mind, let us reflect on what we did in our last Example in order to find the composition <math>G \circ H\!</math> of the dyadic relations <math>G\!</math> and <math>H.\!</math> Here is the setup that we had before: {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix}X & = & \{ 1, 2, 3, 4, 5, 6, 7 \}\end{matrix}</math> |} {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix} G & = & 4:3 & + & 4:4 & + & 4:5 & \subseteq & X \times X \\ H & = & 3:4 & + & 4:4 & + & 5:4 & \subseteq & X \times X \end{matrix}</math> |} Let us recall the rule for finding the relational composition of a pair of dyadic relations. Given the dyadic relations <math>P \subseteq X \times Y\!</math> and <math>Q \subseteq Y \times Z,\!</math> the composition of <math>P ~\text{on}~ Q\!</math> is written as <math>P \circ Q,\!</math> or more simply as <math>PQ,\!</math> and obtained as follows: To compute <math>PQ,\!</math> in general, where <math>P\!</math> and <math>Q\!</math> are dyadic relations, simply multiply out the two sums in the ordinary distributive algebraic way, but subject to the following rule for finding the product of two elementary relations of shapes <math>a:b\!</math> and <math>c:d.\!</math> {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix} (a:b)(c:d) & = & (a:d) & \text{if}~ b = c \\ (a:b)(c:d) & = & 0 & \text{otherwise} \end{matrix}</math> |} To find the relational composition <math>G \circ H,\!</math> one may begin by writing it as a quasi-algebraic product: {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix} G \circ H & = & (4\!:\!3 ~+~ 4\!:\!4 ~+~ 4\!:\!5)(3\!:\!4 ~+~ 4\!:\!4 ~+~ 5\!:\!4) \end{matrix}</math> |} Multiplying this out in accord with the applicable form of distributive law one obtains the following expansion: {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix} G \circ H & = & (4:3)(3:4) & + & (4:3)(4:4) & + & (4:3)(5:4) & + \\ & & (4:4)(3:4) & + & (4:4)(4:4) & + & (4:4)(5:4) & + \\ & & (4:5)(3:4) & + & (4:5)(4:4) & + & (4:5)(5:4) \end{matrix}</math> |} Applying the rule that determines the product of elementary relations produces the following array: {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix} G \circ H & = & 4:4 & + & 0 & + & 0 & + \\ & & 0 & + & 4:4 & + & 0 & + \\ & & 0 & + & 0 & + & 4:4 \end{matrix}</math> |} Since the plus sign in this context represents an operation of logical disjunction or set-theoretic aggregation, all of the positive multiplicites count as one, and this gives the ultimate result: {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix}G \circ H & = & 4:4\end{matrix}</math> |} With an eye toward extracting a general formula for relation composition, viewed here on analogy to algebraic multiplication, let us examine what we did in multiplying the dyadic relations <math>G\!</math> and <math>H\!</math> together to obtain their relational composite <math>G \circ H.\!</math> Given the space <math>X = \{ 1, 2, 3, 4, 5, 6, 7 \},\!</math> whose cardinality <math>|X|\!</math> is <math>7,\!</math> there are <math>|X \times X| = |X| \cdot |X|\!</math> <math>=\!</math> <math>7 \cdot 7 = 49\!</math> elementary relations of the form <math>i:j,\!</math> where <math>i\!</math> and <math>j\!</math> range over the space <math>X.\!</math> Although they might be organized in many different ways, it is convenient to regard the collection of elementary relations as arranged in a lexicographic block of the following form: {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix} 1\!:\!1 & 1\!:\!2 & 1\!:\!3 & 1\!:\!4 & 1\!:\!5 & 1\!:\!6 & 1\!:\!7 \\ 2\!:\!1 & 2\!:\!2 & 2\!:\!3 & 2\!:\!4 & 2\!:\!5 & 2\!:\!6 & 2\!:\!7 \\ 3\!:\!1 & 3\!:\!2 & 3\!:\!3 & 3\!:\!4 & 3\!:\!5 & 3\!:\!6 & 3\!:\!7 \\ 4\!:\!1 & 4\!:\!2 & 4\!:\!3 & 4\!:\!4 & 4\!:\!5 & 4\!:\!6 & 4\!:\!7 \\ 5\!:\!1 & 5\!:\!2 & 5\!:\!3 & 5\!:\!4 & 5\!:\!5 & 5\!:\!6 & 5\!:\!7 \\ 6\!:\!1 & 6\!:\!2 & 6\!:\!3 & 6\!:\!4 & 6\!:\!5 & 6\!:\!6 & 6\!:\!7 \\ 7\!:\!1 & 7\!:\!2 & 7\!:\!3 & 7\!:\!4 & 7\!:\!5 & 7\!:\!6 & 7\!:\!7 \end{matrix}</math> |} The relations <math>G\!</math> and <math>H\!</math> may then be regarded as logical sums of the following forms: {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix} G & = & \displaystyle\sum_{ij} G_{ij} (i\!:\!j) \\[20pt] H & = & \displaystyle\sum_{ij} H_{ij} (i\!:\!j) \end{matrix}\!</math> |} The notation <math>\textstyle\sum_{ij}\!</math> indicates a logical sum over the collection of elementary relations <math>i\!:\!j\!</math> while the factors <math>G_{ij}\!</math> and <math>H_{ij}\!</math> are values in the ''[[boolean domain]]'' <math>\mathbb{B} = \{ 0, 1 \}~\!</math> that are called the ''coefficients'' of the relations <math>G\!</math> and <math>H,\!</math> respectively, with regard to the corresponding elementary relations <math>i\!:\!j.\!</math> In general, for a dyadic relation <math>L,\!</math> the coefficient <math>L_{ij}\!</math> of the elementary relation <math>i\!:\!j\!</math> in the relation <math>L\!</math> will be <math>0\!</math> or <math>1,\!</math> respectively, as <math>i\!:\!j\!</math> is excluded from or included in <math>L.\!</math> With these conventions in place, the expansions of <math>G\!</math> and <math>H\!</math> may be written out as follows: {| align="center" cellpadding="4" width="90%" | <math>\begin{matrix}G & = & 4:3 & + & 4:4 & + & 4:5 & =\end{matrix}</math> |- | <math>\begin{smallmatrix} 0 \cdot (1:1) & + & 0 \cdot (1:2) & + & 0 \cdot (1:3) & + & 0 \cdot (1:4) & + & 0 \cdot (1:5) & + & 0 \cdot (1:6) & + & 0 \cdot (1:7) & + \\ 0 \cdot (2:1) & + & 0 \cdot (2:2) & + & 0 \cdot (2:3) & + & 0 \cdot (2:4) & + & 0 \cdot (2:5) & + & 0 \cdot (2:6) & + & 0 \cdot (2:7) & + \\ 0 \cdot (3:1) & + & 0 \cdot (3:2) & + & 0 \cdot (3:3) & + & 0 \cdot (3:4) & + & 0 \cdot (3:5) & + & 0 \cdot (3:6) & + & 0 \cdot (3:7) & + \\ 0 \cdot (4:1) & + & 0 \cdot (4:2) & + & \mathbf{1} \cdot (4:3) & + & \mathbf{1} \cdot (4:4) & + & \mathbf{1} \cdot (4:5) & + & 0 \cdot (4:6) & + & 0 \cdot (4:7) & + \\ 0 \cdot (5:1) & + & 0 \cdot (5:2) & + & 0 \cdot (5:3) & + & 0 \cdot (5:4) & + & 0 \cdot (5:5) & + & 0 \cdot (5:6) & + & 0 \cdot (5:7) & + \\ 0 \cdot (6:1) & + & 0 \cdot (6:2) & + & 0 \cdot (6:3) & + & 0 \cdot (6:4) & + & 0 \cdot (6:5) & + & 0 \cdot (6:6) & + & 0 \cdot (6:7) & + \\ 0 \cdot (7:1) & + & 0 \cdot (7:2) & + & 0 \cdot (7:3) & + & 0 \cdot (7:4) & + & 0 \cdot (7:5) & + & 0 \cdot (7:6) & + & 0 \cdot (7:7) \end{smallmatrix}</math> |} {| align="center" cellpadding="4" width="90%" | <math>\begin{matrix}H & = & 3:4 & + & 4:4 & + & 5:4 & =\end{matrix}</math> |- | <math>\begin{smallmatrix} 0 \cdot (1:1) & + & 0 \cdot (1:2) & + & 0 \cdot (1:3) & + & 0 \cdot (1:4) & + & 0 \cdot (1:5) & + & 0 \cdot (1:6) & + & 0 \cdot (1:7) & + \\ 0 \cdot (2:1) & + & 0 \cdot (2:2) & + & 0 \cdot (2:3) & + & 0 \cdot (2:4) & + & 0 \cdot (2:5) & + & 0 \cdot (2:6) & + & 0 \cdot (2:7) & + \\ 0 \cdot (3:1) & + & 0 \cdot (3:2) & + & 0 \cdot (3:3) & + & \mathbf{1} \cdot (3:4) & + & 0 \cdot (3:5) & + & 0 \cdot (3:6) & + & 0 \cdot (3:7) & + \\ 0 \cdot (4:1) & + & 0 \cdot (4:2) & + & 0 \cdot (4:3) & + & \mathbf{1} \cdot (4:4) & + & 0 \cdot (4:5) & + & 0 \cdot (4:6) & + & 0 \cdot (4:7) & + \\ 0 \cdot (5:1) & + & 0 \cdot (5:2) & + & 0 \cdot (5:3) & + & \mathbf{1} \cdot (5:4) & + & 0 \cdot (5:5) & + & 0 \cdot (5:6) & + & 0 \cdot (5:7) & + \\ 0 \cdot (6:1) & + & 0 \cdot (6:2) & + & 0 \cdot (6:3) & + & 0 \cdot (6:4) & + & 0 \cdot (6:5) & + & 0 \cdot (6:6) & + & 0 \cdot (6:7) & + \\ 0 \cdot (7:1) & + & 0 \cdot (7:2) & + & 0 \cdot (7:3) & + & 0 \cdot (7:4) & + & 0 \cdot (7:5) & + & 0 \cdot (7:6) & + & 0 \cdot (7:7) \end{smallmatrix}</math> |} Stripping down to the bare essentials, one obtains the following matrices of coefficients for the relations <math>G\!</math> and <math>H.\!</math> {| align="center" cellpadding="8" width="90%" | <math>G ~=~ \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}</math> |} {| align="center" cellpadding="8" width="90%" | <math>H ~=~ \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}</math> |} These are the logical matrix representations of the dyadic relations <math>G\!</math> and <math>H.\!</math> If the dyadic relations <math>G\!</math> and <math>H\!</math> are viewed as logical sums then their relational composition <math>G \circ H\!</math> can be regarded as a product of sums, a fact that can be indicated as follows: {| align="center" cellpadding="8" width="90%" | <math>G \circ H ~=~ (\sum_{ij} G_{ij} (i\!:\!j))(\sum_{ij} H_{ij} (i\!:\!j)).\!</math> |} The composite relation <math>G \circ H\!</math> is itself a dyadic relation over the same space <math>X,\!</math> in other words, <math>G \circ H \subseteq X \times X,\!</math> and this means that <math>G \circ H\!</math> must be amenable to being written as a logical sum of the following form: {| align="center" cellpadding="8" width="90%" | <math>G \circ H ~=~ \sum_{ij} (G \circ H)_{ij} (i\!:\!j).\!</math> |} In this formula, <math>(G \circ H)_{ij}\!</math> is the coefficient of <math>G \circ H\!</math> with respect to the elementary relation <math>i\!:\!j.\!</math> One of the best ways to reason out what <math>G \circ H\!</math> should be is to ask oneself what its coefficient <math>(G \circ H)_{ij}\!</math> should be for each of the elementary relations <math>i\!:\!j\!</math> in turn. So let us pose the question: {| align="center" cellpadding="8" width="90%" | <math>(G \circ H)_{ij} ~=~ ?\!</math> |} In order to answer this question, it helps to realize that the indicated product given above can be written in the following equivalent form: {| align="center" cellpadding="8" width="90%" | <math>G \circ H ~=~ (\sum_{ik} G_{ik} (i\!:\!k))(\sum_{kj} H_{kj} (k\!:\!j)).\!</math> |} A moment's thought will tell us that <math>(G \circ H)_{ij} = 1\!</math> if and only if there is an element <math>k\!</math> in <math>X\!</math> such that <math>G_{ik} = 1\!</math> and <math>H_{kj} = 1.\!</math> Consequently, we have the result: {| align="center" cellpadding="8" width="90%" | <math>(G \circ H)_{ij} ~=~ \sum_{k} G_{ik} H_{kj}.\!</math> |} This follows from the properties of boolean arithmetic, specifically, from the fact that the product <math>G_{ik} H_{kj}\!</math> is <math>1\!</math> if and only if both <math>G_{ik}\!</math> and <math>H_{kj}\!</math> are <math>1\!</math> and from the fact that <math>\textstyle\sum_{k} F_{k}\!</math> is equal to <math>1\!</math> just in case some <math>F_{k}\!</math> is <math>1.\!</math> All that remains in order to obtain a computational formula for the relational composite <math>G \circ H\!</math> of the dyadic relations <math>G\!</math> and <math>H\!</math> is to collect the coefficients <math>(G \circ H)_{ij}\!</math> as <math>i\!</math> and <math>j\!</math> range over <math>X.\!</math> {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix}G \circ H & = & \displaystyle \sum_{ij} (G \circ H)_{ij} (i\!:\!j) & = & \displaystyle \sum_{ij} (\sum_{k} G_{ik} H_{kj}) (i\!:\!j). \end{matrix}</math> |} This is the logical analogue of matrix multiplication in linear algebra, the difference in the logical setting being that all of the operations performed on coefficients take place in a system of boolean arithmetic where summation corresponds to logical disjunction and multiplication corresponds to logical conjunction. By way of disentangling this formula, one may notice that the form <math>\textstyle \sum_{k} G_{ik} H_{kj}\!</math> is what is usually called a ''scalar product''. In this case it is the scalar product of the <math>i^\text{th}\!</math> row of <math>G\!</math> with the <math>j^\text{th}\!</math> column of <math>H.\!</math> To make this statement more concrete, let us go back to the examples of <math>G\!</math> and <math>H\!</math> we came in with: {| align="center" cellpadding="8" width="90%" | <math>G ~=~ \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}</math> |} {| align="center" cellpadding="8" width="90%" | <math>H ~=~ \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}</math> |} The formula for computing <math>G \circ H\!</math> says the following: {| align="center" cellpadding="8" width="90%" | <math>\begin{array}{ccl} (G \circ H)_{ij} & = & \text{the}~ {ij}^\text{th} ~\text{entry in the matrix representation for}~ G \circ H \\[2pt] & = & \text{the entry in the}~ {i}^\text{th} ~\text{row and the}~ {j}^\text{th} ~\text{column of}~ G \circ H \\[2pt] & = & \text{the scalar product of the}~ {i}^\text{th} ~\text{row of}~ G ~\text{with the}~ {j}^\text{th} ~\text{column of}~ H \\[2pt] & = & \sum_{k} G_{ik} H_{kj} \end{array}</math> |} As it happens, it is possible to make exceedingly light work of this example, since there is only one row of <math>G\!</math> and one column of <math>H\!</math> that are not all zeroes. Taking the scalar product, in a logical way, of the fourth row of <math>G\!</math> with the fourth column of <math>H\!</math> produces the sole non-zero entry for the matrix of <math>G \circ H.\!</math> {| align="center" cellpadding="8" width="90%" | <math>G \circ H ~=~ \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}</math> |} ==Graph-theoretic picture== There is another form of representation for dyadic relations that is useful to keep in mind, especially for its ability to render the logic of many complex formulas almost instantly understandable to the mind's eye. This is the representation in terms of ''bipartite graphs'', or ''bigraphs'' for short. Here is what <math>G\!</math> and <math>H\!</math> look like in the bigraph picture: {| align="center" border="0" cellpadding="10" | <pre> o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | /|\ | | / | \ G | | / | \ | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 9. G = 4:3 + 4:4 + 4:5 </pre> |- | <pre> o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | \ | / | | \ | / H | | \|/ | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 10. H = 3:4 + 4:4 + 5:4 </pre> |} These graphs may be read to say: {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix} G ~\text{puts}~ 4 ~\text{in relation to}~ 3, 4, 5. \\[2pt] H ~\text{puts}~ 3, 4, 5 ~\text{in relation to}~ 4. \end{matrix}</math> |} To form the composite relation <math>G \circ H,\!</math> one simply follows the bigraph for <math>G\!</math> by the bigraph for <math>H,\!</math> here arranging the bigraphs in order down the page, and then treats any non-empty set of paths of length two between two nodes as being equivalent to a single directed edge between those nodes in the composite bigraph for <math>G \circ H.\!</math> Here's how it looks in pictures: {| align="center" border="0" cellpadding="10" | <pre> o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | /|\ | | / | \ G | | / | \ | | o o o o o o o X | | \ | / | | \ | / H | | \|/ | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 11. G Followed By H </pre> |- | <pre> o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | | | | | G o H | | | | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 12. G Composed With H </pre> |} Once again we find that <math>G \circ H = 4:4.\!</math> We have now seen three different representations of dyadic relations. If one has a strong preference for letters, or numbers, or pictures, then one may be tempted to take one or another of these as being canonical, but each of them will be found to have its peculiar advantages and disadvantages in any given application, and the maximum advantage is therefore approached by keeping all three of them in mind. To see the promised utility of the bigraph picture of dyadic relations, let us devise a slightly more complex example of a composition problem, and use it to illustrate the logic of the matrix multiplication formula. Keeping to the same space <math>X = \{ 1, 2, 3, 4, 5, 6, 7 \},\!</math> define the dyadic relations <math>M, N \subseteq X \times X\!</math> as follows: {| align="center" cellpadding="8" width="90%" | <math>\begin{array}{*{19}{c}} M & = & 2\!:\!1 & + & 2\!:\!2 & + & 2\!:\!3 & + & 4\!:\!3 & + & 4\!:\!4 & + & 4\!:\!5 & + & 6\!:\!5 & + & 6\!:\!6 & + & 6\!:\!7 \\[2pt] N & = & 1\!:\!1 & + & 2\!:\!1 & + & 3\!:\!3 & + & 4\!:\!3 & ~ & + & ~ & 4\!:\!5 & + & 5\!:\!5 & + & 6\!:\!7 & + & 7\!:\!7 \end{array}</math> |} Here are the bigraph pictures: {| align="center" border="0" cellpadding="10" | <pre> o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | /|\ /|\ /|\ | | / | \ / | \ / | \ M | | / | \ / | \ / | \ | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 13. Dyadic Relation M </pre> |- | <pre> o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | | / | / \ | \ | | | | / | / \ | \ | N | | |/ |/ \| \| | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 14. Dyadic Relation N </pre> |} To form the composite relation <math>M \circ N,\!</math> one simply follows the bigraph for <math>M\!</math> by the bigraph for <math>N,\!</math> arranging the bigraphs in order down the page, and then counts any non-empty set of paths of length two between two nodes as being equivalent to a single directed edge between those two nodes in the composite bigraph for <math>M \circ N.\!</math> Here's how it looks in pictures: {| align="center" border="0" cellpadding="10" | <pre> o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | /|\ /|\ /|\ | | / | \ / | \ / | \ M | | / | \ / | \ / | \ | | o o o o o o o X | | | / | / \ | \ | | | | / | / \ | \ | N | | |/ |/ \| \| | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 15. M Followed By N </pre> |- | <pre> o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | / \ / \ / \ | | / \ / \ / \ M o N | | / \ / \ / \ | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 16. M Composed With N </pre> |} Let us hark back to that mysterious matrix multiplication formula, and see how it appears in the light of the bigraph representation. The coefficient of the composition <math>M \circ N\!</math> between <math>i\!</math> and <math>j\!</math> in <math>X\!</math> is given as follows: {| align="center" cellpadding="8" width="90%" | <math>(M \circ N)_{ij} ~=~ \sum_{k} M_{ik} N_{kj}\!</math> |} Graphically interpreted, this is a ''sum over paths''. Starting at the node <math>i,\!</math> <math>M_{ik}\!</math> being <math>1\!</math> indicates that there is an edge in the bigraph of <math>M\!</math> from node <math>i\!</math> to node <math>k\!</math> and <math>N_{kj}\!</math> being <math>1\!</math> indicates that there is an edge in the bigraph of <math>N\!</math> from node <math>k\!</math> to node <math>j.\!</math> So the <math>\textstyle\sum_{k}\!</math> ranges over all possible intermediaries <math>k,\!</math> ascending from <math>0\!</math> to <math>1\!</math> just as soon as there happens to be a path of length two between nodes <math>i\!</math> and <math>j.\!</math> It is instructive at this point to compute the other possible composition that can be formed from <math>M\!</math> and <math>N,\!</math> namely, the composition <math>N \circ M,\!</math> that takes <math>M\!</math> and <math>N\!</math> in the opposite order. Here is the graphic computation: {| align="center" border="0" cellpadding="10" | <pre> o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | | / | / \ | \ | | | | / | / \ | \ | N | | |/ |/ \| \| | | o o o o o o o X | | /|\ /|\ /|\ | | / | \ / | \ / | \ M | | / | \ / | \ / | \ | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 17. N Followed By M </pre> |- | <pre> o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | | | N o M | | | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 18. N Composed With M </pre> |} In sum, <math>N \circ M = 0.\!</math> This example affords sufficient evidence that relational composition, just like its kindred, matrix multiplication, is a ''non-commutative'' algebraic operation. ==References== * Ulam, S.M., and Bednarek, A.R., &ldquo;On the Theory of Relational Structures and Schemata for Parallel Computation&rdquo; (1977), pp. 477&ndash;508 in A.R. Bednarek and Françoise Ulam (eds.), ''Analogies Between Analogies : The Mathematical Reports of S.M. Ulam and His Los&nbsp;Alamos Collaborators'', University of California Press, Berkeley, CA, 1990. ==Bibliography== * Mathematical Society of Japan, ''Encyclopedic Dictionary of Mathematics'', 2nd edition, 2&nbsp;volumes., Kiyosi Itô (ed.), MIT Press, Cambridge, MA, 1993. * Mili, A., Desharnais, J., Mili, F., with Frappier, M., ''Computer Program Construction'', Oxford University Press, New York, NY, 1994. * Ulam, S.M., ''Analogies Between Analogies : The Mathematical Reports of S.M. Ulam and His Los&nbsp;Alamos Collaborators'', A.R. Bednarek and Françoise Ulam (eds.), University of California Press, Berkeley, CA, 1990. ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Relation_composition Relation Composition @ InterSciWiki] * [http://mywikibiz.com/Relation_composition Relation Composition @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Relation_composition Relation Composition @ Subject Wikis] * [http://en.wikiversity.org/wiki/Relation_composition Relation Composition @ Wikiversity] * [http://beta.wikiversity.org/wiki/Relation_composition Relation Composition @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Relation_composition Relation Composition], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Relation_composition Relation Composition], [http://mywikibiz.com/ MyWikiBiz] * [http://semanticweb.org/wiki/Relation_composition Relation Composition], [http://semanticweb.org/ Semantic Web] * [http://ref.subwiki.org/wiki/Relation_composition Relation Composition], [http://ref.subwiki.org/ Subject Wikis] * [http://en.wikiversity.org/wiki/Relation_composition Relation Composition], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Relation_composition Relation Composition], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Relation_composition&oldid=43467878 Relation Composition], [http://en.wikipedia.org/ Wikipedia] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Database Theory]] [[Category:Discrete Mathematics]] [[Category:Formal Sciences]] [[Category:Inquiry]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Relation Theory]] [[Category:Set Theory]] 35ba75a92d7bfd92a2ae6270b2346b4eab2ede12 0 368 706 2015-10-31T02:02:02Z Jon Awbrey 3 + article wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. '''Logical conjunction''' is an operation on two logical values, typically the values of two propositions, that produces a value of ''true'' if and only if both of its operands are true. The [[truth table]] of <math>p ~\operatorname{AND}~ q,</math> also written <math>p \land q\!</math> or <math>p \cdot q,\!</math> appears below: <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:45%" |+ style="height:30px" | <math>\text{Logical Conjunction}\!</math> |- style="height:40px; background:#f0f0ff" | style="width:33%" | <math>p\!</math> | style="width:33%" | <math>q\!</math> | style="width:33%" | <math>p \land q</math> |- | <math>\operatorname{F}</math> || <math>\operatorname{F}</math> || <math>\operatorname{F}</math> |- | <math>\operatorname{F}</math> || <math>\operatorname{T}</math> || <math>\operatorname{F}</math> |- | <math>\operatorname{T}</math> || <math>\operatorname{F}</math> || <math>\operatorname{F}</math> |- | <math>\operatorname{T}</math> || <math>\operatorname{T}</math> || <math>\operatorname{T}</math> |} <br> ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Logical_conjunction Logical Conjunction @ InterSciWiki] * [http://mywikibiz.com/Logical_conjunction Logical Conjunction @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Logical_conjunction Logical Conjunction @ Subject Wikis] * [http://en.wikiversity.org/wiki/Logical_conjunction Logical Conjunction @ Wikiversity] * [http://beta.wikiversity.org/wiki/Logical_conjunction Logical Conjunction @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Logical_conjunction Logical Conjunction], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Logical_conjunction Logical Conjunction], [http://mywikibiz.com/ MyWikiBiz] * [http://wikinfo.org/w/index.php/Logical_conjunction Logical Conjunction], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Logical_conjunction Logical Conjunction], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Logical_conjunction Logical Conjunction], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Logical_conjunction&oldid=75153420 Logical Conjunction], [http://en.wikipedia.org/ Wikipedia] [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Charles Sanders Peirce]] [[Category:Computer Science]] [[Category:Formal Languages]] [[Category:Formal Sciences]] [[Category:Formal Systems]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Philosophy]] [[Category:Semiotics]] 213f75f197e101166f3e3bf015b0e4d81ad1f564 0 320 707 541 2015-10-31T03:04:03Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. '''Ampheck''', from Greek &#945;&#956;&#966;&#942;&#954;&#951;&#962; double-edged, is a term coined by [[Charles Sanders Peirce]] for either one of the pair of logically dual operators, variously referred to as Peirce arrows, Sheffer strokes, or [[logical NAND|NAND]] and [[logical NNOR|NNOR]]. Either of these logical operators is a ''[[sole sufficient operator]]'' for deriving or generating all of the other operators in the subject matter variously described as [[boolean function]]s, monadic predicate calculus, [[propositional calculus]], sentential calculus, or [[zeroth order logic]]. <blockquote> <p>For example, <math>x \curlywedge y</math> signifies that <math>x\!</math> is <math>\mathbf{f}</math> and <math>y\!</math> is <math>\mathbf{f}</math>. Then <math>(x \curlywedge y) \curlywedge z</math>, or <math>\underline {x \curlywedge y} \curlywedge z</math>, will signify that <math>z\!</math> is <math>\mathbf{f}</math>, but that the statement that <math>x\!</math> and <math>y\!</math> are both <math>\mathbf{f}</math> is itself <math>\mathbf{f}</math>, that is, is ''false''. Hence, the value of <math>x \curlywedge x</math> is the same as that of <math>\overline {x}</math>; and the value of <math>\underline {x \curlywedge x} \curlywedge x</math> is <math>\mathbf{f}</math>, because it is necessarily false; while the value of <math>\underline {x \curlywedge y} \curlywedge \underline {x \curlywedge y}</math> is only <math>\mathbf{f}</math> in case <math>x \curlywedge y</math> is <math>\mathbf{v}</math>; and <math>( \underline {x \curlywedge x} \curlywedge x) \curlywedge (x \curlywedge \underline {x \curlywedge x})</math> is necessarily true, so that its value is <math>\mathbf{v}</math>.</p> <p>With these two signs, the vinculum (with its equivalents, parentheses, brackets, braces, etc.) and the sign <math>\curlywedge</math>, which I will call the ''ampheck'' (from &#945;&#956;&#966;&#951;&#954;&#942;&#962;&nbsp;, cutting both ways), all assertions as to the values of quantities can be expressed. (C.S. Peirce, CP 4.264).</p> </blockquote> In the above passage, Peirce introduces the term ''ampheck'' for the 2-place logical connective or the binary logical operator that is currently called the ''joint denial'' in logic, the NNOR operator in computer science, or indicated by means of phrases like "neither-nor" or "both not" in ordinary language. For this operation he employs a symbol that the typographer most likely set by inverting the zodiac symbol for Aries('''&#9800;'''), but set in the text above by means of the ''curly wedge'' symbol. In the same paper, Peirce introduces a symbol for the logically dual operator. This was rendered by the editors of his ''Collected Papers'' as an inverted Aries symbol with a bar or a serif at the top, in this way denoting the connective or logical operator that is currently called the ''alternative denial'' in logic, the NAND operator in computer science, or invoked by means of phrases like "not-and" or "not both" in ordinary language. It is not clear whether it was Peirce himself or later writers who initiated the practice, but on account of their dual relationship it became common to refer to these two operators in the plural, as the ''amphecks''. ==References and further reading== * Clark, Glenn (1997), &ldquo;New Light on Peirce's Iconic Notation for the Sixteen Binary Connectives&rdquo;, pp. 304&ndash;333 in Houser, Roberts, Van Evra (eds.), ''Studies in the Logic of Charles Sanders Peirce'', Indiana University Press, Bloomington, IN, 1997. * Houser, N., Roberts, Don D., and Van Evra, James (eds., 1997), ''Studies in the Logic of Charles Sanders Peirce'', Indiana University Press, Bloomington, IN. * McCulloch, W.S. (1961), &ldquo;What Is a Number, that a Man May Know It, and a Man, that He May Know a Number?&rdquo; (Ninth Alfred Korzybski Memorial Lecture), ''General Semantics Bulletin'', Nos. 26 & 27, 7&ndash;18, Institute of General Semantics, Lakeville, CT. Reprinted, pp. 1&ndash;18 in ''Embodiments of Mind''. [http://www.vordenker.de/ggphilosophy/mcculloch_what-is-a-number.pdf Online]. * McCulloch, W.S. (1965), ''Embodiments of Mind'', MIT Press, Cambridge, MA. * [[Charles Sanders Peirce (Bibliography)|Peirce, C.S., Bibliography]]. * Peirce, C.S., ''Collected Papers of Charles Sanders Peirce'', vols. 1&ndash;6, Charles Hartshorne and Paul Weiss (eds.), vols. 7&ndash;8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931&ndash;1935, 1958. * Peirce, C.S. (1902), &ldquo;The Simplest Mathematics&rdquo;. First published as CP&nbsp;4.227&ndash;323 in ''Collected Papers''. * Zellweger, Shea (1997), &ldquo;Untapped Potential in Peirce's Iconic Notation for the Sixteen Binary Connectives&rdquo;, pp. 334&ndash;386 in Houser, Roberts, Van Evra (eds.), ''Studies in the Logic of Charles Sanders Peirce'', Indiana University Press, Bloomington, IN, 1997. ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Ampheck Ampheck @ InterSciWiki] * [http://mywikibiz.com/Ampheck Ampheck @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Ampheck Ampheck @ Subject Wikis] * [http://en.wikiversity.org/wiki/Ampheck Ampheck @ Wikiversity] * [http://beta.wikiversity.org/wiki/Ampheck Ampheck @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Ampheck Ampheck], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Ampheck Ampheck], [http://mywikibiz.com/ MyWikiBiz] * [http://planetmath.org/Ampheck Ampheck], [http://planetmath.org/ PlanetMath] * [http://en.wikiversity.org/wiki/Ampheck Ampheck], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Ampheck Ampheck], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Ampheck&oldid=62218032 Ampheck], [http://en.wikipedia.org/ Wikipedia] [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Automata Theory]] [[Category:Charles Sanders Peirce]] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Discrete Mathematics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Neural Networks]] [[Category:Semiotics]] 034f243d4b3c8e34840231a9c2f865030bd47186 717 707 2015-11-05T15:05:35Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. '''Ampheck''', from Greek &#945;&#956;&#966;&#942;&#954;&#951;&#962; double-edged, is a term coined by [[Charles Sanders Peirce]] for either one of the pair of logically dual operators, variously referred to as Peirce arrows, Sheffer strokes, or [[logical NAND|NAND]] and [[logical NNOR|NNOR]]. Either of these logical operators is a ''[[sole sufficient operator]]'' for deriving or generating all of the other operators in the subject matter variously described as [[boolean function]]s, monadic predicate calculus, [[propositional calculus]], sentential calculus, or [[zeroth order logic]]. <blockquote> <p>For example, <math>x \curlywedge y</math> signifies that <math>x\!</math> is <math>\mathbf{f}</math> and <math>y\!</math> is <math>\mathbf{f}</math>. Then <math>(x \curlywedge y) \curlywedge z</math>, or <math>\underline {x \curlywedge y} \curlywedge z</math>, will signify that <math>z\!</math> is <math>\mathbf{f}</math>, but that the statement that <math>x\!</math> and <math>y\!</math> are both <math>\mathbf{f}</math> is itself <math>\mathbf{f}</math>, that is, is ''false''. Hence, the value of <math>x \curlywedge x</math> is the same as that of <math>\overline {x}</math>; and the value of <math>\underline {x \curlywedge x} \curlywedge x</math> is <math>\mathbf{f}</math>, because it is necessarily false; while the value of <math>\underline {x \curlywedge y} \curlywedge \underline {x \curlywedge y}</math> is only <math>\mathbf{f}</math> in case <math>x \curlywedge y</math> is <math>\mathbf{v}</math>; and <math>( \underline {x \curlywedge x} \curlywedge x) \curlywedge (x \curlywedge \underline {x \curlywedge x})</math> is necessarily true, so that its value is <math>\mathbf{v}</math>.</p> <p>With these two signs, the vinculum (with its equivalents, parentheses, brackets, braces, etc.) and the sign <math>\curlywedge</math>, which I will call the ''ampheck'' (from &#945;&#956;&#966;&#951;&#954;&#942;&#962;&nbsp;, cutting both ways), all assertions as to the values of quantities can be expressed. (C.S. Peirce, CP 4.264).</p> </blockquote> In the above passage, Peirce introduces the term ''ampheck'' for the 2-place logical connective or the binary logical operator that is currently called the ''joint denial'' in logic, the NNOR operator in computer science, or indicated by means of phrases like "neither-nor" or "both not" in ordinary language. For this operation he employs a symbol that the typographer most likely set by inverting the zodiac symbol for Aries('''&#9800;'''), but set in the text above by means of the ''curly wedge'' symbol. In the same paper, Peirce introduces a symbol for the logically dual operator. This was rendered by the editors of his ''Collected Papers'' as an inverted Aries symbol with a bar or a serif at the top, in this way denoting the connective or logical operator that is currently called the ''alternative denial'' in logic, the NAND operator in computer science, or invoked by means of phrases like "not-and" or "not both" in ordinary language. It is not clear whether it was Peirce himself or later writers who initiated the practice, but on account of their dual relationship it became common to refer to these two operators in the plural, as the ''amphecks''. ==References and further reading== * Clark, Glenn (1997), &ldquo;New Light on Peirce's Iconic Notation for the Sixteen Binary Connectives&rdquo;, pp. 304&ndash;333 in Houser, Roberts, Van Evra (eds.), ''Studies in the Logic of Charles Sanders Peirce'', Indiana University Press, Bloomington, IN, 1997. * Houser, N., Roberts, Don D., and Van Evra, James (eds., 1997), ''Studies in the Logic of Charles Sanders Peirce'', Indiana University Press, Bloomington, IN. * McCulloch, W.S. (1961), &ldquo;What Is a Number, that a Man May Know It, and a Man, that He May Know a Number?&rdquo; (Ninth Alfred Korzybski Memorial Lecture), ''General Semantics Bulletin'', Nos. 26 & 27, 7&ndash;18, Institute of General Semantics, Lakeville, CT. Reprinted, pp. 1&ndash;18 in ''Embodiments of Mind''. [http://www.vordenker.de/ggphilosophy/mcculloch_what-is-a-number.pdf Online]. * McCulloch, W.S. (1965), ''Embodiments of Mind'', MIT Press, Cambridge, MA. * [[Charles Sanders Peirce (Bibliography)|Peirce, C.S., Bibliography]]. * Peirce, C.S., ''Collected Papers of Charles Sanders Peirce'', vols. 1&ndash;6, Charles Hartshorne and Paul Weiss (eds.), vols. 7&ndash;8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931&ndash;1935, 1958. * Peirce, C.S. (1902), &ldquo;The Simplest Mathematics&rdquo;. First published as CP&nbsp;4.227&ndash;323 in ''Collected Papers''. * Zellweger, Shea (1997), &ldquo;Untapped Potential in Peirce's Iconic Notation for the Sixteen Binary Connectives&rdquo;, pp. 334&ndash;386 in Houser, Roberts, Van Evra (eds.), ''Studies in the Logic of Charles Sanders Peirce'', Indiana University Press, Bloomington, IN, 1997. ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Ampheck Ampheck @ InterSciWiki] * [http://mywikibiz.com/Ampheck Ampheck @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Ampheck Ampheck @ Subject Wikis] * [http://en.wikiversity.org/wiki/Ampheck Ampheck @ Wikiversity] * [http://beta.wikiversity.org/wiki/Ampheck Ampheck @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Ampheck Ampheck], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Ampheck Ampheck], [http://mywikibiz.com/ MyWikiBiz] * [http://planetmath.org/Ampheck Ampheck], [http://planetmath.org/ PlanetMath] * [http://wikinfo.org/w/index.php/Ampheck Ampheck], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Ampheck Ampheck], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Ampheck Ampheck], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Ampheck&oldid=62218032 Ampheck], [http://en.wikipedia.org/ Wikipedia] [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Automata Theory]] [[Category:Charles Sanders Peirce]] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Discrete Mathematics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Neural Networks]] [[Category:Semiotics]] b624b3c7a5eb172c83abfadee6cf038a490f3a8b 0 369 708 2015-11-02T00:56:48Z Jon Awbrey 3 + article wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. '''Logical disjunction''', also called '''logical alternation''', is an operation on two logical values, typically the values of two propositions, that produces a value of ''false'' if and only if both of its operands are false. The [[truth table]] of <math>p ~\operatorname{OR}~ q,</math> also written <math>p \lor q,\!</math> appears below: <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:45%" |+ style="height:30px" | <math>\text{Logical Disjunction}\!</math> |- style="height:40px; background:#f0f0ff" | style="width:33%" | <math>p\!</math> | style="width:33%" | <math>q\!</math> | style="width:33%" | <math>p \lor q</math> |- | <math>\operatorname{F}</math> || <math>\operatorname{F}</math> || <math>\operatorname{F}</math> |- | <math>\operatorname{F}</math> || <math>\operatorname{T}</math> || <math>\operatorname{T}</math> |- | <math>\operatorname{T}</math> || <math>\operatorname{F}</math> || <math>\operatorname{T}</math> |- | <math>\operatorname{T}</math> || <math>\operatorname{T}</math> || <math>\operatorname{T}</math> |} <br> ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Logical_disjunction Logical Disjunction @ InterSciWiki] * [http://mywikibiz.com/Logical_disjunction Logical Disjunction @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Logical_disjunction Logical Disjunction @ Subject Wikis] * [http://en.wikiversity.org/wiki/Logical_disjunction Logical Disjunction @ Wikiversity] * [http://beta.wikiversity.org/wiki/Logical_disjunction Logical Disjunction @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Logical_disjunction Logical Disjunction], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Logical_disjunction Logical Disjunction], [http://mywikibiz.com/ MyWikiBiz] * [http://wikinfo.org/w/index.php/Logical_disjunction Logical Disjunction], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Logical_disjunction Logical Disjunction], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Logical_disjunction Logical Disjunction], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Logical_disjunction&oldid=75154551 Logical Disjunction], [http://en.wikipedia.org/ Wikipedia] [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Charles Sanders Peirce]] [[Category:Computer Science]] [[Category:Formal Languages]] [[Category:Formal Sciences]] [[Category:Formal Systems]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Philosophy]] [[Category:Semiotics]] 619d7511a25825a6877f4aabd83c378b6f8ba77b 0 344 709 699 2015-11-03T21:30:07Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. '''''Semeiotic''''' is one of the terms that Charles Sanders Peirce used to describe his theory of [[triadic relation|triadic]] [[sign relations]], along with ''semiotic'' and the plural variants of both terms. The form ''semeiotic'' is often used to distinguish Peirce's theory, since it is less often used by other writers to denote their particular approaches to the subject. ==Types of signs== There are three principal ways that a sign can denote its objects. These are usually described as ''kinds'', ''species'', or ''types'' of signs, but it is important to recognize that these are not ontological species, that is, they are not mutually exclusive features of description, since the same thing can be a sign in several different ways. Beginning very roughly, the three main ways of being a sign can be described as follows: :* An ''icon'' is a sign that denotes its objects by virtue of a quality that it shares with its objects. :* An ''index'' is a sign that denotes its objects by virtue of an existential connection that it has with its objects. :* A ''symbol'' is a sign that denotes its objects solely by virtue of the fact that it is interpreted to do so. One of Peirce's early delineations of the three types of signs is still quite useful as a first approach to understanding their differences and their relationships to each other: {| align="center" cellpadding="8" width="90%" | <p>In the first place there are likenesses or copies &mdash; such as ''statues'', ''pictures'', ''emblems'', ''hieroglyphics'', and the like. Such representations stand for their objects only so far as they have an actual resemblance to them &mdash; that is agree with them in some characters. The peculiarity of such representations is that they do not determine their objects &mdash; they stand for anything more or less; for they stand for whatever they resemble and they resemble everything more or less.</p> <p>The second kind of representations are such as are set up by a convention of men or a decree of God. Such are ''tallies'', ''proper names'', &c. The peculiarity of these ''conventional signs'' is that they represent no character of their objects. Likenesses denote nothing in particular; ''conventional signs'' connote nothing in particular.</p> <p>The third and last kind of representations are ''symbols'' or general representations. They connote attributes and so connote them as to determine what they denote. To this class belong all ''words'' and all ''conceptions''. Most combinations of words are also symbols. A proposition, an argument, even a whole book may be, and should be, a single symbol. (Peirce 1866, &ldquo;Lowell Lecture 7&rdquo;, CE&nbsp;1, 467&ndash;468).</p> |} ==References== * Peirce, C.S., [[Charles Sanders Peirce (Bibliography)|Bibliography]]. * Peirce, C.S., ''Writings of Charles S. Peirce : A Chronological Edition, Volume&nbsp;1, 1857&ndash;1866'', Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1982. Cited as CE&nbsp;1. * Peirce, C.S. (1865), "On the Logic of Science", Harvard University Lectures, CE&nbsp;1, 161&ndash;302. * Peirce, C.S. (1866), "The Logic of Science, or, Induction and Hypothesis", Lowell Institute Lectures, CE&nbsp;1, 357&ndash;504. ==Readings== * Awbrey, J.L., and Awbrey, S.M. (1995), &ldquo;Interpretation as Action : The Risk of Inquiry&rdquo;, ''Inquiry : Critical Thinking Across the Disciplines'' 15(1), pp. 40&ndash;52. [http://web.archive.org/web/19970626071826/http://chss.montclair.edu/inquiry/fall95/awbrey.html Archive]. [http://independent.academia.edu/JonAwbrey/Papers/1302117/Interpretation_as_Action_The_Risk_of_Inquiry Online]. ==Resources== * [http://vectors.usc.edu/thoughtmesh/publish/142.php Semeiotic &rarr; ThoughtMesh] * Bergman & Paavola (eds.), ''Commens Dictionary of Peirce's Terms'', [http://www.helsinki.fi/science/commens/dictionary.html Webpage] ** ''[http://www.helsinki.fi/science/commens/terms/semeiotic.html Semeiotic]'' ** ''[http://www.helsinki.fi/science/commens/terms/icon.html Icon]'' ** ''[http://www.helsinki.fi/science/commens/terms/index2.html Index]'' ** ''[http://www.helsinki.fi/science/commens/terms/symbol.html Symbol]'' ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Semeiotic Semeiotic @ InterSciWiki] * [http://mywikibiz.com/Semeiotic Semeiotic @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Semeiotic Semeiotic @ Subject Wikis] * [http://en.wikiversity.org/wiki/Semeiotic Semeiotic @ Wikiversity] * [http://beta.wikiversity.org/wiki/Semeiotic Semeiotic @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Semiotic_Information Semiotic Information] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Semeiotic Semeiotic], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Semeiotic Semeiotic], [http://mywikibiz.com/ MyWikiBiz] * [http://semanticweb.org/wiki/Semeiotic Semeiotic], [http://semanticweb.org/ Semantic Web] * [http://ref.subwiki.org/wiki/Semeiotic Semeiotic], [http://ref.subwiki.org/ Subject Wikis] * [http://vectors.usc.edu/thoughtmesh/publish/142.php Semeiotic], [http://vectors.usc.edu/thoughtmesh/ ThoughtMesh] * [http://wikinfo.org/w/index.php/Semeiotic Semeiotic], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Semeiotic Semeiotic], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Semeiotic Semeiotic], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Semeiotic&oldid=246563989 Semeiotic], [http://en.wikipedia.org/ Wikipedia] [[Category:Artificial Intelligence]] [[Category:Charles Sanders Peirce]] [[Category:Critical Thinking]] [[Category:Cybernetics]] [[Category:Education]] [[Category:Hermeneutics]] [[Category:Information Systems]] [[Category:Inquiry]] [[Category:Intelligence Amplification]] [[Category:Learning Organizations]] [[Category:Knowledge Representation]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Philosophy]] [[Category:Pragmatics]] [[Category:Semantics]] [[Category:Semiotics]] [[Category:Systems Science]] 9e305ceae7287a542e0c78436e9e84aa102d9125 0 370 710 2015-11-04T04:24:54Z Jon Awbrey 3 add article wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. '''Logical equality''' is an operation on two logical values, typically the values of two propositions, that produces a value of ''true'' if and only if both operands are false or both operands are true. The [[truth table]] of <math>p ~\operatorname{EQ}~ q,</math> also written <math>p = q,\!</math> <math>p \Leftrightarrow q,\!</math> or <math>p \equiv q,\!</math> appears below: <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:45%" |+ style="height:30px" | <math>\text{Logical Equality}\!</math> |- style="height:40px; background:#f0f0ff" | style="width:33%" | <math>p\!</math> | style="width:33%" | <math>q\!</math> | style="width:33%" | <math>p = q\!</math> |- | <math>\operatorname{F}</math> || <math>\operatorname{F}</math> || <math>\operatorname{T}</math> |- | <math>\operatorname{F}</math> || <math>\operatorname{T}</math> || <math>\operatorname{F}</math> |- | <math>\operatorname{T}</math> || <math>\operatorname{F}</math> || <math>\operatorname{F}</math> |- | <math>\operatorname{T}</math> || <math>\operatorname{T}</math> || <math>\operatorname{T}</math> |} <br> ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Logical_equality Logical Equality @ InterSciWiki] * [http://mywikibiz.com/Logical_equality Logical Equality @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Logical_equality Logical Equality @ Subject Wikis] * [http://en.wikiversity.org/wiki/Logical_equality Logical Equality @ Wikiversity] * [http://beta.wikiversity.org/wiki/Logical_equality Logical Equality @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Logical_equality Logical Equality], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Logical_equality Logical Equality], [http://mywikibiz.com/ MyWikiBiz] * [http://wikinfo.org/w/index.php/Logical_equality Logical Equality], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Logical_equality Logical Equality], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Logical_equality Logical Equality], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Logical_equality&oldid=77110577 Logical Equality], [http://en.wikipedia.org/ Wikipedia] [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Charles Sanders Peirce]] [[Category:Computer Science]] [[Category:Formal Languages]] [[Category:Formal Sciences]] [[Category:Formal Systems]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Philosophy]] [[Category:Semiotics]] c8cf1165925b12e6562081848ab342d83d212970 0 312 711 700 2015-11-04T05:24:04Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. This article treats relations from the perspective of combinatorics, in other words, as a subject matter in discrete mathematics, with special attention to finite structures and concrete set-theoretic constructions, many of which arise quite naturally in applications. This approach to relation theory, or the theory of relations, is distinguished from, though closely related to, its study from the perspectives of abstract algebra on the one hand and formal logic on the other. ==Preliminaries== Two definitions of the relation concept are common in the literature. Although it is usually clear in context which definition is being used at a given time, it tends to become less clear as contexts collide, or as discussion moves from one context to another. The same sort of ambiguity arose in the development of the function concept and it may save some effort to follow the pattern of resolution that worked itself out there. When we speak of a function <math>f : X \to Y\!</math> we are thinking of a mathematical object whose articulation requires three pieces of data, specifying the set <math>X,\!</math> the set <math>Y,\!</math> and a particular subset of their cartesian product <math>{X \times Y}.\!</math> So far so good. Let us write <math>f = (\mathrm{obj_1}f, \mathrm{obj_2}f, \mathrm{obj_{12}}f)\!</math> to express what has been said so far. When it comes to parsing the notation <math>{}^{\backprime\backprime} f : X \to Y {}^{\prime\prime},\!</math> everyone takes the part <math>{}^{\backprime\backprime} X \to Y {}^{\prime\prime}\!</math> to specify the ''type'' of the function, that is, the pair <math>(\mathrm{obj_1}f, \mathrm{obj_2}f),\!</math> but <math>{}^{\backprime\backprime} f {}^{\prime\prime}\!</math> is used equivocally to denote both the triple and the subset <math>\mathrm{obj_{12}}f\!</math> that forms one part of it. One way to resolve the ambiguity is to formalize a distinction between a function and its ''graph'', letting <math>\mathrm{graph}(f) := \mathrm{obj_{12}}f.\!</math> Another tactic treats the whole notation <math>{}^{\backprime\backprime} f : X \to Y {}^{\prime\prime}\!</math> as sufficient denotation for the triple, letting <math>{}^{\backprime\backprime} f {}^{\prime\prime}\!</math> denote <math>\mathrm{graph}(f).\!</math> In categorical and computational contexts, at least initially, the type is regarded as an essential attribute or an integral part of the function itself. In other contexts it may be desirable to use a more abstract concept of function, treating a function as a mathematical object that appears in connection with many different types. Following the pattern of the functional case, let the notation <math>{}^{\backprime\backprime} L \subseteq X \times Y {}^{\prime\prime}\!</math> bring to mind a mathematical object that is specified by three pieces of data, the set <math>X,\!</math> the set <math>Y,\!</math> and a particular subset of their cartesian product <math>{X \times Y}.\!</math> As before we have two choices, either let <math>L = (X, Y, \mathrm{graph}(L))\!</math> or let <math>{}^{\backprime\backprime} L {}^{\prime\prime}\!</math> denote <math>\mathrm{graph}(L)\!</math> and choose another name for the triple. ==Definition== It is convenient to begin with the definition of a ''<math>k\!</math>-place relation'', where <math>k\!</math> is a positive integer. '''Definition.''' A ''<math>k\!</math>-place relation'' <math>L \subseteq X_1 \times \ldots \times X_k\!</math> over the nonempty sets <math>X_1, \ldots, X_k\!</math> is a <math>(k+1)\!</math>-tuple <math>(X_1, \ldots, X_k, L)\!</math> where <math>L\!</math> is a subset of the cartesian product <math>X_1 \times \ldots \times X_k.\!</math> ==Remarks== Though usage varies as usage will, there are several bits of optional language that are frequently useful in discussing relations. The sets <math>X_1, \ldots, X_k\!</math> are called the ''domains'' of the relation <math>L \subseteq X_1 \times \ldots \times X_k,\!</math> with <math>{X_j}\!</math> being the <math>j^\text{th}\!</math> domain. If all of the <math>{X_j}\!</math> are the same set <math>X\!</math> then <math>L \subseteq X_1 \times \ldots \times X_k\!</math> is more simply described as a <math>k\!</math>-place relation over <math>X.\!</math> The set <math>L\!</math> is called the ''graph'' of the relation <math>L \subseteq X_1 \times \ldots \times X_k,\!</math> on analogy with the graph of a function. If the sequence of sets <math>X_1, \ldots, X_k\!</math> is constant throughout a given discussion or is otherwise determinate in context, then the relation <math>L \subseteq X_1 \times \ldots \times X_k\!</math> is determined by its graph <math>L,\!</math> making it acceptable to denote the relation by referring to its graph. Other synonyms for the adjective ''<math>k\!</math>-place'' are ''<math>k\!</math>-adic'' and ''<math>k\!</math>-ary'', all of which leads to the integer <math>k\!</math> being called the ''dimension'', ''adicity'', or ''arity'' of the relation <math>L.\!</math> ==Local incidence properties== A ''local incidence property'' (LIP) of a relation <math>L\!</math> is a property that depends in turn on the properties of special subsets of <math>L\!</math> that are known as its ''local flags''. The local flags of a relation are defined in the following way: Let <math>L\!</math> be a <math>k\!</math>-place relation <math>L \subseteq X_1 \times \ldots \times X_k.\!</math> Select a relational domain <math>{X_j}\!</math> and one of its elements <math>x.\!</math> Then <math>L_{x \operatorname{at} j}\!</math> is a subset of <math>L\!</math> that is referred to as the ''flag'' of <math>L\!</math> with <math>x\!</math> at <math>{j},\!</math> or the ''<math>x \operatorname{at} j\!</math>-flag'' of <math>L,\!</math> an object that has the following definition: {| align="center" cellpadding="8" style="text-align:center" | <math>L_{x \operatorname{at} j} ~=~ \{ (x_1, \ldots, x_j, \ldots, x_k) \in L ~:~ x_j = x \}.\!</math> |} Any property <math>C\!</math> of the local flag <math>L_{x \operatorname{at} j} \subseteq L\!</math> is said to be a ''local incidence property'' of <math>L\!</math> with respect to the ''locus'' <math>x \operatorname{at} j.\!</math> A <math>k\!</math>-adic relation <math>L \subseteq X_1 \times \ldots \times X_k\!</math> is said to be ''<math>C\!</math>-regular'' at <math>j\!</math> if and only if every flag of <math>L\!</math> with <math>x\!</math> at <math>j\!</math> has the property <math>C,\!</math> where <math>x\!</math> is taken to vary over the ''theme'' of the fixed domain <math>X_j.\!</math> Expressed in symbols, <math>L\!</math> is ''<math>C\!</math>-regular'' at <math>j\!</math> if and only if <math>C(L_{x \operatorname{at} j})\!</math> is true for all <math>x\!</math> in <math>X_j.\!</math> ==Regional incidence properties== The definition of a local flag can be broadened from a point <math>x\!</math> in <math>{X_j}\!</math> to a subset <math>M\!</math> of <math>X_j,\!</math> arriving at the definition of a ''regional flag'' in the following way: Suppose that <math>L \subseteq X_1 \times \ldots \times X_k,\!</math> and choose a subset <math>M \subseteq X_j.\!</math> Then <math>L_{M \operatorname{at} j}\!</math> is a subset of <math>L\!</math> that is said to be the ''flag'' of <math>L\!</math> with <math>M\!</math> at <math>{j},\!</math> or the ''<math>M \operatorname{at} j\!</math>-flag'' of <math>L,\!</math> an object which has the following definition: {| align="center" cellpadding="8" style="text-align:center" | <math>L_{M \operatorname{at} j} ~=~ \{ (x_1, \ldots, x_j, \ldots, x_k) \in L ~:~ x_j \in M \}.\!</math> |} ==Numerical incidence properties== A ''numerical incidence property'' (NIP) of a relation is a local incidence property that depends on the cardinalities of its local flags. For example, <math>L\!</math> is said to be ''<math>c\!</math>-regular at <math>j\!</math>'' if and only if the cardinality of the local flag <math>L_{x \operatorname{at} j}\!</math> is <math>c\!</math> for all <math>x\!</math> in <math>{X_j},\!</math> or, to write it in symbols, if and only if <math>|L_{x \operatorname{at} j}| = c\!</math> for all <math>{x \in X_j}.\!</math> In a similar fashion, one can define the NIPs, ''<math>(<\!c)\!</math>-regular at <math>{j},\!</math>'' ''<math>(>\!c)\!</math>-regular at <math>{j},\!</math>'' and so on. For ease of reference, a few of these definitions are recorded here: {| align="center" cellpadding="8" style="text-align:center" | <math>\begin{matrix} L & \text{is} & c\textit{-regular} & \text{at}\ j & \text{if and only if} & |L_{x \operatorname{at} j}| & = & c & \text{for all}\ x \in X_j. \\[4pt] L & \text{is} & (<\!c)\textit{-regular} & \text{at}\ j & \text{if and only if} & |L_{x \operatorname{at} j}| & < & c & \text{for all}\ x \in X_j. \\[4pt] L & \text{is} & (>\!c)\textit{-regular} & \text{at}\ j & \text{if and only if} & |L_{x \operatorname{at} j}| & > & c & \text{for all}\ x \in X_j. \end{matrix}</math> |} ==Species of 2-adic relations== Returning to 2-adic relations, it is useful to describe some familiar classes of objects in terms of their local and numerical incidence properties. Let <math>L \subseteq S \times T\!</math> be an arbitrary 2-adic relation. The following properties of <math>L\!</math> can be defined: {| align="center" cellpadding="8" style="text-align:center" | <math>\begin{matrix} L & \text{is} & \textit{total} & \text{at}~ S & \text{if and only if} & L & \text{is} & (\ge 1)\text{-regular} & \text{at}~ S. \\[4pt] L & \text{is} & \textit{total} & \text{at}~ T & \text{if and only if} & L & \text{is} & (\ge 1)\text{-regular} & \text{at}~ T. \\[4pt] L & \text{is} & \textit{tubular} & \text{at}~ S & \text{if and only if} & L & \text{is} & (\le 1)\text{-regular} & \text{at}\ S. \\[4pt] L & \text{is} & \textit{tubular} & \text{at}~ T & \text{if and only if} & L & \text{is} & (\le 1)\text{-regular} & \text{at}~ T. \end{matrix}</math> |} If <math>L \subseteq S \times T\!</math> is tubular at <math>S\!</math> then <math>L\!</math> is called a ''partial function'' or a ''prefunction'' from <math>S\!</math> to <math>T.\!</math> This is sometimes indicated by giving <math>L\!</math> an alternate name, say, <math>{}^{\backprime\backprime} p {}^{\prime\prime},~\!</math> and writing <math>L = p : S \rightharpoonup T.\!</math> Just by way of formalizing the definition: {| align="center" cellpadding="8" style="text-align:center" | <math>\begin{matrix} L & = & p : S \rightharpoonup T & \text{if and only if} & L & \text{is} & \text{tubular} & \text{at}~ S. \end{matrix}</math> |} If <math>L\!</math> is a prefunction <math>p : S \rightharpoonup T\!</math> that happens to be total at <math>S,\!</math> then <math>L\!</math> is called a ''function'' from <math>S\!</math> to <math>T,\!</math> indicated by writing <math>L = f : S \to T.\!</math> To say that a relation <math>L \subseteq S \times T\!</math> is ''totally tubular'' at <math>S\!</math> is to say that it is <math>1\!</math>-regular at <math>S.\!</math> Thus, we may formalize the following definition: {| align="center" cellpadding="8" style="text-align:center" | <math>\begin{matrix} L & = & f : S \to T & \text{if and only if} & L & \text{is} & 1\text{-regular} & \text{at}~ S. \end{matrix}</math> |} In the case of a function <math>f : S \to T,\!</math> one has the following additional definitions: {| align="center" cellpadding="8" style="text-align:center" | <math>\begin{matrix} f & \text{is} & \textit{surjective} & \text{if and only if} & f & \text{is} & \text{total} & \text{at}~ T. \\[4pt] f & \text{is} & \textit{injective} & \text{if and only if} & f & \text{is} & \text{tubular} & \text{at}~ T. \\[4pt] f & \text{is} & \textit{bijective} & \text{if and only if} & f & \text{is} & 1\text{-regular} & \text{at}~ T. \end{matrix}</math> |} ==Variations== Because the concept of a relation has been developed quite literally from the beginnings of logic and mathematics, and because it has incorporated contributions from a diversity of thinkers from many different times and intellectual climes, there is a wide variety of terminology that the reader may run across in connection with the subject. One dimension of variation is reflected in the names that are given to <math>k\!</math>-place relations, for <math>k = 0, 1, 2, 3, \ldots,\!</math> with some writers using the Greek forms, ''medadic'', ''monadic'', ''dyadic'', ''triadic'', ''<math>k\!</math>-adic'', and other writers using the Latin forms, ''nullary'', ''unary'', ''binary'', ''ternary'', ''<math>k\!</math>-ary''. The number of relational domains may be referred to as the ''adicity'', ''arity'', or ''dimension'' of the relation. Accordingly, one finds a relation on a finite number of domains described as a ''polyadic'' relation or a ''finitary'' relation, but others count infinitary relations among the polyadic. If the number of domains is finite, say equal to <math>k,\!</math> then the relation may be described as a ''<math>k\!</math>-adic'' relation, a ''<math>k\!</math>-ary'' relation, or a ''<math>k\!</math>-dimensional'' relation, respectively. A more conceptual than nominal variation depends on whether one uses terms like ''predicate'', ''relation'', and even ''term'' to refer to the formal object proper or else to the allied syntactic items that are used to denote them. Compounded with this variation is still another, frequently associated with philosophical differences over the status in reality accorded formal objects. Among those who speak of numbers, functions, properties, relations, and sets as being real, that is to say, as having objective properties, there are divergences as to whether some things are more real than others, especially whether particulars or properties are equally real or else one is derivative in relationship to the other. Historically speaking, just about every combination of modalities has been used by one school of thought or another, but it suffices here merely to indicate how the options are generated. ==Examples== : ''See the articles on [[relation (mathematics)|relations]], [[relation composition]], [[relation reduction]], [[sign relation]]s, and [[triadic relation]]s for concrete examples of relations.'' Many relations of the greatest interest in mathematics are triadic relations, but this fact is somewhat disguised by the circumstance that many of them are referred to as ''binary operations'', and because the most familiar of these have very specific properties that are dictated by their axioms. This makes it practical to study these operations for quite some time by focusing on their dyadic aspects before being forced to consider their proper characters as triadic relations. ==References== * Peirce, C.S., &ldquo;Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic&rdquo;, ''Memoirs of the American Academy of Arts and Sciences'', 9, 317&ndash;378, 1870. Reprinted, ''Collected Papers'' CP 3.45&ndash;149, ''Chronological Edition'' CE 2, 359&ndash;429. * Ulam, S.M. and Bednarek, A.R., &ldquo;On the Theory of Relational Structures and Schemata for Parallel Computation&rdquo;, pp. 477&ndash;508 in A.R. Bednarek and Françoise Ulam (eds.), ''Analogies Between Analogies : The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators'', University of California Press, Berkeley, CA, 1990. ==Bibliography== * Barr, Michael, and Wells, Charles (1990), ''Category Theory for Computing Science'', Prentice Hall, Hemel Hempstead, UK. * Bourbaki, Nicolas (1994), ''Elements of the History of Mathematics'', John Meldrum (trans.), Springer-Verlag, Berlin, Germany. * Carnap, Rudolf (1958), ''Introduction to Symbolic Logic with Applications'', Dover Publications, New York, NY. * Chang, C.C., and Keisler, H.J. (1973), ''Model Theory'', North-Holland, Amsterdam, Netherlands. * van Dalen, Dirk (1980), ''Logic and Structure'', 2nd edition, Springer-Verlag, Berlin, Germany. * Devlin, Keith J. (1993), ''The Joy of Sets : Fundamentals of Contemporary Set Theory'', 2nd edition, Springer-Verlag, New York, NY. * Halmos, Paul Richard (1960), ''Naive Set Theory'', D. Van Nostrand Company, Princeton, NJ. * van Heijenoort, Jean (1967/1977), ''From Frege to Gödel : A Source Book in Mathematical Logic, 1879&ndash;1931'', Harvard University Press, Cambridge, MA, 1967. Reprinted with corrections, 1977. * Kelley, John L. (1955), ''General Topology'', Van Nostrand Reinhold, New York, NY. * Kneale, William; and Kneale, Martha (1962/1975), ''The Development of Logic'', Oxford University Press, Oxford, UK, 1962. Reprinted with corrections, 1975. * Lawvere, Francis William; and Rosebrugh, Robert (2003), ''Sets for Mathematics'', Cambridge University Press, Cambridge, UK. * Lawvere, Francis William; and Schanuel, Stephen H. (1997/2000), ''Conceptual Mathematics : A First Introduction to Categories'', Cambridge University Press, Cambridge, UK, 1997. Reprinted with corrections, 2000. * Manin, Yu. I. (1977), ''A Course in Mathematical Logic'', Neal Koblitz (trans.), Springer-Verlag, New York, NY. * Mathematical Society of Japan (1993), ''Encyclopedic Dictionary of Mathematics'', 2nd edition, 2 vols., Kiyosi Itô (ed.), MIT Press, Cambridge, MA. * Mili, A., Desharnais, J., Mili, F., with Frappier, M. (1994), ''Computer Program Construction'', Oxford University Press, New York, NY. (Introduction to Tarskian relation theory and relational programming.) * Mitchell, John C. (1996), ''Foundations for Programming Languages'', MIT Press, Cambridge, MA. * Peirce, Charles Sanders (1870), ``Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic", ''Memoirs of the American Academy of Arts and Sciences'' 9 (1870), 317&ndash;378. Reprinted (CP 3.45&ndash;149), (CE 2, 359&ndash;429). * Peirce, Charles Sanders (1931&ndash;1935, 1958), ''Collected Papers of Charles Sanders Peirce'', vols. 1&ndash;6, Charles Hartshorne and Paul Weiss (eds.), vols. 7&ndash;8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA. Cited as (CP volume.paragraph). * Peirce, Charles Sanders (1981&ndash;), ''Writings of Charles S. Peirce : A Chronological Edition'', Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN. Cited as (CE volume, page). * Poizat, Bruno (2000), ''A Course in Model Theory : An Introduction to Contemporary Mathematical Logic'', Moses Klein (trans.), Springer-Verlag, New York, NY. * Quine, Willard Van Orman (1940/1981), ''Mathematical Logic'', 1940. Revised edition, Harvard University Press, Cambridge, MA, 1951. New preface, 1981. * Royce, Josiah (1961), ''The Principles of Logic'', Philosophical Library, New York, NY. * Runes, Dagobert D. (ed., 1962), ''Dictionary of Philosophy'', Littlefield, Adams, and Company, Totowa, NJ. * Shoenfield, Joseph R. (1967), ''Mathematical Logic'', Addison-Wesley, Reading, MA. * Styazhkin, N.I. (1969), ''History of Mathematical Logic from Leibniz to Peano'', MIT Press, Cambridge, MA. * Suppes, Patrick (1957/1999), ''Introduction to Logic'', 1st published 1957. Reprinted, Dover Publications, New York, NY, 1999. * Suppes, Patrick (1960/1972), ''Axiomatic Set Theory'', 1st published 1960. Reprinted, Dover Publications, New York, NY, 1972. * Tarski, Alfred (1956/1983), ''Logic, Semantics, Metamathematics : Papers from 1923 to 1938'', J.H. Woodger (trans.), Oxford University Press, 1956. 2nd edition, J. Corcoran (ed.), Hackett Publishing, Indianapolis, IN, 1983. * Ulam, Stanislaw Marcin; and Bednarek, A.R. (1977), &ldquo;On the Theory of Relational Structures and Schemata for Parallel Computation&rdquo;, pp. 477&ndash;508 in A.R. Bednarek and Françoise Ulam (eds.), ''Analogies Between Analogies : The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators'', University of California Press, Berkeley, CA, 1990. * Ulam, Stanislaw Marcin (1990), ''Analogies Between Analogies : The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators'', A.R. Bednarek and Françoise Ulam (eds.), University of California Press, Berkeley, CA. * Ullman, Jeffrey D. (1980), ''Principles of Database Systems'', Computer Science Press, Rockville, MD. * Venetus, Paulus (1472/1984), ''Logica Parva : Translation of the 1472 Edition with Introduction and Notes'', Alan R. Perreiah (trans.), Philosophia Verlag, Munich, Germany. ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Relation_theory Relation Theory @ InterSciWiki] * [http://mywikibiz.com/Relation_theory Relation Theory @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Relation_theory Relation Theory @ Subject Wikis] * [http://beta.wikiversity.org/wiki/Relation_theory Relation Theory @ Wikiversity Beta] * [http://en.wikiversity.org/wiki/Relation_theory Relation Theory @ Wikiversity] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Relation_theory Relation Theory], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Relation_theory Relation Theory], [http://mywikibiz.com/ MyWikiBiz] * [http://planetmath.org/RelationTheory Relation Theory], [http://planetmath.org/ PlanetMath] * [http://semanticweb.org/wiki/Relation_theory Relation Theory], [http://semanticweb.org/ Semantic Web] * [http://wikinfo.org/w/index.php/Relation_Theory Relation Theory], [http://wikinfo.org/w/ Wikinfo] * [http://beta.wikiversity.org/wiki/Relation_theory Relation Theory], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikiversity.org/wiki/Relation_theory Relation Theory], [http://en.wikiversity.org/ Wikiversity] * [http://en.wikipedia.org/w/index.php?title=Theory_of_relations&oldid=45042729 Relation Theory], [http://en.wikipedia.org/ Wikipedia] [[Category:Charles Sanders Peirce]] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Database Theory]] [[Category:Discrete Mathematics]] [[Category:Formal Sciences]] [[Category:Inquiry]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Relation Theory]] [[Category:Set Theory]] 91158f05b861fa16a5fda4d4f020856b9291354b 712 711 2015-11-04T19:24:11Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. This article treats relations from the perspective of combinatorics, in other words, as a subject matter in discrete mathematics, with special attention to finite structures and concrete set-theoretic constructions, many of which arise quite naturally in applications. This approach to relation theory, or the theory of relations, is distinguished from, though closely related to, its study from the perspectives of abstract algebra on the one hand and formal logic on the other. ==Preliminaries== Two definitions of the relation concept are common in the literature. Although it is usually clear in context which definition is being used at a given time, it tends to become less clear as contexts collide, or as discussion moves from one context to another. The same sort of ambiguity arose in the development of the function concept and it may save some effort to follow the pattern of resolution that worked itself out there. When we speak of a function <math>f : X \to Y\!</math> we are thinking of a mathematical object whose articulation requires three pieces of data, specifying the set <math>X,\!</math> the set <math>Y,\!</math> and a particular subset of their cartesian product <math>{X \times Y}.\!</math> So far so good. Let us write <math>f = (\mathrm{obj_1}f, \mathrm{obj_2}f, \mathrm{obj_{12}}f)\!</math> to express what has been said so far. When it comes to parsing the notation <math>{}^{\backprime\backprime} f : X \to Y {}^{\prime\prime},\!</math> everyone takes the part <math>{}^{\backprime\backprime} X \to Y {}^{\prime\prime}\!</math> to specify the ''type'' of the function, that is, the pair <math>(\mathrm{obj_1}f, \mathrm{obj_2}f),\!</math> but <math>{}^{\backprime\backprime} f {}^{\prime\prime}\!</math> is used equivocally to denote both the triple and the subset <math>\mathrm{obj_{12}}f\!</math> that forms one part of it. One way to resolve the ambiguity is to formalize a distinction between a function and its ''graph'', letting <math>\mathrm{graph}(f) := \mathrm{obj_{12}}f.\!</math> Another tactic treats the whole notation <math>{}^{\backprime\backprime} f : X \to Y {}^{\prime\prime}\!</math> as sufficient denotation for the triple, letting <math>{}^{\backprime\backprime} f {}^{\prime\prime}\!</math> denote <math>\mathrm{graph}(f).\!</math> In categorical and computational contexts, at least initially, the type is regarded as an essential attribute or an integral part of the function itself. In other contexts it may be desirable to use a more abstract concept of function, treating a function as a mathematical object that appears in connection with many different types. Following the pattern of the functional case, let the notation <math>{}^{\backprime\backprime} L \subseteq X \times Y {}^{\prime\prime}\!</math> bring to mind a mathematical object that is specified by three pieces of data, the set <math>X,\!</math> the set <math>Y,\!</math> and a particular subset of their cartesian product <math>{X \times Y}.\!</math> As before we have two choices, either let <math>L = (X, Y, \mathrm{graph}(L))\!</math> or let <math>{}^{\backprime\backprime} L {}^{\prime\prime}\!</math> denote <math>\mathrm{graph}(L)\!</math> and choose another name for the triple. ==Definition== It is convenient to begin with the definition of a ''<math>k\!</math>-place relation'', where <math>k\!</math> is a positive integer. '''Definition.''' A ''<math>k\!</math>-place relation'' <math>L \subseteq X_1 \times \ldots \times X_k\!</math> over the nonempty sets <math>X_1, \ldots, X_k\!</math> is a <math>(k+1)\!</math>-tuple <math>(X_1, \ldots, X_k, L)\!</math> where <math>L\!</math> is a subset of the cartesian product <math>X_1 \times \ldots \times X_k.\!</math> ==Remarks== Though usage varies as usage will, there are several bits of optional language that are frequently useful in discussing relations. The sets <math>X_1, \ldots, X_k\!</math> are called the ''domains'' of the relation <math>L \subseteq X_1 \times \ldots \times X_k,\!</math> with <math>{X_j}\!</math> being the <math>j^\text{th}\!</math> domain. If all of the <math>{X_j}\!</math> are the same set <math>X\!</math> then <math>L \subseteq X_1 \times \ldots \times X_k\!</math> is more simply described as a <math>k\!</math>-place relation over <math>X.\!</math> The set <math>L\!</math> is called the ''graph'' of the relation <math>L \subseteq X_1 \times \ldots \times X_k,\!</math> on analogy with the graph of a function. If the sequence of sets <math>X_1, \ldots, X_k\!</math> is constant throughout a given discussion or is otherwise determinate in context, then the relation <math>L \subseteq X_1 \times \ldots \times X_k\!</math> is determined by its graph <math>L,\!</math> making it acceptable to denote the relation by referring to its graph. Other synonyms for the adjective ''<math>k\!</math>-place'' are ''<math>k\!</math>-adic'' and ''<math>k\!</math>-ary'', all of which leads to the integer <math>k\!</math> being called the ''dimension'', ''adicity'', or ''arity'' of the relation <math>L.\!</math> ==Local incidence properties== A ''local incidence property'' (LIP) of a relation <math>L\!</math> is a property that depends in turn on the properties of special subsets of <math>L\!</math> that are known as its ''local flags''. The local flags of a relation are defined in the following way: Let <math>L\!</math> be a <math>k\!</math>-place relation <math>L \subseteq X_1 \times \ldots \times X_k.\!</math> Select a relational domain <math>{X_j}\!</math> and one of its elements <math>x.\!</math> Then <math>L_{x \operatorname{at} j}\!</math> is a subset of <math>L\!</math> that is referred to as the ''flag'' of <math>L\!</math> with <math>x\!</math> at <math>{j},\!</math> or the ''<math>x \operatorname{at} j\!</math>-flag'' of <math>L,\!</math> an object that has the following definition: {| align="center" cellpadding="8" style="text-align:center" | <math>L_{x \operatorname{at} j} ~=~ \{ (x_1, \ldots, x_j, \ldots, x_k) \in L ~:~ x_j = x \}.\!</math> |} Any property <math>C\!</math> of the local flag <math>L_{x \operatorname{at} j} \subseteq L\!</math> is said to be a ''local incidence property'' of <math>L\!</math> with respect to the ''locus'' <math>x \operatorname{at} j.\!</math> A <math>k\!</math>-adic relation <math>L \subseteq X_1 \times \ldots \times X_k\!</math> is said to be ''<math>C\!</math>-regular'' at <math>j\!</math> if and only if every flag of <math>L\!</math> with <math>x\!</math> at <math>j\!</math> has the property <math>C,\!</math> where <math>x\!</math> is taken to vary over the ''theme'' of the fixed domain <math>X_j.\!</math> Expressed in symbols, <math>L\!</math> is ''<math>C\!</math>-regular'' at <math>j\!</math> if and only if <math>C(L_{x \operatorname{at} j})\!</math> is true for all <math>x\!</math> in <math>X_j.\!</math> ==Regional incidence properties== The definition of a local flag can be broadened from a point <math>x\!</math> in <math>{X_j}\!</math> to a subset <math>M\!</math> of <math>X_j,\!</math> arriving at the definition of a ''regional flag'' in the following way: Suppose that <math>L \subseteq X_1 \times \ldots \times X_k,\!</math> and choose a subset <math>M \subseteq X_j.\!</math> Then <math>L_{M \operatorname{at} j}\!</math> is a subset of <math>L\!</math> that is said to be the ''flag'' of <math>L\!</math> with <math>M\!</math> at <math>{j},\!</math> or the ''<math>M \operatorname{at} j\!</math>-flag'' of <math>L,\!</math> an object which has the following definition: {| align="center" cellpadding="8" style="text-align:center" | <math>L_{M \operatorname{at} j} ~=~ \{ (x_1, \ldots, x_j, \ldots, x_k) \in L ~:~ x_j \in M \}.\!</math> |} ==Numerical incidence properties== A ''numerical incidence property'' (NIP) of a relation is a local incidence property that depends on the cardinalities of its local flags. For example, <math>L\!</math> is said to be ''<math>c\!</math>-regular at <math>j\!</math>'' if and only if the cardinality of the local flag <math>L_{x \operatorname{at} j}\!</math> is <math>c\!</math> for all <math>x\!</math> in <math>{X_j},\!</math> or, to write it in symbols, if and only if <math>|L_{x \operatorname{at} j}| = c\!</math> for all <math>{x \in X_j}.\!</math> In a similar fashion, one can define the NIPs, ''<math>(<\!c)\!</math>-regular at <math>{j},\!</math>'' ''<math>(>\!c)\!</math>-regular at <math>{j},\!</math>'' and so on. For ease of reference, a few of these definitions are recorded here: {| align="center" cellpadding="8" style="text-align:center" | <math>\begin{matrix} L & \text{is} & c\textit{-regular} & \text{at}\ j & \text{if and only if} & |L_{x \operatorname{at} j}| & = & c & \text{for all}\ x \in X_j. \\[4pt] L & \text{is} & (<\!c)\textit{-regular} & \text{at}\ j & \text{if and only if} & |L_{x \operatorname{at} j}| & < & c & \text{for all}\ x \in X_j. \\[4pt] L & \text{is} & (>\!c)\textit{-regular} & \text{at}\ j & \text{if and only if} & |L_{x \operatorname{at} j}| & > & c & \text{for all}\ x \in X_j. \end{matrix}</math> |} ==Species of 2-adic relations== Returning to 2-adic relations, it is useful to describe some familiar classes of objects in terms of their local and numerical incidence properties. Let <math>L \subseteq S \times T\!</math> be an arbitrary 2-adic relation. The following properties of <math>L\!</math> can be defined: {| align="center" cellpadding="8" style="text-align:center" | <math>\begin{matrix} L & \text{is} & \textit{total} & \text{at}~ S & \text{if and only if} & L & \text{is} & (\ge 1)\text{-regular} & \text{at}~ S. \\[4pt] L & \text{is} & \textit{total} & \text{at}~ T & \text{if and only if} & L & \text{is} & (\ge 1)\text{-regular} & \text{at}~ T. \\[4pt] L & \text{is} & \textit{tubular} & \text{at}~ S & \text{if and only if} & L & \text{is} & (\le 1)\text{-regular} & \text{at}\ S. \\[4pt] L & \text{is} & \textit{tubular} & \text{at}~ T & \text{if and only if} & L & \text{is} & (\le 1)\text{-regular} & \text{at}~ T. \end{matrix}</math> |} If <math>L \subseteq S \times T\!</math> is tubular at <math>S\!</math> then <math>L\!</math> is called a ''partial function'' or a ''prefunction'' from <math>S\!</math> to <math>T.\!</math> This is sometimes indicated by giving <math>L\!</math> an alternate name, say, <math>{}^{\backprime\backprime} p {}^{\prime\prime},~\!</math> and writing <math>L = p : S \rightharpoonup T.\!</math> Just by way of formalizing the definition: {| align="center" cellpadding="8" style="text-align:center" | <math>\begin{matrix} L & = & p : S \rightharpoonup T & \text{if and only if} & L & \text{is} & \text{tubular} & \text{at}~ S. \end{matrix}</math> |} If <math>L\!</math> is a prefunction <math>p : S \rightharpoonup T\!</math> that happens to be total at <math>S,\!</math> then <math>L\!</math> is called a ''function'' from <math>S\!</math> to <math>T,\!</math> indicated by writing <math>L = f : S \to T.\!</math> To say that a relation <math>L \subseteq S \times T\!</math> is ''totally tubular'' at <math>S\!</math> is to say that it is <math>1\!</math>-regular at <math>S.\!</math> Thus, we may formalize the following definition: {| align="center" cellpadding="8" style="text-align:center" | <math>\begin{matrix} L & = & f : S \to T & \text{if and only if} & L & \text{is} & 1\text{-regular} & \text{at}~ S. \end{matrix}</math> |} In the case of a function <math>f : S \to T,\!</math> one has the following additional definitions: {| align="center" cellpadding="8" style="text-align:center" | <math>\begin{matrix} f & \text{is} & \textit{surjective} & \text{if and only if} & f & \text{is} & \text{total} & \text{at}~ T. \\[4pt] f & \text{is} & \textit{injective} & \text{if and only if} & f & \text{is} & \text{tubular} & \text{at}~ T. \\[4pt] f & \text{is} & \textit{bijective} & \text{if and only if} & f & \text{is} & 1\text{-regular} & \text{at}~ T. \end{matrix}</math> |} ==Variations== Because the concept of a relation has been developed quite literally from the beginnings of logic and mathematics, and because it has incorporated contributions from a diversity of thinkers from many different times and intellectual climes, there is a wide variety of terminology that the reader may run across in connection with the subject. One dimension of variation is reflected in the names that are given to <math>k\!</math>-place relations, for <math>k = 0, 1, 2, 3, \ldots,\!</math> with some writers using the Greek forms, ''medadic'', ''monadic'', ''dyadic'', ''triadic'', ''<math>k\!</math>-adic'', and other writers using the Latin forms, ''nullary'', ''unary'', ''binary'', ''ternary'', ''<math>k\!</math>-ary''. The number of relational domains may be referred to as the ''adicity'', ''arity'', or ''dimension'' of the relation. Accordingly, one finds a relation on a finite number of domains described as a ''polyadic'' relation or a ''finitary'' relation, but others count infinitary relations among the polyadic. If the number of domains is finite, say equal to <math>k,\!</math> then the relation may be described as a ''<math>k\!</math>-adic'' relation, a ''<math>k\!</math>-ary'' relation, or a ''<math>k\!</math>-dimensional'' relation, respectively. A more conceptual than nominal variation depends on whether one uses terms like ''predicate'', ''relation'', and even ''term'' to refer to the formal object proper or else to the allied syntactic items that are used to denote them. Compounded with this variation is still another, frequently associated with philosophical differences over the status in reality accorded formal objects. Among those who speak of numbers, functions, properties, relations, and sets as being real, that is to say, as having objective properties, there are divergences as to whether some things are more real than others, especially whether particulars or properties are equally real or else one is derivative in relationship to the other. Historically speaking, just about every combination of modalities has been used by one school of thought or another, but it suffices here merely to indicate how the options are generated. ==Examples== : ''See the articles on [[relation (mathematics)|relations]], [[relation composition]], [[relation reduction]], [[sign relation]]s, and [[triadic relation]]s for concrete examples of relations.'' Many relations of the greatest interest in mathematics are triadic relations, but this fact is somewhat disguised by the circumstance that many of them are referred to as ''binary operations'', and because the most familiar of these have very specific properties that are dictated by their axioms. This makes it practical to study these operations for quite some time by focusing on their dyadic aspects before being forced to consider their proper characters as triadic relations. ==References== * Peirce, C.S., &ldquo;Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic&rdquo;, ''Memoirs of the American Academy of Arts and Sciences'', 9, 317&ndash;378, 1870. Reprinted, ''Collected Papers'' CP 3.45&ndash;149, ''Chronological Edition'' CE 2, 359&ndash;429. * Ulam, S.M. and Bednarek, A.R., &ldquo;On the Theory of Relational Structures and Schemata for Parallel Computation&rdquo;, pp. 477&ndash;508 in A.R. Bednarek and Françoise Ulam (eds.), ''Analogies Between Analogies : The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators'', University of California Press, Berkeley, CA, 1990. ==Bibliography== * Barr, Michael, and Wells, Charles (1990), ''Category Theory for Computing Science'', Prentice Hall, Hemel Hempstead, UK. * Bourbaki, Nicolas (1994), ''Elements of the History of Mathematics'', John Meldrum (trans.), Springer-Verlag, Berlin, Germany. * Carnap, Rudolf (1958), ''Introduction to Symbolic Logic with Applications'', Dover Publications, New York, NY. * Chang, C.C., and Keisler, H.J. (1973), ''Model Theory'', North-Holland, Amsterdam, Netherlands. * van Dalen, Dirk (1980), ''Logic and Structure'', 2nd edition, Springer-Verlag, Berlin, Germany. * Devlin, Keith J. (1993), ''The Joy of Sets : Fundamentals of Contemporary Set Theory'', 2nd edition, Springer-Verlag, New York, NY. * Halmos, Paul Richard (1960), ''Naive Set Theory'', D. Van Nostrand Company, Princeton, NJ. * van Heijenoort, Jean (1967/1977), ''From Frege to Gödel : A Source Book in Mathematical Logic, 1879&ndash;1931'', Harvard University Press, Cambridge, MA, 1967. Reprinted with corrections, 1977. * Kelley, John L. (1955), ''General Topology'', Van Nostrand Reinhold, New York, NY. * Kneale, William; and Kneale, Martha (1962/1975), ''The Development of Logic'', Oxford University Press, Oxford, UK, 1962. Reprinted with corrections, 1975. * Lawvere, Francis William; and Rosebrugh, Robert (2003), ''Sets for Mathematics'', Cambridge University Press, Cambridge, UK. * Lawvere, Francis William; and Schanuel, Stephen H. (1997/2000), ''Conceptual Mathematics : A First Introduction to Categories'', Cambridge University Press, Cambridge, UK, 1997. Reprinted with corrections, 2000. * Manin, Yu. I. (1977), ''A Course in Mathematical Logic'', Neal Koblitz (trans.), Springer-Verlag, New York, NY. * Mathematical Society of Japan (1993), ''Encyclopedic Dictionary of Mathematics'', 2nd edition, 2 vols., Kiyosi Itô (ed.), MIT Press, Cambridge, MA. * Mili, A., Desharnais, J., Mili, F., with Frappier, M. (1994), ''Computer Program Construction'', Oxford University Press, New York, NY. (Introduction to Tarskian relation theory and relational programming.) * Mitchell, John C. (1996), ''Foundations for Programming Languages'', MIT Press, Cambridge, MA. * Peirce, Charles Sanders (1870), ``Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic", ''Memoirs of the American Academy of Arts and Sciences'' 9 (1870), 317&ndash;378. Reprinted (CP 3.45&ndash;149), (CE 2, 359&ndash;429). * Peirce, Charles Sanders (1931&ndash;1935, 1958), ''Collected Papers of Charles Sanders Peirce'', vols. 1&ndash;6, Charles Hartshorne and Paul Weiss (eds.), vols. 7&ndash;8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA. Cited as (CP volume.paragraph). * Peirce, Charles Sanders (1981&ndash;), ''Writings of Charles S. Peirce : A Chronological Edition'', Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN. Cited as (CE volume, page). * Poizat, Bruno (2000), ''A Course in Model Theory : An Introduction to Contemporary Mathematical Logic'', Moses Klein (trans.), Springer-Verlag, New York, NY. * Quine, Willard Van Orman (1940/1981), ''Mathematical Logic'', 1940. Revised edition, Harvard University Press, Cambridge, MA, 1951. New preface, 1981. * Royce, Josiah (1961), ''The Principles of Logic'', Philosophical Library, New York, NY. * Runes, Dagobert D. (ed., 1962), ''Dictionary of Philosophy'', Littlefield, Adams, and Company, Totowa, NJ. * Shoenfield, Joseph R. (1967), ''Mathematical Logic'', Addison-Wesley, Reading, MA. * Styazhkin, N.I. (1969), ''History of Mathematical Logic from Leibniz to Peano'', MIT Press, Cambridge, MA. * Suppes, Patrick (1957/1999), ''Introduction to Logic'', 1st published 1957. Reprinted, Dover Publications, New York, NY, 1999. * Suppes, Patrick (1960/1972), ''Axiomatic Set Theory'', 1st published 1960. Reprinted, Dover Publications, New York, NY, 1972. * Tarski, Alfred (1956/1983), ''Logic, Semantics, Metamathematics : Papers from 1923 to 1938'', J.H. Woodger (trans.), Oxford University Press, 1956. 2nd edition, J. Corcoran (ed.), Hackett Publishing, Indianapolis, IN, 1983. * Ulam, Stanislaw Marcin; and Bednarek, A.R. (1977), &ldquo;On the Theory of Relational Structures and Schemata for Parallel Computation&rdquo;, pp. 477&ndash;508 in A.R. Bednarek and Françoise Ulam (eds.), ''Analogies Between Analogies : The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators'', University of California Press, Berkeley, CA, 1990. * Ulam, Stanislaw Marcin (1990), ''Analogies Between Analogies : The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators'', A.R. Bednarek and Françoise Ulam (eds.), University of California Press, Berkeley, CA. * Ullman, Jeffrey D. (1980), ''Principles of Database Systems'', Computer Science Press, Rockville, MD. * Venetus, Paulus (1472/1984), ''Logica Parva : Translation of the 1472 Edition with Introduction and Notes'', Alan R. Perreiah (trans.), Philosophia Verlag, Munich, Germany. ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Relation_theory Relation Theory @ InterSciWiki] * [http://mywikibiz.com/Relation_theory Relation Theory @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Relation_theory Relation Theory @ Subject Wikis] * [http://en.wikiversity.org/wiki/Relation_theory Relation Theory @ Wikiversity] * [http://beta.wikiversity.org/wiki/Relation_theory Relation Theory @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Relation_theory Relation Theory], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Relation_theory Relation Theory], [http://mywikibiz.com/ MyWikiBiz] * [http://planetmath.org/RelationTheory Relation Theory], [http://planetmath.org/ PlanetMath] * [http://semanticweb.org/wiki/Relation_theory Relation Theory], [http://semanticweb.org/ Semantic Web] * [http://wikinfo.org/w/index.php/Relation_Theory Relation Theory], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Relation_theory Relation Theory], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Relation_theory Relation Theory], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Theory_of_relations&oldid=45042729 Relation Theory], [http://en.wikipedia.org/ Wikipedia] [[Category:Charles Sanders Peirce]] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Database Theory]] [[Category:Discrete Mathematics]] [[Category:Formal Sciences]] [[Category:Inquiry]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Relation Theory]] [[Category:Set Theory]] 1114be3acce3d2abe611d2335806d0fbc35be0f6 0 360 713 630 2015-11-04T20:20:11Z Jon Awbrey 3 add article wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. The concept of '''logical implication''' encompasses a specific logical [[function (mathematics)|function]], a specific logical [[relation (mathematics)|relation]], and the various symbols that are used to denote this function and this relation. In order to define the specific function, relation, and symbols in question it is first necessary to establish a few ideas about the connections among them. Close approximations to the concept of logical implication are expressed in ordinary language by means of linguistic forms like the following: {| align="center" cellspacing="10" width="90%" | <math>\begin{array}{l} p ~\text{implies}~ q. \\[6pt] \text{if}~ p ~\text{then}~ q. \end{array}</math> |} Here <math>p\!</math> and <math>q\!</math> are propositional variables that stand for any propositions in a given language. In a statement of the form <math>{}^{\backprime\backprime} \text{if}~ p ~\text{then}~ q {}^{\prime\prime},</math> the first term, <math>p,\!</math> is called the ''antecedent'' and the second term, <math>q,\!</math> is called the ''consequent'', while the statement as a whole is called either the ''conditional'' or the ''consequence''. Assuming that the conditional statement is true, then the truth of the antecedent is a ''sufficient condition'' for the truth of the consequent, while the truth of the consequent is a ''necessary condition'' for the truth of the antecedent. '''Note.''' Many writers draw a technical distinction between the form <math>{}^{\backprime\backprime} p ~\text{implies}~ q {}^{\prime\prime}</math> and the form <math>{}^{\backprime\backprime} \text{if}~ p ~\text{then}~ q {}^{\prime\prime}.</math> In this usage, writing <math>{}^{\backprime\backprime} p ~\text{implies}~ q {}^{\prime\prime}</math> asserts the existence of a certain relation between the logical value of <math>p\!</math> and the logical value of <math>q,\!</math> whereas writing <math>{}^{\backprime\backprime} \text{if}~ p ~\text{then}~ q {}^{\prime\prime}</math> merely forms a compound statement whose logical value is a function of the logical values of <math>p\!</math> and <math>q.\!</math> This will be discussed in detail below. ==Definition== The concept of logical implication is associated with an operation on two logical values, typically the values of two propositions, that produces a value of ''false'' just in case the first operand is true and the second operand is false. In the interpretation where <math>0 = \operatorname{false}</math> and <math>1 = \operatorname{true}</math>, the truth table associated with the statement <math>{}^{\backprime\backprime} p ~\text{implies}~ q {}^{\prime\prime},</math> symbolized as <math>{}^{\backprime\backprime} p \Rightarrow q {}^{\prime\prime},</math> appears below: <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:45%" |+ style="height:30px" | <math>\text{Logical Implication}\!</math> |- style="height:40px; background:#f0f0ff" | style="width:33%" | <math>p\!</math> | style="width:33%" | <math>q\!</math> | style="width:33%" | <math>p \Rightarrow q\!</math> |- | <math>0\!</math> || <math>0\!</math> || <math>1\!</math> |- | <math>0\!</math> || <math>1\!</math> || <math>1\!</math> |- | <math>1\!</math> || <math>0\!</math> || <math>0\!</math> |- | <math>1\!</math> || <math>1\!</math> || <math>1\!</math> |} <br> ==Discussion== The usage of the terms '''''logical implication''''' and '''''material conditional''''' varies from field to field and even across different contexts of discussion. One way to minimize the potential confusion is to begin with a focus on the various types of formal objects that are being discussed, of which there are only a few, taking up the variations in language as a secondary matter. The main formal object under discussion is a logical operation on two logical values, typically the values of two propositions, that produces a value of <math>\operatorname{false}</math> just in case the first operand is true and the second operand is false. By way of a temporary name, the logical operation in question may be written as <math>\operatorname{Cond}(p, q),</math> where <math>p\!</math> and <math>q\!</math> are logical values. The [[truth table]] associated with this operation appears below: <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:60%" |+ style="height:30px" | <math>\text{Conditional Operation} : \mathbb{B} \times \mathbb{B} \to \mathbb{B}\!</math> |- style="height:40px; background:#f0f0ff" | style="width:33%" | <math>p\!</math> | style="width:33%" | <math>q\!</math> | style="width:33%" | <math>\operatorname{Cond}(p, q)</math> |- | <math>\operatorname{F}</math> || <math>\operatorname{F}</math> || <math>\operatorname{T}</math> |- | <math>\operatorname{F}</math> || <math>\operatorname{T}</math> || <math>\operatorname{T}</math> |- | <math>\operatorname{T}</math> || <math>\operatorname{F}</math> || <math>\operatorname{F}</math> |- | <math>\operatorname{T}</math> || <math>\operatorname{T}</math> || <math>\operatorname{T}</math> |} <br> Some logicians draw a firm distinction between the conditional connective, the symbol <math>{}^{\backprime\backprime} \rightarrow {}^{\prime\prime},</math> and the implication relation, the object denoted by the symbol <math>{}^{\backprime\backprime} \Rightarrow {}^{\prime\prime}.</math> These logicians use the phrase ''if&ndash;then'' for the conditional connective and the term ''implies'' for the implication relation. Some explain the difference by saying that the conditional is the ''contemplated'' relation while the implication is the ''asserted'' relation. In most fields of mathematics, it is treated as a variation in the usage of the single sign <math>{}^{\backprime\backprime} \Rightarrow {}^{\prime\prime},</math> not requiring two separate signs. Not all of those who use the sign <math>{}^{\backprime\backprime} \rightarrow {}^{\prime\prime}</math> for the conditional connective regard it as a sign that denotes any kind of object, but treat it as a so-called ''syncategorematic sign'', that is, a sign with a purely syntactic function. For the sake of clarity and simplicity in the present introduction, it is convenient to use the two-sign notation, but allow the sign <math>{}^{\backprime\backprime} \rightarrow {}^{\prime\prime}</math> to denote the [[boolean function]] that is associated with the [[truth table]] of the material conditional. These considerations result in the following scheme of notation. {| align="center" cellspacing="10" width="90%" | <math>\begin{matrix} p \rightarrow q & \quad & \quad & p \Rightarrow q \\ \text{if}~ p ~\text{then}~ q & \quad & \quad & p ~\text{implies}~ q \end{matrix}</math> |} Let <math>\mathbb{B} = \{ \operatorname{F}, \operatorname{T} \}</math> be the ''[[boolean domain]]'' consisting of two logical values. The truth table shows the ordered triples of a [[triadic relation]] <math>L \subseteq \mathbb{B} \times \mathbb{B} \times \mathbb{B}\!</math> that is defined as follows: {| align="center" cellspacing="10" width="90%" | <math>L = \{ (p, q, r) \in \mathbb{B} \times \mathbb{B} \times \mathbb{B} : \operatorname{Cond}(p, q) = r \}.</math> |} Regarded as a set, this triadic relation is the same thing as the binary operation: {| align="center" cellspacing="10" width="90%" | <math>\operatorname{Cond} : \mathbb{B} \times \mathbb{B} \to \mathbb{B}.</math> |} The relationship between <math>\operatorname{Cond}</math> and <math>L\!</math> exemplifies the standard association that exists between any binary operation and its corresponding triadic relation. The conditional sign <math>{}^{\backprime\backprime} \rightarrow {}^{\prime\prime}</math> denotes the same formal object as the function name <math>{}^{\backprime\backprime} \operatorname{Cond} {}^{\prime\prime},</math> the only difference being that the first is written infix while the second is written prefix. Thus we have the following equation: {| align="center" cellspacing="10" width="90%" | <math>(p \rightarrow q) = \operatorname{Cond}(p, q).</math> |} Consider once again the triadic relation <math>L \subseteq \mathbb{B} \times \mathbb{B} \times \mathbb{B}\!</math> that is defined in the following equivalent fashion: {| align="center" cellspacing="10" width="90%" | <math>L = \{ (p, q, \operatorname{Cond}(p, q) ) : (p, q) \in \mathbb{B} \times \mathbb{B} \}.</math> |} Associated with the triadic relation <math>L\!</math> is a binary relation <math>L_{\underline{~} \, \underline{~} \, \operatorname{T}} \subseteq \mathbb{B} \times \mathbb{B}</math> that is called the ''fiber'' of <math>L\!</math> with <math>\operatorname{T}</math> in the third place. This object is defined as follows: {| align="center" cellspacing="10" width="90%" | <math>L_{\underline{~} \, \underline{~} \, \operatorname{T}} = \{ (p, q) \in \mathbb{B} \times \mathbb{B} : (p, q, \operatorname{T}) \in L \}.</math> |} The same object is achieved in the following way. Begin with the binary operation: {| align="center" cellspacing="10" width="90%" | <math>\operatorname{Cond} : \mathbb{B} \times \mathbb{B} \to \mathbb{B}.</math> |} Form the binary relation that is called the ''fiber'' of <math>\operatorname{Cond}</math> at <math>\operatorname{T},</math> notated as follows: {| align="center" cellspacing="10" width="90%" | <math>\operatorname{Cond}^{-1}(\operatorname{T}) \subseteq \mathbb{B} \times \mathbb{B}.</math> |} This object is defined as follows: {| align="center" cellspacing="10" width="90%" | <math>\operatorname{Cond}^{-1}(\operatorname{T}) = \{ (p, q) \in \mathbb{B} \times \mathbb{B} : \operatorname{Cond}(p, q) = \operatorname{T} \}.</math> |} The implication sign <math>{}^{\backprime\backprime} \rightarrow {}^{\prime\prime}</math> denotes the same formal object as the relation names <math>{}^{\backprime\backprime} L_{\underline{~} \, \underline{~} \, \operatorname{T}} {}^{\prime\prime}</math> and <math>{}^{\backprime\backprime} \operatorname{Cond}^{-1}(T) {}^{\prime\prime},</math> the only differences being purely syntactic. Thus we have the following logical equivalence: {| align="center" cellspacing="10" width="90%" | <math>(p \Rightarrow q) \iff (p, q) \in L_{\underline{~} \, \underline{~} \, \operatorname{T}} \iff (p, q) \in \operatorname{Cond}^{-1}(T).</math> |} This completes the derivation of the mathematical objects that are denoted by the signs <math>{}^{\backprime\backprime} \rightarrow {}^{\prime\prime}</math> and <math>{}^{\backprime\backprime} \Rightarrow {}^{\prime\prime}</math> in this discussion. It needs to be remembered, though, that not all writers observe this distinction in every context. Especially in mathematics, where the single arrow sign <math>{}^{\backprime\backprime} \rightarrow {}^{\prime\prime}</math> is reserved for function notation, it is common to see the double arrow sign <math>{}^{\backprime\backprime} \Rightarrow {}^{\prime\prime}</math> being used for both concepts. ==References== * [[Frank Markham Brown|Brown, Frank Markham]] (2003), ''Boolean Reasoning: The Logic of Boolean Equations'', 1st edition, Kluwer Academic Publishers, Norwell, MA. 2nd edition, Dover Publications, Mineola, NY, 2003. * Edgington, Dorothy (2001), "Conditionals", in Lou Goble (ed.), ''The Blackwell Guide to Philosophical Logic'', Blackwell. * Edgington, Dorothy (2006), "Conditionals", in Edward N. Zalta (ed.), ''The Stanford Encyclopedia of Philosophy'', [http://plato.stanford.edu/entries/conditionals/ Eprint]. * [[W.V. Quine|Quine, W.V.]] (1982), ''Methods of Logic'', (1st ed. 1950), (2nd ed. 1959), (3rd ed. 1972), 4th edition, Harvard University Press, Cambridge, MA. ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Logical_implication Logical Implication @ InterSciWiki] * [http://mywikibiz.com/Logical_implication Logical Implication @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Logical_implication Logical Implication @ Subject Wikis] * [http://en.wikiversity.org/wiki/Logical_implication Logical Implication @ Wikiversity] * [http://beta.wikiversity.org/wiki/Logical_implication Logical Implication @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Logical_implication Logical Implication], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Logical_implication Logical Implication], [http://mywikibiz.com/ MyWikiBiz] * [http://wikinfo.org/w/index.php/Logical_implication Logical Implication], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Logical_implication Logical Implication], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Logical_implication Logical Implication], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Logical_implication&oldid=77109738 Logical Implication], [http://en.wikipedia.org/ Wikipedia] [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Charles Sanders Peirce]] [[Category:Computer Science]] [[Category:Formal Languages]] [[Category:Formal Sciences]] [[Category:Formal Systems]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Philosophy]] [[Category:Semiotics]] 25ccf2c0680838f0672586d73fda505a458c31dc 0 371 714 2015-11-04T23:00:10Z Jon Awbrey 3 add article wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. The '''logical NAND''' is an operation on two logical values, typically the values of two propositions, that produces a value of ''false'' if and only if both of its operands are true. In other words, it produces a value of ''true'' if and only if at least one of its operands is false. The [[truth table]] of <math>p ~\operatorname{NAND}~ q,</math> also written <math>p \stackrel{\circ}{\curlywedge} q\!</math> or <math>p \barwedge q,\!</math> appears below: <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:45%" |+ style="height:30px" | <math>\text{Logical NAND}\!</math> |- style="height:40px; background:#f0f0ff" | style="width:33%" | <math>p\!</math> | style="width:33%" | <math>q\!</math> | style="width:33%" | <math>p \stackrel{\circ}{\curlywedge} q\!</math> |- | <math>\operatorname{F}</math> || <math>\operatorname{F}</math> || <math>\operatorname{T}</math> |- | <math>\operatorname{F}</math> || <math>\operatorname{T}</math> || <math>\operatorname{T}</math> |- | <math>\operatorname{T}</math> || <math>\operatorname{F}</math> || <math>\operatorname{T}</math> |- | <math>\operatorname{T}</math> || <math>\operatorname{T}</math> || <math>\operatorname{F}</math> |} <br> ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Logical_NAND Logical NAND @ InterSciWiki] * [http://mywikibiz.com/Logical_NAND Logical NAND @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Logical_NAND Logical NAND @ Subject Wikis] * [http://en.wikiversity.org/wiki/Logical_NAND Logical NAND @ Wikiversity] * [http://beta.wikiversity.org/wiki/Logical_NAND Logical NAND @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Logical_NAND Logical NAND], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Logical_NAND Logical NAND], [http://mywikibiz.com/ MyWikiBiz] * [http://wikinfo.org/w/index.php/Logical_NAND Logical NAND], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Logical_NAND Logical NAND], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Logical_NAND Logical NAND], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Logical_NAND&oldid=75155132 Logical NAND], [http://en.wikipedia.org/ Wikipedia] [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Charles Sanders Peirce]] [[Category:Computer Science]] [[Category:Formal Languages]] [[Category:Formal Sciences]] [[Category:Formal Systems]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Philosophy]] [[Category:Semiotics]] e59f785170398d8f532031a6b279bc2f5b6cacb2 0 372 715 2015-11-05T03:54:03Z Jon Awbrey 3 add article wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. The '''logical NNOR''' (&ldquo;Neither Nor&rdquo;) is an operation on two logical values, typically the values of two propositions, that produces a value of ''true'' if and only if both of its operands are false. In other words, it produces a value of ''false'' if and only if at least one of its operands is true. The [[truth table]] of <math>p ~\operatorname{NNOR}~ q,</math> also written <math>p \curlywedge q,\!</math> appears below: <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:45%" |+ style="height:30px" | <math>\text{Logical NNOR}\!</math> |- style="height:40px; background:#f0f0ff" | style="width:33%" | <math>p\!</math> | style="width:33%" | <math>q\!</math> | style="width:33%" | <math>p \curlywedge q\!</math> |- | <math>\operatorname{F}</math> || <math>\operatorname{F}</math> || <math>\operatorname{T}</math> |- | <math>\operatorname{F}</math> || <math>\operatorname{T}</math> || <math>\operatorname{F}</math> |- | <math>\operatorname{T}</math> || <math>\operatorname{F}</math> || <math>\operatorname{F}</math> |- | <math>\operatorname{T}</math> || <math>\operatorname{T}</math> || <math>\operatorname{F}</math> |} <br> ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Logical_NNOR Logical NNOR @ InterSciWiki] * [http://mywikibiz.com/Logical_NNOR Logical NNOR @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Logical_NNOR Logical NNOR @ Subject Wikis] * [http://en.wikiversity.org/wiki/Logical_NNOR Logical NNOR @ Wikiversity] * [http://beta.wikiversity.org/wiki/Logical_NNOR Logical NNOR @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Logical_NNOR Logical NNOR], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Logical_NNOR Logical NNOR], [http://mywikibiz.com/ MyWikiBiz] * [http://wikinfo.org/w/index.php/Logical_NNOR Logical NNOR], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Logical_NNOR Logical NNOR], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Logical_NNOR Logical NNOR], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Logical_NNOR&oldid=75155433 Logical NNOR], [http://en.wikipedia.org/ Wikipedia] [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Charles Sanders Peirce]] [[Category:Computer Science]] [[Category:Formal Languages]] [[Category:Formal Sciences]] [[Category:Formal Systems]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Philosophy]] [[Category:Semiotics]] 0a7211430f021b820e7faba0cfa17af2c9f8e555 0 373 716 2015-11-05T14:18:22Z Jon Awbrey 3 add article wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. '''Logical negation''' is an operation on one logical value, typically the value of a proposition, that produces a value of ''true'' when its operand is false and a value of ''false'' when its operand is true. The [[truth table]] of <math>\operatorname{NOT}~ p,</math> also written <math>\lnot p,\!</math> appears below: <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:45%" |+ style="height:30px" | <math>\text{Logical Negation}\!</math> |- style="height:40px; background:#f0f0ff" | style="width:50%" | <math>p\!</math> | style="width:50%" | <math>\lnot p\!</math> |- | <math>\operatorname{F}</math> || <math>\operatorname{T}</math> |- | <math>\operatorname{T}</math> || <math>\operatorname{F}</math> |} <br> The negation of a proposition <math>p\!</math> may be found notated in various ways in various contexts of application, often merely for typographical convenience. Among these variants are the following: <br> {| align="center" border="1" cellpadding="8" cellspacing="0" width="45%" |+ style="height:30px" | <math>\text{Variant Notations}\!</math> |- style="height:40px; background:#f0f0ff" | width="50%" align="center" | <math>\text{Notation}\!</math> | width="50%" | <math>\text{Vocalization}\!</math> |- | align="center" | <math>\bar{p}\!</math> | <math>p\!</math> bar |- | align="center" | <math>\tilde{p}\!</math> | <math>p\!</math> tilde |- | align="center" | <math>p'\!</math> | <math>p\!</math> prime<br> <math>p\!</math> complement |- | align="center" | <math>!p\!</math> | bang <math>p\!</math> |} <br> ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Logical_negation Logical Negation @ InterSciWiki] * [http://mywikibiz.com/Logical_negation Logical Negation @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Logical_negation Logical Negation @ Subject Wikis] * [http://en.wikiversity.org/wiki/Logical_negation Logical Negation @ Wikiversity] * [http://beta.wikiversity.org/wiki/Logical_negation Logical Negation @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Logical_negation Logical Negation], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Logical_negation Logical Negation], [http://mywikibiz.com/ MyWikiBiz] * [http://wikinfo.org/w/index.php/Logical_negation Logical Negation], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Logical_negation Logical Negation], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Logical_negation Logical Negation], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Logical_negation&oldid=77111608 Logical Negation], [http://en.wikipedia.org/ Wikipedia] [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Charles Sanders Peirce]] [[Category:Computer Science]] [[Category:Formal Languages]] [[Category:Formal Sciences]] [[Category:Formal Systems]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Philosophy]] [[Category:Semiotics]] ad59af80c9fa69d16788c9ca83d31287214208b5 0 321 718 542 2015-11-05T19:56:28Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''boolean domain''' <math>\mathbb{B}</math> is a generic 2-element [[set]], say, <math>\mathbb{B} = \{ 0, 1 \},</math> whose elements are interpreted as [[logical value]]s, typically, <math>0 = \operatorname{false}</math> and <math>1 = \operatorname{true}.</math> A '''boolean variable''' <math>x\!</math> is a [[variable]] that takes its value from a boolean domain, as <math>x \in \mathbb{B}.</math> ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Boolean_domain Boolean Domain @ InterSciWiki] * [http://mywikibiz.com/Boolean_domain Boolean Domain @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Boolean_domain Boolean Domain @ Subject Wikis] * [http://en.wikiversity.org/wiki/Boolean_domain Boolean Domain @ Wikiversity] * [http://beta.wikiversity.org/wiki/Boolean_domain Boolean Domain @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Boolean_domain Boolean Domain], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Boolean_domain Boolean Domain], [http://mywikibiz.com/ MyWikiBiz] * [http://planetmath.org/BooleanDomain Boolean Domain], [http://planetmath.org/ PlanetMath] * [http://wikinfo.org/w/index.php/Boolean_domain Boolean Domain], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Boolean_domain Boolean Domain], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Boolean_domain Boolean Domain], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Boolean_domain&oldid=71168300 Boolean Domain], [http://en.wikipedia.org/ Wikipedia] [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Discrete Mathematics]] [[Category:Logic]] [[Category:Mathematics]] 45363ebbdaff90a8fc49894b2507d9aaa215b809 0 322 719 543 2015-11-05T20:25:45Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''finitary boolean function''' is a [[function (mathematics)|function]] of the form <math>f : \mathbb{B}^k \to \mathbb{B},</math> where <math>\mathbb{B} = \{ 0, 1 \}</math> is a [[boolean domain]] and where <math>k\!</math> is a nonnegative integer. In the case where <math>k = 0,\!</math> the function is simply a constant element of <math>\mathbb{B}.</math> There are <math>2^{2^k}</math> such functions. These play a basic role in questions of [[complexity theory]] as well as the design of circuits and chips for digital computers. ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Boolean_function Boolean Function @ InterSciWiki] * [http://mywikibiz.com/Boolean_function Boolean Function @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Boolean_function Boolean Function @ Subject Wikis] * [http://en.wikiversity.org/wiki/Boolean_function Boolean Function @ Wikiversity] * [http://beta.wikiversity.org/wiki/Boolean_function Boolean Function @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Boolean_function Boolean Function], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Boolean_function Boolean Function], [http://mywikibiz.com/ MyWikiBiz] * [http://planetmath.org/BooleanValuedFunction Boolean Function], [http://planetmath.org/ PlanetMath] * [http://wikinfo.org/w/index.php/Boolean_function Boolean Function], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Boolean_function Boolean Function], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Boolean_function Boolean Function], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Boolean_function&oldid=60886833 Boolean Function], [http://en.wikipedia.org/ Wikipedia] [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Discrete Mathematics]] [[Category:Logic]] [[Category:Mathematics]] 97996a6cc4a23b6122b2f88712e64adef068edb5 0 323 720 544 2015-11-05T21:16:46Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''boolean-valued function''' is a function of the type <math>f : X \to \mathbb{B},</math> where <math>X\!</math> is an arbitrary set and where <math>\mathbb{B}</math> is a [[boolean domain]]. In the formal sciences &mdash; mathematics, mathematical logic, statistics &mdash; and their applied disciplines, a boolean-valued function may also be referred to as a characteristic function, indicator function, predicate, or proposition. In all of these uses it is understood that the various terms refer to a mathematical object and not the corresponding sign or syntactic expression. In formal semantic theories of truth, a '''truth predicate''' is a predicate on the sentences of a formal language, interpreted for logic, that formalizes the intuitive concept that is normally expressed by saying that a sentence is true. A truth predicate may have additional domains beyond the formal language domain, if that is what is required to determine a final truth value. ==Examples== A '''binary sequence''' is a boolean-valued function <math>f : \mathbb{N}^+ \to \mathbb{B}</math>, where <math>\mathbb{N}^+ = \{ 1, 2, 3, \ldots \},</math>. In other words, <math>f\!</math> is an infinite sequence of 0's and 1's. A '''binary sequence''' of '''length''' <math>k\!</math> is a boolean-valued function <math>f : [k] \to \mathbb{B}</math>, where <math>[k] = \{ 1, 2, \ldots k \}.</math> ==References== * Brown, Frank Markham (2003), ''Boolean Reasoning : The Logic of Boolean Equations'', 1st edition, Kluwer Academic Publishers, Norwell, MA. 2nd edition, Dover Publications, Mineola, NY, 2003. * Kohavi, Zvi (1978), ''Switching and Finite Automata Theory'', 1st edition, McGraw–Hill, 1970. 2nd edition, McGraw–Hill, 1978. * Korfhage, Robert R. (1974), ''Discrete Computational Structures'', Academic Press, New York, NY. * Mathematical Society of Japan, ''Encyclopedic Dictionary of Mathematics'', 2nd edition, 2 vols., Kiyosi Itô (ed.), MIT Press, Cambridge, MA, 1993. Cited as EDM. * Minsky, Marvin L., and Papert, Seymour, A. (1988), ''Perceptrons, An Introduction to Computational Geometry'', MIT Press, Cambridge, MA, 1969. Revised, 1972. Expanded edition, 1988. ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Boolean-valued_function Boolean-Valued Function @ InterSciWiki] * [http://mywikibiz.com/Boolean-valued_function Boolean-Valued Function @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Boolean-valued_function Boolean-Valued Function @ Subject Wikis] * [http://en.wikiversity.org/wiki/Boolean-valued_function Boolean-Valued Function @ Wikiversity] * [http://beta.wikiversity.org/wiki/Boolean-valued_function Boolean-Valued Function @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Boolean-valued_function Boolean-Valued Function], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Boolean-valued_function Boolean-Valued Function], [http://mywikibiz.com/ MyWikiBiz] * [http://planetmath.org/BooleanValuedFunction Boolean-Valued Function], [http://planetmath.org/ PlanetMath] * [http://wikinfo.org/w/index.php/Boolean-valued_function Boolean-Valued Function], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Boolean-valued_function Boolean-Valued Function], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Boolean-valued_function Boolean-Valued Function], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Boolean-valued_function&oldid=67166584 Boolean-Valued Function], [http://en.wikipedia.org/ Wikipedia] [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Discrete Mathematics]] [[Category:Logic]] [[Category:Mathematics]] 6fb16a1175e831d06873ecaeeea15be5050b8330 0 324 721 545 2015-11-05T22:34:13Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. '''Differential logic''' is the component of logic whose object is the successful description of variation &mdash; for example, the aspects of change, difference, distribution, and diversity &mdash; in [[universes of discourse]] that are subject to logical description. In formal logic, differential logic treats the principles that govern the use of a ''differential logical calculus'', that is, a formal system with the expressive capacity to describe change and diversity in logical universes of discourse. A simple example of a differential logical calculus is furnished by a [[differential propositional calculus]]. This augments ordinary [[propositional calculus]] in the same way that the differential calculus of Leibniz and Newton augments the analytic geometry of Descartes. ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Differential_logic Differential Logic @ InterSciWiki] * [http://mywikibiz.com/Differential_logic Differential Logic @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Differential_logic Differential Logic @ Subject Wikis] * [http://en.wikiversity.org/wiki/Differential_logic Differential Logic @ Wikiversity] * [http://beta.wikiversity.org/wiki/Differential_logic Differential Logic @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Differential_logic Differential Logic], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Differential_logic Differential Logic], [http://mywikibiz.com/ MyWikiBiz] * [http://planetmath.org/DifferentialLogic Differential Logic], [http://planetmath.org/ PlanetMath] * [http://semanticweb.org/wiki/Differential_logic Differential Logic], [http://semanticweb.org/ Semantic Web] * [http://wikinfo.org/w/index.php/Differential_logic Differential Logic], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Differential_logic Differential Logic], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Differential_logic Differential Logic], [http://beta.wikiversity.org/ Wikiversity Beta] [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Artificial Intelligence]] [[Category:Charles Sanders Peirce]] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Cybernetics]] [[Category:Differential Logic]] [[Category:Dynamical Systems]] [[Category:Equational Reasoning]] [[Category:Formal Languages]] [[Category:Formal Sciences]] [[Category:Formal Systems]] [[Category:Graph Theory]] [[Category:History of Logic]] [[Category:History of Mathematics]] [[Category:Knowledge Representation]] [[Category:Logic]] [[Category:Logical Graphs]] [[Category:Mathematics]] [[Category:Mathematical Systems Theory]] [[Category:Philosophy]] [[Category:Semiotics]] [[Category:Systems Science]] [[Category:Visualization]] 259bf510559aef546681789645f320c12a4f09f1 0 187 722 666 2015-11-06T14:38:32Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''logical graph''' is a graph-theoretic structure in one of the systems of graphical syntax that Charles Sanders Peirce developed for logic. In his papers on ''qualitative logic'', ''entitative graphs'', and ''existential graphs'', Peirce developed several versions of a graphical formalism, or a graph-theoretic formal language, designed to be interpreted for logic. In the century since Peirce initiated this line of development, a variety of formal systems have branched out from what is abstractly the same formal base of graph-theoretic structures. This article examines the common basis of these formal systems from a bird's eye view, focusing on those aspects of form that are shared by the entire family of algebras, calculi, or languages, however they happen to be viewed in a given application. ==Abstract point of view== {| width="100%" cellpadding="2" cellspacing="0" | width="60%" | &nbsp; | width="40%" | ''Wollust ward dem Wurm gegeben &hellip;'' |- | &nbsp; | align="right" | &mdash; Friedrich Schiller, ''An die Freude'' |} The bird's eye view in question is more formally known as the perspective of formal equivalence, from which remove one cannot see many distinctions that appear momentous from lower levels of abstraction. In particular, expressions of different formalisms whose syntactic structures are isomorphic from the standpoint of algebra or topology are not recognized as being different from each other in any significant sense. Though we may note in passing such historical details as the circumstance that Charles Sanders Peirce used a ''streamer-cross symbol'' where George Spencer Brown used a ''carpenter's square marker'', the theme of principal interest at the abstract level of form is neutral with regard to variations of that order. ==In lieu of a beginning== Consider the formal equations indicated in Figures&nbsp;1 and 2. {| align="center" border="0" cellpadding="10" cellspacing="0" | [[Image:Logical_Graph_Figure_1_Visible_Frame.jpg|500px]] || (1) |- | [[Image:Logical_Graph_Figure_2_Visible_Frame.jpg|500px]] || (2) |} For the time being these two forms of transformation may be referred to as ''axioms'' or ''initial equations''. ==Duality : logical and topological== There are two types of duality that have to be kept separately mind in the use of logical graphs &mdash; logical duality and topological duality. There is a standard way that graphs of the order that Peirce considered, those embedded in a continuous manifold like that commonly represented by a plane sheet of paper &mdash; with or without the paper bridges that Peirce used to augment its topological genus &mdash; can be represented in linear text as what are called ''parse strings'' or ''traversal strings'' and parsed into ''pointer structures'' in computer memory. A blank sheet of paper can be represented in linear text as a blank space, but that way of doing it tends to be confusing unless the logical expression under consideration is set off in a separate display. For example, consider the axiom or initial equation that is shown below: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_3_Visible_Frame.jpg|500px]] || (3) |} This can be written inline as <math>{}^{\backprime\backprime} \texttt{(~(~)~)} = \quad {}^{\prime\prime}\!</math> or set off in a text display as follows: {| align="center" cellpadding="10" | width="33%" | <math>\texttt{(~(~)~)}\!</math> | width="34%" | <math>=\!</math> | width="33%" | &nbsp; |} When we turn to representing the corresponding expressions in computer memory, where they can be manipulated with utmost facility, we begin by transforming the planar graphs into their topological duals. The planar regions of the original graph correspond to nodes (or points) of the dual graph, and the boundaries between planar regions in the original graph correspond to edges (or lines) between the nodes of the dual graph. For example, overlaying the corresponding dual graphs on the plane-embedded graphs shown above, we get the following composite picture: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_4_Visible_Frame.jpg|500px]] || (4) |} Though it's not really there in the most abstract topology of the matter, for all sorts of pragmatic reasons we find ourselves compelled to single out the outermost region of the plane in a distinctive way and to mark it as the ''root node'' of the corresponding dual graph. In the present style of Figure the root nodes are marked by horizontal strike-throughs. Extracting the dual graphs from their composite matrices, we get this picture: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_5_Visible_Frame.jpg|500px]] || (5) |} It is easy to see the relationship between the parenthetical expressions of Peirce's logical graphs, that somewhat clippedly picture the ordered containments of their formal contents, and the associated dual graphs, that constitute the species of rooted trees here to be described. In the case of our last example, a moment's contemplation of the following picture will lead us to see that we can get the corresponding parenthesis string by starting at the root of the tree, climbing up the left side of the tree until we reach the top, then climbing back down the right side of the tree until we return to the root, all the while reading off the symbols, in this case either <math>{}^{\backprime\backprime} \texttt{(} {}^{\prime\prime}\!</math> or <math>{}^{\backprime\backprime} \texttt{)} {}^{\prime\prime},\!</math> that we happen to encounter in our travels. {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_6_Visible_Frame.jpg|500px]] || (6) |} This ritual is called ''traversing'' the tree, and the string read off is called the ''traversal string'' of the tree. The reverse ritual, that passes from the string to the tree, is called ''parsing'' the string, and the tree constructed is called the ''parse graph'' of the string. The speakers thereof tend to be a bit loose in this language, often using ''parse string'' to mean the string that gets parsed into the associated graph. We have treated in some detail various forms of the initial equation or logical axiom that is formulated in string form as <math>{}^{\backprime\backprime} \texttt{(~(~)~)} = \quad {}^{\prime\prime}.~\!</math> For the sake of comparison, let's record the plane-embedded and topological dual forms of the axiom that is formulated in string form as <math>{}^{\backprime\backprime} \texttt{(~)(~)} = \texttt{(~)} {}^{\prime\prime}.\!</math> First the plane-embedded maps: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_7_Visible_Frame.jpg|500px]] || (7) |} Next the plane-embedded maps and their dual trees superimposed: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_8_Visible_Frame.jpg|500px]] || (8) |} Finally the dual trees by themselves: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_9_Visible_Frame.jpg|500px]] || (9) |} And here are the parse trees with their traversal strings indicated: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_10_Visible_Frame.jpg|500px]] || (10) |} We have at this point enough material to begin thinking about the forms of analogy, iconicity, metaphor, morphism, whatever you want to call them, that are pertinent to the use of logical graphs in their various logical interpretations, for instance, those that Peirce described as ''entitative graphs'' and ''existential graphs''. ==Computational representation== The parse graphs that we've been looking at so far bring us one step closer to the pointer graphs that it takes to make these maps and trees live in computer memory, but they are still a couple of steps too abstract to detail the concrete species of dynamic data structures that we need. The time has come to flesh out the skeletons that we've drawn up to this point. Nodes in a graph represent ''records'' in computer memory. A record is a collection of data that can be conceived to reside at a specific ''address''. The address of a record is analogous to a demonstrative pronoun, on which account programmers commonly describe it as a ''pointer'' and semioticians recognize it as a type of sign called an ''index''. At the next level of concreteness, a pointer-record structure is represented as follows: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_11_Visible_Frame.jpg|500px]] || (11) |} This portrays the pointer <math>\mathit{index}_0\!</math> as the address of a record that contains the following data: {| align="center" cellpadding="10" | <math>\mathit{datum}_1, \mathit{datum}_2, \mathit{datum}_3, \ldots,\!</math> and so on. |} What makes it possible to represent graph-theoretical structures as data structures in computer memory is the fact that an address is just another datum, and so we may have a state of affairs like the following: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_12_Visible_Frame.jpg|500px]] || (12) |} Returning to the abstract level, it takes three nodes to represent the three data records illustrated above: one root node connected to a couple of adjacent nodes. The items of data that do not point any further up the tree are then treated as labels on the record-nodes where they reside, as shown below: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_13_Visible_Frame.jpg|500px]] || (13) |} Notice that drawing the arrows is optional with rooted trees like these, since singling out a unique node as the root induces a unique orientation on all the edges of the tree, with ''up'' being the same direction as ''away from the root''. ==Quick tour of the neighborhood== This much preparation allows us to take up the founding axioms or initial equations that determine the entire system of logical graphs. ===Primary arithmetic as semiotic system=== Though it may not seem too exciting, logically speaking, there are many reasons to make oneself at home with the system of forms that is represented indifferently, topologically speaking, by rooted trees, balanced strings of parentheses, or finite sets of non-intersecting simple closed curves in the plane. :* One reason is that it gives us a respectable example of a sign domain on which to cut our semiotic teeth, non-trivial in the sense that it contains a countable infinity of signs. :* Another reason is that it allows us to study a simple form of computation that is recognizable as a species of ''[[semiosis]]'', or sign-transforming process. This space of forms, along with the two axioms that induce its partition into exactly two equivalence classes, is what George Spencer Brown called the ''primary arithmetic''. The axioms of the primary arithmetic are shown below, as they appear in both graph and string forms, along with pairs of names that come in handy for referring to the two opposing directions of applying the axioms. {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_14_Banner.jpg|500px]] || (14) |- | [[Image:Logical_Graph_Figure_15_Banner.jpg|500px]] || (15) |} Let <math>S\!</math> be the set of rooted trees and let <math>S_0\!</math> be the 2-element subset of <math>S\!</math> that consists of a rooted node and a rooted edge. {| align="center" cellpadding="10" style="text-align:center" | <math>S\!</math> | <math>=\!</math> | <math>\{ \text{rooted trees} \}\!</math> |- | <math>S_0\!</math> | <math>=\!</math> | <math>\{ \ominus, \vert \} = \{\!</math>[[Image:Rooted Node.jpg|16px]], [[Image:Rooted Edge.jpg|12px]]<math>\}\!</math> |} Simple intuition, or a simple inductive proof, assures us that any rooted tree can be reduced by way of the arithmetic initials either to a root node [[Image:Rooted Node.jpg|16px]] or else to a rooted edge [[Image:Rooted Edge.jpg|12px]]&nbsp;. For example, consider the reduction that proceeds as follows: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_16.jpg|500px]] || (16) |} Regarded as a semiotic process, this amounts to a sequence of signs, every one after the first serving as the ''interpretant'' of its predecessor, ending in a final sign that may be taken as the canonical sign for their common object, in the upshot being the result of the computation process. Simple as it is, this exhibits the main features of any computation, namely, a semiotic process that proceeds from an obscure sign to a clear sign of the same object, being in its aim and effect an action on behalf of clarification. ===Primary algebra as pattern calculus=== Experience teaches that complex objects are best approached in a gradual, laminar, modular fashion, one step, one layer, one piece at a time, and it's just as much the case when the complexity of the object is irreducible, that is, when the articulations of the representation are necessarily at joints that are cloven disjointedly from nature, with some assembly required in the synthetic integrity of the intuition. That's one good reason for spending so much time on the first half of [[zeroth order logic]], represented here by the primary arithmetic, a level of formal structure that C.S. Peirce verged on intuiting at numerous points and times in his work on logical graphs, and that Spencer Brown named and brought more completely to life. There is one other reason for lingering a while longer in these primitive forests, and this is that an acquaintance with &ldquo;bare trees&rdquo;, those as yet unadorned with literal or numerical labels, will provide a firm basis for understanding what's really at issue in such problems as the &ldquo;ontological status of variables&rdquo;. It is probably best to illustrate this theme in the setting of a concrete case, which we can do by reviewing the previous example of reductive evaluation shown in Figure&nbsp;16. The observation of several ''semioses'', or sign-transformations, of roughly this shape will most likely lead an observer with any observational facility whatever to notice that it doesn't really matter what sorts of branches happen to sprout from the side of the root aside from the lone edge that also grows there &mdash; the end result will always be the same. Our observer might think to summarize the results of many such observations by introducing a label or variable to signify any shape of branch whatever, writing something like the following: {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_17.jpg|500px]] || (17) |} Observations like that, made about an arithmetic of any variety, germinated by their summarizations, are the root of all algebra. Speaking of algebra, and having encountered already one example of an algebraic law, we might as well introduce the axioms of the ''primary algebra'', once again deriving their substance and their name from the works of Charles Sanders Peirce and George Spencer Brown, respectively. {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_18.jpg|500px]] || (18) |- | [[Image:Logical_Graph_Figure_19.jpg|500px]] || (19) |} The choice of axioms for any formal system is to some degree a matter of aesthetics, as it is commonly the case that many different selections of formal rules will serve as axioms to derive all the rest as theorems. As it happens, the example of an algebraic law that we noticed first, <math>a(~) = (~),\!</math> as simple as it appears, proves to be provable as a theorem on the grounds of the foregoing axioms. We might also notice at this point a subtle difference between the primary arithmetic and the primary algebra with respect to the grounds of justification that we have naturally if tacitly adopted for their respective sets of axioms. The arithmetic axioms were introduced by fiat, in an ''a&nbsp;priori'' fashion, though of course it is only long prior experience with the practical uses of comparably developed generations of formal systems that would actually induce us to such a quasi-primal move. The algebraic axioms, in contrast, can be seen to derive their motive and their justice from the observation and summarization of patterns that are visible in the arithmetic spectrum. ==Formal development== What precedes this point is intended as an informal introduction to the axioms of the primary arithmetic and primary algebra, and hopefully provides the reader with an intuitive sense of their motivation and rationale. The next order of business is to give the exact forms of the axioms that are used in the following more formal development, devolving from Peirce's various systems of logical graphs via Spencer-Brown's ''Laws of Form'' (LOF). In formal proofs, a variation of the annotation scheme from LOF will be used to mark each step of the proof according to which axiom, or ''initial'', is being invoked to justify the corresponding step of syntactic transformation, whether it applies to graphs or to strings. ===Axioms=== The axioms are just four in number, divided into the ''arithmetic initials'', <math>I_1\!</math> and <math>I_2,\!</math> and the ''algebraic initials'', <math>J_1\!</math> and <math>J_2.\!</math> {| align="center" cellpadding="10" | [[Image:Logical_Graph_Figure_20.jpg|500px]] || (20) |- | [[Image:Logical_Graph_Figure_21.jpg|500px]] || (21) |- | [[Image:Logical_Graph_Figure_22.jpg|500px]] || (22) |- | [[Image:Logical_Graph_Figure_23.jpg|500px]] || (23) |} One way of assigning logical meaning to the initial equations is known as the ''entitative interpretation'' (EN). Under EN, the axioms read as follows: {| align="center" cellpadding="10" | <math>\begin{array}{ccccc} I_1 & : & \operatorname{true}\ \operatorname{or}\ \operatorname{true} & = & \operatorname{true} \\ I_2 & : & \operatorname{not}\ \operatorname{true}\ & = & \operatorname{false} \\ J_1 & : & a\ \operatorname{or}\ \operatorname{not}\ a & = & \operatorname{true} \\ J_2 & : & (a\ \operatorname{or}\ b)\ \operatorname{and}\ (a\ \operatorname{or}\ c) & = & a\ \operatorname{or}\ (b\ \operatorname{and}\ c) \\ \end{array}</math> |} Another way of assigning logical meaning to the initial equations is known as the ''existential interpretation'' (EX). Under EX, the axioms read as follows: {| align="center" cellpadding="10" | <math>\begin{array}{ccccc} I_1 & : & \operatorname{false}\ \operatorname{and}\ \operatorname{false} & = & \operatorname{false} \\ I_2 & : & \operatorname{not}\ \operatorname{false} & = & \operatorname{true} \\ J_1 & : & a\ \operatorname{and}\ \operatorname{not}\ a & = & \operatorname{false} \\ J_2 & : & (a\ \operatorname{and}\ b)\ \operatorname{or}\ (a\ \operatorname{and}\ c) & = & a\ \operatorname{and}\ (b\ \operatorname{or}\ c) \\ \end{array}</math> |} All of the axioms in this set have the form of equations. This means that all of the inference steps that they allow are reversible. The proof annotation scheme employed below makes use of a double bar <math>\overline{\underline{~~~~~~}}\!</math> to mark this fact, although it will often be left to the reader to decide which of the two possible directions is the one required for applying the indicated axiom. ===Frequently used theorems=== The actual business of proof is a far more strategic affair than the simple cranking of inference rules might suggest. Part of the reason for this lies in the circumstance that the usual brands of inference rules combine the moving forward of a state of inquiry with the losing of information along the way that doesn't appear to be immediately relevant, at least, not as viewed in the local focus and the short run of the moment to moment proceedings of the proof in question. Over the long haul, this has the pernicious side-effect that one is forever strategically required to reconstruct much of the information that one had strategically thought to forget in earlier stages of the proof, if "before the proof started" can be counted as an earlier stage of the proof in view. For this reason, among others, it is very instructive to study equational inference rules of the sort that our axioms have just provided. Although equational forms of reasoning are paramount in mathematics, they are less familiar to the student of conventional logic textbooks, who may find a few surprises here. By way of gaining a minimal experience with how equational proofs look in the present forms of syntax, let us examine the proofs of a few essential theorems in the primary algebra. ====C<sub>1</sub>. Double negation==== The first theorem goes under the names of ''Consequence&nbsp;1'' <math>(C_1)\!</math>, the ''double negation theorem'' (DNT), or ''Reflection''. {| align="center" cellpadding="10" | [[Image:Double Negation 1.0 Splash Page.png|500px]] || (24) |} The proof that follows is adapted from the one that was given by George Spencer Brown in his book ''Laws of Form'' (LOF) and credited to two of his students, John Dawes and D.A. Utting. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Double Negation 1.0 Marquee Title.png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 1.png|500px]] |- | [[Image:Equational Inference I2 Elicit (( )).png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 2.png|500px]] |- | [[Image:Equational Inference J1 Insert (a).png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 3.png|500px]] |- | [[Image:Equational Inference J2 Distribute ((a)).png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 4.png|500px]] |- | [[Image:Equational Inference J1 Delete (a).png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 5.png|500px]] |- | [[Image:Equational Inference J1 Insert a.png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 6.png|500px]] |- | [[Image:Equational Inference J2 Collect a.png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 7.png|500px]] |- | [[Image:Equational Inference J1 Delete ((a)).png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 8.png|500px]] |- | [[Image:Equational Inference I2 Cancel (( )).png|500px]] |- | [[Image:Double Negation 1.0 Storyboard 9.png|500px]] |- | [[Image:Equational Inference Marquee QED.png|500px]] |} | (25) |} The steps of this proof are replayed in the following animation. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Double Negation 2.0 Animation.gif]] |} | (26) |} ====C<sub>2</sub>. Generation theorem==== One theorem of frequent use goes under the nickname of the ''weed and seed theorem'' (WAST). The proof is just an exercise in mathematical induction, once a suitable basis is laid down, and it will be left as an exercise for the reader. What the WAST says is that a label can be freely distributed or freely erased anywhere in a subtree whose root is labeled with that label. The second in our list of frequently used theorems is in fact the base case of this weed and seed theorem. In LOF, it goes by the names of ''Consequence&nbsp;2'' <math>(C_2)\!</math> or ''Generation''. {| align="center" cellpadding="10" | [[Image:Generation Theorem 1.0 Splash Page.png|500px]] || (27) |} Here is a proof of the Generation Theorem. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Generation Theorem 1.0 Marquee Title.png|500px]] |- | [[Image:Generation Theorem 1.0 Storyboard 1.png|500px]] |- | [[Image:Equational Inference C1 Reflect a(b).png|500px]] |- | [[Image:Generation Theorem 1.0 Storyboard 2.png|500px]] |- | [[Image:Equational Inference I2 Elicit (( )).png|500px]] |- | [[Image:Generation Theorem 1.0 Storyboard 3.png|500px]] |- | [[Image:Equational Inference J1 Insert a.png|500px]] |- | [[Image:Generation Theorem 1.0 Storyboard 4.png|500px]] |- | [[Image:Equational Inference J2 Collect a.png|500px]] |- | [[Image:Generation Theorem 1.0 Storyboard 5.png|500px]] |- | [[Image:Equational Inference C1 Reflect a, b.png|500px]] |- | [[Image:Generation Theorem 1.0 Storyboard 6.png|500px]] |- | [[Image:Equational Inference Marquee QED.png|500px]] |} | (28) |} The steps of this proof are replayed in the following animation. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Generation Theorem 2.0 Animation.gif]] |} | (29) |} ====C<sub>3</sub>. Dominant form theorem==== The third of the frequently used theorems of service to this survey is one that Spencer-Brown annotates as ''Consequence&nbsp;3'' <math>(C_3)\!</math> or ''Integration''. A better mnemonic might be ''dominance and recession theorem'' (DART), but perhaps the brevity of ''dominant form theorem'' (DFT) is sufficient reminder of its double-edged role in proofs. {| align="center" cellpadding="10" | [[Image:Dominant Form 1.0 Splash Page.png|500px]] || (30) |} Here is a proof of the Dominant Form Theorem. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Dominant Form 1.0 Marquee Title.png|500px]] |- | [[Image:Dominant Form 1.0 Storyboard 1.png|500px]] |- | [[Image:Equational Inference C2 Regenerate a.png|500px]] |- | [[Image:Dominant Form 1.0 Storyboard 2.png|500px]] |- | [[Image:Equational Inference J1 Delete a.png|500px]] |- | [[Image:Dominant Form 1.0 Storyboard 3.png|500px]] |- | [[Image:Equational Inference Marquee QED.png|500px]] |} | (31) |} The following animation provides an instant re*play. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Dominant Form 2.0 Animation.gif]] |} | (32) |} ===Exemplary proofs=== Based on the axioms given at the outest, and aided by the theorems recorded so far, it is possible to prove a multitude of much more complex theorems. A couple of all-time favorites are given next. ====Peirce's law==== : ''Main article'' : [[Peirce's law]] Peirce's law is commonly written in the following form: {| align="center" cellpadding="10" | <math>((p \Rightarrow q) \Rightarrow p) \Rightarrow p\!</math> |} The existential graph representation of Peirce's law is shown in Figure&nbsp;33. {| align="center" cellpadding="10" | [[Image:Peirce's Law 1.0 Splash Page.png|500px]] || (33) |} A graphical proof of Peirce's law is shown in Figure&nbsp;34. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Peirce's Law 1.0 Marquee Title.png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 1.png|500px]] |- | [[Image:Equational Inference Band Collect p.png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 2.png|500px]] |- | [[Image:Equational Inference Band Quit ((q)).png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 3.png|500px]] |- | [[Image:Equational Inference Band Cancel (( )).png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 4.png|500px]] |- | [[Image:Equational Inference Band Delete p.png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 5.png|500px]] |- | [[Image:Equational Inference Band Cancel (( )).png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 6.png|500px]] |- | [[Image:Equational Inference Marquee QED.png|500px]] |} | (34) |} The following animation replays the steps of the proof. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Peirce's Law 2.0 Animation.gif]] |} | (35) |} ====Praeclarum theorema==== An illustrious example of a propositional theorem is the ''praeclarum theorema'', the ''admirable'', ''shining'', or ''splendid'' theorem of Leibniz. {| align="center" cellspacing="6" width="90%" <!--QUOTE--> | <p>If ''a'' is ''b'' and ''d'' is ''c'', then ''ad'' will be ''bc''.</p> <p>This is a fine theorem, which is proved in this way:</p> <p>''a'' is ''b'', therefore ''ad'' is ''bd'' (by what precedes),</p> <p>''d'' is ''c'', therefore ''bd'' is ''bc'' (again by what precedes),</p> <p>''ad'' is ''bd'', and ''bd'' is ''bc'', therefore ''ad'' is ''bc''. Q.E.D.</p> <p>(Leibniz, ''Logical Papers'', p. 41).</p> |} Under the existential interpretation, the praeclarum theorema is represented by means of the following logical graph. {| align="center" cellpadding="10" | [[Image:Praeclarum Theorema 1.0 Splash Page.png|500px]] || (36) |} And here's a neat proof of that nice theorem. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Praeclarum Theorema 1.0 Marquee Title.png|500px]] |- | [[Image:Praeclarum Theorema 1.0 Storyboard 1.png|500px]] |- | [[Image:Equational Inference Rule Reflect ad(bc).png|500px]] |- | [[Image:Praeclarum Theorema 1.0 Storyboard 2.png|500px]] |- | [[Image:Equational Inference Rule Weed a, d.png|500px]] |- | [[Image:Praeclarum Theorema 1.0 Storyboard 3.png|500px]] |- | [[Image:Equational Inference Rule Reflect b, c.png|500px]] |- | [[Image:Praeclarum Theorema 1.0 Storyboard 4.png|500px]] |- | [[Image:Equational Inference Rule Weed bc.png|500px]] |- | [[Image:Praeclarum Theorema 1.0 Storyboard 5.png|500px]] |- | [[Image:Equational Inference Rule Quit abcd.png|500px]] |- | [[Image:Praeclarum Theorema 1.0 Storyboard 6.png|500px]] |- | [[Image:Equational Inference Rule Cancel (( )).png|500px]] |- | [[Image:Praeclarum Theorema 1.0 Storyboard 7.png|500px]] |- | [[Image:Equational Inference Marquee QED.png|500px]] |} | (37) |} The steps of the proof are replayed in the following animation. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Praeclarum Theorema 2.0 Animation.gif]] |} | (38) |} ====Two-thirds majority function==== Consider the following equation in boolean algebra, posted as a [http://mathoverflow.net/questions/9292/newbie-boolean-algebra-question problem for proof] at [http://mathoverflow.net/ MathOverFlow]. {| align="center" cellpadding="20" | <math>\begin{matrix} a b \bar{c} + a \bar{b} c + \bar{a} b c + a b c \\[6pt] \iff \\[6pt] a b + a c + b c \end{matrix}</math> | &nbsp;&nbsp;&nbsp;&nbsp; |} The required equation can be proven in the medium of logical graphs as shown in the following Figure. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Two-Thirds Majority Eq 1 Pf 1 Banner Title.png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 1.png|500px]] |- | [[Image:Equational Inference Bar Reflect ab, ac, bc.png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 2.png|500px]] |- | [[Image:Equational Inference Bar Distribute (abc).png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 3.png|500px]] |- | [[Image:Equational Inference Bar Collect ab, ac, bc.png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 4.png|500px]] |- | [[Image:Equational Inference Bar Quit (a), (b), (c).png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 5.png|500px]] |- | [[Image:Equational Inference Bar Cancel (( )).png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 6.png|500px]] |- | [[Image:Equational Inference Bar Weed ab, ac, bc.png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 7.png|500px]] |- | [[Image:Equational Inference Bar Delete a, b, c.png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 8.png|500px]] |- | [[Image:Equational Inference Bar Cancel (( )).png|500px]] |- | [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 9.png|500px]] |- | [[Image:Equational Inference Banner QED.png|500px]] |} | (39) |} Here's an animated recap of the graphical transformations that occur in the above proof: {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Two-Thirds Majority Function 500 x 250 Animation.gif]] |} | (40) |} ==Bibliography== * Leibniz, G.W. (1679&ndash;1686 ?), &ldquo;Addenda to the Specimen of the Universal Calculus&rdquo;, pp. 40&ndash;46 in G.H.R. Parkinson (ed. and trans., 1966), ''Leibniz : Logical Papers'', Oxford University Press, London, UK. * Peirce, C.S. (1931&ndash;1935, 1958), ''Collected Papers of Charles Sanders Peirce'', vols. 1&ndash;6, Charles Hartshorne and Paul Weiss (eds.), vols. 7&ndash;8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA. Cited as (CP&nbsp;volume.paragraph). * Peirce, C.S. (1981&ndash;), ''Writings of Charles S. Peirce : A Chronological Edition'', Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN. Cited as (CE&nbsp;volume, page). * Peirce, C.S. (1885), &ldquo;On the Algebra of Logic : A Contribution to the Philosophy of Notation&rdquo;, ''American Journal of Mathematics'' 7 (1885), 180&ndash;202. Reprinted as CP&nbsp;3.359&ndash;403 and CE&nbsp;5, 162&ndash;190. * Peirce, C.S. (''c.'' 1886), &ldquo;Qualitative Logic&rdquo;, MS 736. Published as pp. 101&ndash;115 in Carolyn Eisele (ed., 1976), ''The New Elements of Mathematics by Charles S. Peirce, Volume 4, Mathematical Philosophy'', Mouton, The Hague. * Peirce, C.S. (1886 a), &ldquo;Qualitative Logic&rdquo;, MS 582. Published as pp. 323&ndash;371 in ''Writings of Charles S. Peirce : A Chronological Edition, Volume 5, 1884&ndash;1886'', Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1993. * Peirce, C.S. (1886 b), &ldquo;The Logic of Relatives : Qualitative and Quantitative&rdquo;, MS 584. Published as pp. 372&ndash;378 in ''Writings of Charles S. Peirce : A Chronological Edition, Volume 5, 1884&ndash;1886'', Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1993. * Spencer Brown, George (1969), ''Laws of Form'', George Allen and Unwin, London, UK. ==Resources== * [http://planetmath.org/ PlanetMath] ** [http://planetmath.org/LogicalGraphIntroduction Logical Graph : Introduction] ** [http://planetmath.org/LogicalGraphFormalDevelopment Logical Graph : Formal Development] * Bergman and Paavola (eds.), [http://www.helsinki.fi/science/commens/dictionary.html Commens Dictionary of Peirce's Terms] ** [http://www.helsinki.fi/science/commens/terms/graphexis.html Existential Graph] ** [http://www.helsinki.fi/science/commens/terms/graphlogi.html Logical Graph] * [http://dr-dau.net/index.shtml Dau, Frithjof] ** [http://dr-dau.net/eg_readings.shtml Peirce's Existential Graphs : Readings and Links] ** [http://web.archive.org/web/20070706192257/http://dr-dau.net/pc.shtml Existential Graphs as Moving Pictures of Thought] &mdash; Computer Animated Proof of Leibniz's Praeclarum Theorema * [http://www.math.uic.edu/~kauffman/ Kauffman, Louis H.] ** [http://www.math.uic.edu/~kauffman/Arithmetic.htm Box Algebra, Boundary Mathematics, Logic, and Laws of Form] * [http://mathworld.wolfram.com/ MathWorld : A Wolfram Web Resource] ** [http://mathworld.wolfram.com/about/author.html Weisstein, Eric W.], [http://mathworld.wolfram.com/Spencer-BrownForm.html Spencer-Brown Form] * Shoup, Richard (ed.), [http://www.lawsofform.org/ Laws of Form Web Site] ** Spencer-Brown, George (1973), [http://www.lawsofform.org/aum/session1.html Transcript Session One], [http://www.lawsofform.org/aum/ AUM Conference], Esalen, CA. ==Translations== * [http://pt.wikipedia.org/wiki/Grafo_l%C3%B3gico Grafo lógico], [http://pt.wikipedia.org/ Portuguese Wikipedia]. ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Logical_graph Logical Graph @ InterSciWiki] * [http://mywikibiz.com/Logical_graph Logical Graph @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Logical_graph Logical Graph @ Subject Wikis] * [http://en.wikiversity.org/wiki/Logical_graph Logical Graph @ Wikiversity] * [http://beta.wikiversity.org/wiki/Logical_graph Logical Graph @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Logical_graph Logical Graph], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Logical_graph Logical Graph], [http://mywikibiz.com/ MyWikiBiz] * [http://planetmath.org/LogicalGraphIntroduction Logical Graph 1], [http://planetmath.org/ PlanetMath] * [http://planetmath.org/LogicalGraphFormalDevelopment Logical Graph 2], [http://planetmath.org/ PlanetMath] * [http://semanticweb.org/wiki/Logical_graph Logical Graph], [http://semanticweb.org/ Semantic Web] * [http://wikinfo.org/w/index.php/Logical_graph Logical Graph], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Logical_graph Logical Graph], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Logical_graph Logical Graph], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Logical_graph&oldid=67277491 Logical Graph], [http://en.wikipedia.org/ Wikipedia] [[Category:Artificial Intelligence]] [[Category:Boolean Functions]] [[Category:Charles Sanders Peirce]] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Cybernetics]] [[Category:Equational Reasoning]] [[Category:Formal Languages]] [[Category:Formal Systems]] [[Category:George Spencer Brown]] [[Category:Graph Theory]] [[Category:History of Logic]] [[Category:History of Mathematics]] [[Category:Inquiry]] [[Category:Knowledge Representation]] [[Category:Laws of Form]] [[Category:Logic]] [[Category:Logical Graphs]] [[Category:Mathematics]] [[Category:Philosophy]] [[Category:Propositional Calculus]] [[Category:Semiotics]] [[Category:Visualization]] b0ec5856b5869754db4015bbe82a505b1ced7ca3 0 315 723 667 2015-11-06T19:22:24Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''minimal negation operator''' <math>(\texttt{Mno})\!</math> is a logical connective that says &ldquo;just one false&rdquo; of its logical arguments. If the list of arguments is empty, as expressed in the form <math>\texttt{Mno}(),\!</math> then it cannot be true that exactly one of the arguments is false, so <math>\texttt{Mno}() = \texttt{False}.\!</math> If <math>p\!</math> is the only argument, then <math>\texttt{Mno}(p)\!</math> says that <math>p\!</math> is false, so <math>\texttt{Mno}(p)\!</math> expresses the logical negation of the proposition <math>p.\!</math> Wrtten in several different notations, <math>\texttt{Mno}(p) = \texttt{Not}(p) = \lnot p = \tilde{p} = p^\prime.\!</math> If <math>p\!</math> and <math>q\!</math> are the only two arguments, then <math>\texttt{Mno}(p, q)\!</math> says that exactly one of <math>p, q\!</math> is false, so <math>\texttt{Mno}(p, q)\!</math> says the same thing as <math>p \neq q.\!</math> Expressing <math>\texttt{Mno}(p, q)\!</math> in terms of ands <math>(\cdot),\!</math> ors <math>(\lor),\!</math> and nots <math>(\tilde{~})\!</math> gives the following form. {| align="center" cellpadding="8" | <math>\texttt{Mno}(p, q) = \tilde{p} \cdot q ~\lor~ p \cdot \tilde{q}.\!</math> |} As usual, one drops the dots <math>(\cdot)~\!</math> in contexts where they are understood, giving the following form. {| align="center" cellpadding="8" | <math>\texttt{Mno}(p, q) = \tilde{p}q \lor p\tilde{q}.\!</math> |} The venn diagram for <math>\texttt{Mno}(p, q)\!</math> is shown in Figure&nbsp;1. {| align="center" cellpadding="8" style="text-align:center" | <p>[[Image:Venn Diagram (P,Q).jpg|500px]]</p> <p><math>\text{Figure 1.}~~\texttt{Mno}(p, q)\!</math></p> |} The venn diagram for <math>\texttt{Mno}(p, q, r)\!</math> is shown in Figure&nbsp;2. {| align="center" cellpadding="8" style="text-align:center" | <p>[[Image:Venn Diagram (P,Q,R).jpg|500px]]</p> <p><math>\text{Figure 2.}~~\texttt{Mno}(p, q, r)\!</math></p> |} The center cell is the region where all three arguments <math>p, q, r\!</math> hold true, so <math>\texttt{Mno}(p, q, r)\!</math> holds true in just the three neighboring cells. In other words, <math>\texttt{Mno}(p, q, r) = \tilde{p}qr \lor p\tilde{q}r \lor pq\tilde{r}.\!</math> ==Initial definition== The '''minimal negation operator''' <math>\nu\!</math> is a [[multigrade operator]] <math>(\nu_k)_{k \in \mathbb{N}}\!</math> where each <math>\nu_k\!</math> is a <math>k\!</math>-ary [[boolean function]] defined in such a way that <math>\nu_k (x_1, \ldots , x_k) = 1\!</math> in just those cases where exactly one of the arguments <math>x_j\!</math> is <math>0.\!</math> In contexts where the initial letter <math>\nu\!</math> is understood, the minimal negation operators can be indicated by argument lists in parentheses. In the following text a distinctive typeface will be used for logical expressions based on minimal negation operators, for example, <math>\texttt{(x, y, z)}\!</math> = <math>\nu (x, y, z).\!</math> The first four members of this family of operators are shown below, with paraphrases in a couple of other notations, where tildes and primes, respectively, indicate logical negation. {| align="center" cellpadding="8" style="text-align:center" | <math>\begin{matrix} \texttt{()} & = & \nu_0 & = & 0 & = & \mathrm{false} \\[6pt] \texttt{(x)} & = & \nu_1 (x) & = & \tilde{x} & = & x^\prime \\[6pt] \texttt{(x, y)} & = & \nu_2 (x, y) & = & \tilde{x}y \lor x\tilde{y} & = & x^\prime y \lor x y^\prime \\[6pt] \texttt{(x, y, z)} & = & \nu_3 (x, y, z) & = & \tilde{x}yz \lor x\tilde{y}z \lor xy\tilde{z} & = & x^\prime y z \lor x y^\prime z \lor x y z^\prime \end{matrix}</math> |} ==Formal definition== To express the general case of <math>\nu_k\!</math> in terms of familiar operations, it helps to introduce an intermediary concept: '''Definition.''' Let the function <math>\lnot_j : \mathbb{B}^k \to \mathbb{B}\!</math> be defined for each integer <math>j\!</math> in the interval <math>[1, k]\!</math> by the following equation: {| align="center" cellpadding="8" | <math>\lnot_j (x_1, \ldots, x_j, \ldots, x_k) ~=~ x_1 \land \ldots \land x_{j-1} \land \lnot x_j \land x_{j+1} \land \ldots \land x_k.\!</math> |} Then <math>{\nu_k : \mathbb{B}^k \to \mathbb{B}}\!</math> is defined by the following equation: {| align="center" cellpadding="8" | <math>\nu_k (x_1, \ldots, x_k) ~=~ \lnot_1 (x_1, \ldots, x_k) \lor \ldots \lor \lnot_j (x_1, \ldots, x_k) \lor \ldots \lor \lnot_k (x_1, \ldots, x_k).\!</math> |} If we think of the point <math>x = (x_1, \ldots, x_k) \in \mathbb{B}^k\!</math> as indicated by the boolean product <math>x_1 \cdot \ldots \cdot x_k\!</math> or the logical conjunction <math>x_1 \land \ldots \land x_k,\!</math> then the minimal negation <math>\texttt{(} x_1, \ldots, x_k \texttt{)}\!</math> indicates the set of points in <math>\mathbb{B}^k\!</math> that differ from <math>x\!</math> in exactly one coordinate. This makes <math>\texttt{(} x_1, \ldots, x_k \texttt{)}\!</math> a discrete functional analogue of a ''point omitted neighborhood'' in analysis, more exactly, a ''point omitted distance one neighborhood''. In this light, the minimal negation operator can be recognized as a differential construction, an observation that opens a very wide field. It also serves to explain a variety of other names for the same concept, for example, ''logical boundary operator'', ''limen operator'', ''least action operator'', or ''hedge operator'', to name but a few. The rationale for these names is visible in the venn diagrams of the corresponding operations on sets. The remainder of this discussion proceeds on the ''algebraic boolean convention'' that the plus sign <math>(+)\!</math> and the summation symbol <math>(\textstyle\sum)\!</math> both refer to addition modulo 2. Unless otherwise noted, the boolean domain <math>\mathbb{B} = \{ 0, 1 \}~\!</math> is interpreted so that <math>0 = \mathrm{false}\!</math> and <math>1 = \mathrm{true}.\!</math> This has the following consequences: {| align="center" cellpadding="4" width="90%" | valign="top" | <big>&bull;</big> | The operation <math>x + y\!</math> is a function equivalent to the exclusive disjunction of <math>x\!</math> and <math>y,\!</math> while its fiber of 1 is the relation of inequality between <math>x\!</math> and <math>y.\!</math> |- | valign="top" | <big>&bull;</big> | The operation <math>\textstyle\sum_{j=1}^k x_j\!</math> maps the bit sequence <math>(x_1, \ldots, x_k)\!</math> to its ''parity''. |} The following properties of the minimal negation operators <math>{\nu_k : \mathbb{B}^k \to \mathbb{B}}\!</math> may be noted: {| align="center" cellpadding="4" width="90%" | valign="top" | <big>&bull;</big> | The function <math>\texttt{(x, y)}\!</math> is the same as that associated with the operation <math>x + y\!</math> and the relation <math>x \ne y.\!</math> |- | valign="top" | <big>&bull;</big> | In contrast, <math>\texttt{(x, y, z)}\!</math> is not identical to <math>x + y + z.\!</math> |- | valign="top" | <big>&bull;</big> | More generally, the function <math>\nu_k (x_1, \dots, x_k)\!</math> for <math>k > 2\!</math> is not identical to the boolean sum <math>\textstyle\sum_{j=1}^k x_j.\!</math> |- | valign="top" | <big>&bull;</big> | The inclusive disjunctions indicated for the <math>\nu_k\!</math> of more than one argument may be replaced with exclusive disjunctions without affecting the meaning, since the terms disjoined are already disjoint. |} ==Truth tables== Table&nbsp;3 is a [[truth table]] for the sixteen boolean functions of type <math>f : \mathbb{B}^3 \to \mathbb{B}\!</math> whose fibers of 1 are either the boundaries of points in <math>\mathbb{B}^3\!</math> or the complements of those boundaries. <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:70%" |+ <math>\text{Table 3.} ~~ \text{Logical Boundaries and Their Complements}\!</math> |- style="background:ghostwhite" | <math>\mathcal{L}_1\!</math> | <math>\mathcal{L}_2\!</math> | <math>\mathcal{L}_3\!</math> | <math>\mathcal{L}_4\!</math> |- style="background:ghostwhite" | &nbsp; | align="right" | <math>p\colon\!</math> | <math>1~1~1~1~0~0~0~0\!</math> | &nbsp; |- style="background:ghostwhite" | &nbsp; | align="right" | <math>q\colon\!</math> | <math>1~1~0~0~1~1~0~0\!</math> | &nbsp; |- style="background:ghostwhite" | &nbsp; | align="right" | <math>r\colon\!</math> | <math>1~0~1~0~1~0~1~0\!</math> | &nbsp; |- | <math>\begin{matrix} f_{104} \\[4pt] f_{148} \\[4pt] f_{146} \\[4pt] f_{97} \\[4pt] f_{134} \\[4pt] f_{73} \\[4pt] f_{41} \\[4pt] f_{22} \end{matrix}</math> | <math>\begin{matrix} f_{01101000} \\[4pt] f_{10010100} \\[4pt] f_{10010010} \\[4pt] f_{01100001} \\[4pt] f_{10000110} \\[4pt] f_{01001001} \\[4pt] f_{00101001} \\[4pt] f_{00010110} \end{matrix}</math> | <math>\begin{matrix} 0~1~1~0~1~0~0~0 \\[4pt] 1~0~0~1~0~1~0~0 \\[4pt] 1~0~0~1~0~0~1~0 \\[4pt] 0~1~1~0~0~0~0~1 \\[4pt] 1~0~0~0~0~1~1~0 \\[4pt] 0~1~0~0~1~0~0~1 \\[4pt] 0~0~1~0~1~0~0~1 \\[4pt] 0~0~0~1~0~1~1~0 \end{matrix}</math> | <math>\begin{matrix} \texttt{(~p~,~q~,~r~)} \\[4pt] \texttt{(~p~,~q~,(r))} \\[4pt] \texttt{(~p~,(q),~r~)} \\[4pt] \texttt{(~p~,(q),(r))} \\[4pt] \texttt{((p),~q~,~r~)} \\[4pt] \texttt{((p),~q~,(r))} \\[4pt] \texttt{((p),(q),~r~)} \\[4pt] \texttt{((p),(q),(r))} \end{matrix}</math> |- | <math>\begin{matrix} f_{233} \\[4pt] f_{214} \\[4pt] f_{182} \\[4pt] f_{121} \\[4pt] f_{158} \\[4pt] f_{109} \\[4pt] f_{107} \\[4pt] f_{151} \end{matrix}</math> | <math>\begin{matrix} f_{11101001} \\[4pt] f_{11010110} \\[4pt] f_{10110110} \\[4pt] f_{01111001} \\[4pt] f_{10011110} \\[4pt] f_{01101101} \\[4pt] f_{01101011} \\[4pt] f_{10010111} \end{matrix}</math> | <math>\begin{matrix} 1~1~1~0~1~0~0~1 \\[4pt] 1~1~0~1~0~1~1~0 \\[4pt] 1~0~1~1~0~1~1~0 \\[4pt] 0~1~1~1~1~0~0~1 \\[4pt] 1~0~0~1~1~1~1~0 \\[4pt] 0~1~1~0~1~1~0~1 \\[4pt] 0~1~1~0~1~0~1~1 \\[4pt] 1~0~0~1~0~1~1~1 \end{matrix}</math> | <math>\begin{matrix} \texttt{(((p),(q),(r)))} \\[4pt] \texttt{(((p),(q),~r~))} \\[4pt] \texttt{(((p),~q~,(r)))} \\[4pt] \texttt{(((p),~q~,~r~))} \\[4pt] \texttt{((~p~,(q),(r)))} \\[4pt] \texttt{((~p~,(q),~r~))} \\[4pt] \texttt{((~p~,~q~,(r)))} \\[4pt] \texttt{((~p~,~q~,~r~))} \end{matrix}</math> |} <br> ==Charts and graphs== This Section focuses on visual representations of minimal negation operators. A few bits of terminology are useful in describing the pictures, but the formal details are tedious reading, and may be familiar to many readers, so the full definitions of the terms marked in ''italics'' are relegated to a Glossary at the end of the article. Two ways of visualizing the space <math>\mathbb{B}^k\!</math> of <math>2^k\!</math> points are the [[hypercube]] picture and the [[venn diagram]] picture. The hypercube picture associates each point of <math>\mathbb{B}^k\!</math> with a unique point of the <math>k\!</math>-dimensional hypercube. The venn diagram picture associates each point of <math>\mathbb{B}^k\!</math> with a unique "cell" of the venn diagram on <math>k\!</math> "circles". In addition, each point of <math>\mathbb{B}^k\!</math> is the unique point in the ''[[fiber (mathematics)|fiber]] of truth'' <math>[|s|]\!</math> of a ''singular proposition'' <math>s : \mathbb{B}^k \to \mathbb{B},\!</math> and thus it is the unique point where a ''singular conjunction'' of <math>k\!</math> ''literals'' is <math>1.\!</math> For example, consider two cases at opposite vertices of the cube: {| align="center" cellpadding="4" width="90%" | valign="top" | <big>&bull;</big> | The point <math>(1, 1, \ldots , 1, 1)\!</math> with all 1's as coordinates is the point where the conjunction of all posited variables evaluates to <math>1,\!</math> namely, the point where: |- | &nbsp; | align="center" | <math>x_1 ~ x_2 ~\ldots~ x_{n-1} ~ x_n ~=~ 1.\!</math> |- | valign="top" | <big>&bull;</big> | The point <math>(0, 0, \ldots , 0, 0)\!</math> with all 0's as coordinates is the point where the conjunction of all negated variables evaluates to <math>1,\!</math> namely, the point where: |- | &nbsp; | align="center" | <math>\texttt{(} x_1 \texttt{)(} x_2 \texttt{)} \ldots \texttt{(} x_{n-1} \texttt{)(} x_n \texttt{)} ~=~ 1.\!</math> |} To pass from these limiting examples to the general case, observe that a singular proposition <math>s : \mathbb{B}^k \to \mathbb{B}\!</math> can be given canonical expression as a conjunction of literals, <math>s = e_1 e_2 \ldots e_{k-1} e_k\!</math>. Then the proposition <math>\nu (e_1, e_2, \ldots, e_{k-1}, e_k)\!</math> is <math>1\!</math> on the points adjacent to the point where <math>s\!</math> is <math>1,\!</math> and 0 everywhere else on the cube. For example, consider the case where <math>k = 3.\!</math> Then the minimal negation operation <math>\nu (p, q, r)\!</math> &mdash; written more simply as <math>\texttt{(p, q, r)}\!</math> &mdash; has the following venn diagram: {| align="center" cellpadding="8" style="text-align:center" | <p>[[Image:Venn Diagram (P,Q,R).jpg|500px]]</p> <p><math>\text{Figure 4.}~~\texttt{(p, q, r)}\!</math></p> |} For a contrasting example, the boolean function expressed by the form <math>\texttt{((p),(q),(r))}\!</math> has the following venn diagram: {| align="center" cellpadding="8" style="text-align:center" | <p>[[Image:Venn Diagram ((P),(Q),(R)).jpg|500px]]</p> <p><math>\text{Figure 5.}~~\texttt{((p),(q),(r))}\!</math></p> |} ==Glossary of basic terms== ; Boolean domain : A ''[[boolean domain]]'' <math>\mathbb{B}\!</math> is a generic 2-element set, for example, <math>\mathbb{B} = \{ 0, 1 \},\!</math> whose elements are interpreted as logical values, usually but not invariably with <math>0 = \mathrm{false}\!</math> and <math>1 = \mathrm{true}.\!</math> ; Boolean variable : A ''[[boolean variable]]'' <math>x\!</math> is a variable that takes its value from a boolean domain, as <math>x \in \mathbb{B}.\!</math> ; Proposition : In situations where boolean values are interpreted as logical values, a [[boolean-valued function]] <math>f : X \to \mathbb{B}\!</math> or a [[boolean function]] <math>g : \mathbb{B}^k \to \mathbb{B}\!</math> is frequently called a ''[[proposition]]''. ; Basis element, Coordinate projection : Given a sequence of <math>k\!</math> boolean variables, <math>x_1, \ldots, x_k,\!</math> each variable <math>x_j\!</math> may be treated either as a ''basis element'' of the space <math>\mathbb{B}^k\!</math> or as a ''coordinate projection'' <math>x_j : \mathbb{B}^k \to \mathbb{B}.\!</math> ; Basic proposition : This means that the set of objects <math>\{ x_j : 1 \le j \le k \}\!</math> is a set of boolean functions <math>\{ x_j : \mathbb{B}^k \to \mathbb{B} \}\!</math> subject to logical interpretation as a set of ''basic propositions'' that collectively generate the complete set of <math>2^{2^k}\!</math> propositions over <math>\mathbb{B}^k.\!</math> ; Literal : A ''literal'' is one of the <math>2k\!</math> propositions <math>x_1, \ldots, x_k, \texttt{(} x_1 \texttt{)}, \ldots, \texttt{(} x_k \texttt{)},\!</math> in other words, either a ''posited'' basic proposition <math>x_j\!</math> or a ''negated'' basic proposition <math>\texttt{(} x_j \texttt{)},\!</math> for some <math>j = 1 ~\text{to}~ k.\!</math> ; Fiber : In mathematics generally, the ''[[fiber (mathematics)|fiber]]'' of a point <math>y \in Y\!</math> under a function <math>f : X \to Y\!</math> is defined as the inverse image <math>f^{-1}(y) \subseteq X.\!</math> : In the case of a boolean function <math>f : \mathbb{B}^k \to \mathbb{B},\!</math> there are just two fibers: : The fiber of <math>0\!</math> under <math>f,\!</math> defined as <math>f^{-1}(0),\!</math> is the set of points where the value of <math>f\!</math> is <math>0.\!</math> : The fiber of <math>1\!</math> under <math>f,\!</math> defined as <math>f^{-1}(1),\!</math> is the set of points where the value of <math>f\!</math> is <math>1.\!</math> ; Fiber of truth : When <math>1\!</math> is interpreted as the logical value <math>\mathrm{true},\!</math> then <math>f^{-1}(1)\!</math> is called the ''fiber of truth'' in the proposition <math>f.\!</math> Frequent mention of this fiber makes it useful to have a shorter way of referring to it. This leads to the definition of the notation <math>[|f|] = f^{-1}(1)\!</math> for the fiber of truth in the proposition <math>f.\!</math> ; Singular boolean function : A ''singular boolean function'' <math>s : \mathbb{B}^k \to \mathbb{B}\!</math> is a boolean function whose fiber of <math>1\!</math> is a single point of <math>\mathbb{B}^k.\!</math> ; Singular proposition : In the interpretation where <math>1\!</math> equals <math>\mathrm{true},\!</math> a singular boolean function is called a ''singular proposition''. : Singular boolean functions and singular propositions serve as functional or logical representatives of the points in <math>\mathbb{B}^k.\!</math> ; Singular conjunction : A ''singular conjunction'' in <math>\mathbb{B}^k \to \mathbb{B}\!</math> is a conjunction of <math>k\!</math> literals that includes just one conjunct of the pair <math>\{ x_j, ~\nu(x_j) \}\!</math> for each <math>j = 1 ~\text{to}~ k.\!</math> : A singular proposition <math>s : \mathbb{B}^k \to \mathbb{B}\!</math> can be expressed as a singular conjunction: {| align="center" cellspacing"10" width="90%" | height="36" | <math>s ~=~ e_1 e_2 \ldots e_{k-1} e_k\!</math>, |- | <math>\begin{array}{llll} \text{where} & e_j & = & x_j \\[6pt] \text{or} & e_j & = & \nu (x_j), \\[6pt] \text{for} & j & = & 1 ~\text{to}~ k. \end{array}</math> |} ==Resources== * [http://atlas.wolfram.com/ Wolfram Atlas of Simple Programs] ** [http://atlas.wolfram.com/01/01/ Elementary Cellular Automata Rules (ECARs)] ** [http://atlas.wolfram.com/01/01/rulelist.html ECAR Index] ** [http://atlas.wolfram.com/01/01/views/3/TableView.html ECAR Icons] ** [http://atlas.wolfram.com/01/01/views/87/TableView.html ECAR Examples] ** [http://atlas.wolfram.com/01/01/views/172/TableView.html ECAR Formulas] ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Minimal_negation_operator Minimal Negation Operator @ InterSciWiki] * [http://mywikibiz.com/Minimal_negation_operator Minimal Negation Operator @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Minimal_negation_operator Minimal Negation Operator @ Subject Wikis] * [http://en.wikiversity.org/wiki/Minimal_negation_operator Minimal Negation Operator @ Wikiversity] * [http://beta.wikiversity.org/wiki/Minimal_negation_operator Minimal Negation Operator @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Minimal_negation_operator Minimal Negation Operator], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Minimal_negation_operator Minimal Negation Operator], [http://mywikibiz.com/ MyWikiBiz] * [http://planetmath.org/MinimalNegationOperator Minimal Negation Operator], [http://planetmath.org/ PlanetMath] * [http://wikinfo.org/w/index.php/Minimal_negation_operator Minimal Negation Operator], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Minimal_negation_operator Minimal Negation Operator], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Minimal_negation_operator Minimal Negation Operator], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://web.archive.org/web/20060913000000/http://en.wikipedia.org/wiki/Minimal_negation_operator Minimal Negation Operator], [http://en.wikipedia.org/ Wikipedia] [[Category:Automata Theory]] [[Category:Boolean Functions]] [[Category:Charles Sanders Peirce]] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Differential Logic]] [[Category:Equational Reasoning]] [[Category:Formal Languages]] [[Category:Formal Sciences]] [[Category:Formal Systems]] [[Category:Inquiry]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Logical Graphs]] [[Category:Mathematics]] [[Category:Neural Networks]] [[Category:Philosophy]] [[Category:Propositional Calculus]] [[Category:Semiotics]] 7a4914f01b9f29bf608ad2f103356218478fafcd 0 316 725 537 2015-11-07T03:32:40Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''multigrade operator''' <math>\Omega\!</math> is a ''[[parametric operator]]'' with ''parameter'' <math>k\!</math> in the set <math>\mathbb{N}\!</math> of non-negative integers. The application of a multigrade operator <math>\Omega\!</math> to a finite sequence of operands <math>(x_1, \ldots, x_k)\!</math> is typically denoted with the parameter <math>k\!</math> left tacit, as the appropriate application is implicit in the number of operands listed. Thus <math>\Omega (x_1, \ldots, x_k)\!</math> may be taken for <math>\Omega_k (x_1, \ldots, x_k).\!</math> ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Multigrade_operator Multigrade Operator @ InterSciWiki] * [http://mywikibiz.com/Multigrade_operator Multigrade Operator @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Multigrade_operator Multigrade Operator @ Subject Wikis] * [http://en.wikiversity.org/wiki/Multigrade_operator Multigrade Operator @ Wikiversity] * [http://beta.wikiversity.org/wiki/Multigrade_operator Multigrade Operator @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Multigrade_operator Multigrade Operator], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Multigrade_operator Multigrade Operator], [http://mywikibiz.com/ MyWikiBiz] * [http://planetmath.org/MultigradeOperator Multigrade Operator], [http://planetmath.org/ PlanetMath] * [http://wikinfo.org/w/index.php/Multigrade_operator Multigrade Operator], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Multigrade_operator Multigrade Operator], [http://beta.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Multigrade_operator Multigrade Operator], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Multigrade_operator&oldid=40451309 Multigrade Operator], [http://en.wikipedia.org/ Wikipedia] [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Automata Theory]] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Differential Logic]] [[Category:Equational Reasoning]] [[Category:Formal Languages]] [[Category:Formal Sciences]] [[Category:Formal Systems]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Neural Networks]] [[Category:Philosophy]] [[Category:Semiotics]] d1c22abeaea660700f1b54dc7214104d91a96dea 726 725 2015-11-07T04:12:40Z Jon Awbrey 3 /* Document history */ wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''multigrade operator''' <math>\Omega\!</math> is a ''[[parametric operator]]'' with ''parameter'' <math>k\!</math> in the set <math>\mathbb{N}\!</math> of non-negative integers. The application of a multigrade operator <math>\Omega\!</math> to a finite sequence of operands <math>(x_1, \ldots, x_k)\!</math> is typically denoted with the parameter <math>k\!</math> left tacit, as the appropriate application is implicit in the number of operands listed. Thus <math>\Omega (x_1, \ldots, x_k)\!</math> may be taken for <math>\Omega_k (x_1, \ldots, x_k).\!</math> ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Multigrade_operator Multigrade Operator @ InterSciWiki] * [http://mywikibiz.com/Multigrade_operator Multigrade Operator @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Multigrade_operator Multigrade Operator @ Subject Wikis] * [http://en.wikiversity.org/wiki/Multigrade_operator Multigrade Operator @ Wikiversity] * [http://beta.wikiversity.org/wiki/Multigrade_operator Multigrade Operator @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Multigrade_operator Multigrade Operator], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Multigrade_operator Multigrade Operator], [http://mywikibiz.com/ MyWikiBiz] * [http://planetmath.org/MultigradeOperator Multigrade Operator], [http://planetmath.org/ PlanetMath] * [http://wikinfo.org/w/index.php/Multigrade_operator Multigrade Operator], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Multigrade_operator Multigrade Operator], [http://beta.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Multigrade_operator Multigrade Operator], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://web.archive.org/web/20060913000000/http://en.wikipedia.org/wiki/Multigrade_operator Multigrade Operator], [http://en.wikipedia.org/ Wikipedia] [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Automata Theory]] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Differential Logic]] [[Category:Equational Reasoning]] [[Category:Formal Languages]] [[Category:Formal Sciences]] [[Category:Formal Systems]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Neural Networks]] [[Category:Philosophy]] [[Category:Semiotics]] b0df86fe86e0b9f09516600d49e8f68bd51ecbe1 0 317 727 538 2015-11-07T04:18:37Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''parametric operator''' <math>\Omega\!</math> with '''parameter''' <math>\alpha\!</math> in the '''parameter set''' <math>\Alpha\!</math> is an indexed family of operators <math>(\Omega_\alpha)_\Alpha = \{ \Omega_\alpha : \alpha \in \Alpha \}\!</math> with index <math>\alpha\!</math> in the index set <math>\Alpha.\!</math> ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Parametric_operator Parametric Operator @ InterSciWiki] * [http://mywikibiz.com/Parametric_operator Parametric Operator @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Parametric_operator Parametric Operator @ Subject Wikis] * [http://en.wikiversity.org/wiki/Parametric_operator Parametric Operator @ Wikiversity] * [http://beta.wikiversity.org/wiki/Parametric_operator Parametric Operator @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Parametric_operator Parametric Operator], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Parametric_operator Parametric Operator], [http://mywikibiz.com/ MyWikiBiz] * [http://planetmath.org/ParametricOperator Parametric Operator], [http://planetmath.org/ PlanetMath] * [http://wikinfo.org/w/index.php/Parametric_operator Parametric Operator], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Parametric_operator Parametric Operator], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Parametric_operator Parametric Operator], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Parametric_operator&oldid=40451935 Parametric Operator], [http://en.wikipedia.org/ Wikipedia] [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Automata Theory]] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Differential Logic]] [[Category:Equational Reasoning]] [[Category:Formal Languages]] [[Category:Formal Sciences]] [[Category:Formal Systems]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Neural Networks]] [[Category:Philosophy]] [[Category:Semiotics]] 51f08f0f97bcd6b6ce86a5e62135362d40dcac6a 0 301 728 532 2015-11-07T15:34:53Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. '''Peirce's law''' is a formula in [[propositional calculus]] that is commonly expressed in the following form: {| align="center" cellpadding="10" | <math>((p \Rightarrow q) \Rightarrow p) \Rightarrow p</math> |} Peirce's law holds in classical propositional calculus, but not in intuitionistic propositional calculus. The precise axiom system that one chooses for classical propositional calculus determines whether Peirce's law is taken as an axiom or proven as a theorem. ==History== Here is Peirce's own statement and proof of the law: {| align="center" cellpadding="8" width="90%" | <p>A ''fifth icon'' is required for the principle of excluded middle and other propositions connected with it. One of the simplest formulae of this kind is:</p> <center> <p><math>\{ (x \,-\!\!\!< y) \,-\!\!\!< x \} \,-\!\!\!< x.</math></p> </center> <p>This is hardly axiomatical. That it is true appears as follows. It can only be false by the final consequent <math>x\!</math> being false while its antecedent <math>(x \,-\!\!\!< y) \,-\!\!\!< x</math> is true. If this is true, either its consequent, <math>x,\!</math> is true, when the whole formula would be true, or its antecedent <math>x \,-\!\!\!< y</math> is false. But in the last case the antecedent of <math>x \,-\!\!\!< y,</math> that is <math>x,\!</math> must be true. (Peirce, CP&nbsp;3.384).</p> |} Peirce goes on to point out an immediate application of the law: {| align="center" cellpadding="8" width="90%" | <p>From the formula just given, we at once get:</p> <center> <p><math>\{ (x \,-\!\!\!< y) \,-\!\!\!< a \} \,-\!\!\!< x,</math></p> </center> <p>where the <math>a\!</math> is used in such a sense that <math>(x \,-\!\!\!< y) \,-\!\!\!< a</math> means that from <math>(x \,-\!\!\!< y)</math> every proposition follows. With that understanding, the formula states the principle of excluded middle, that from the falsity of the denial of <math>x\!</math> follows the truth of <math>x.\!</math> (Peirce, CP&nbsp;3.384).</p> |} '''Note.''' Peirce uses the ''sign of illation'' “<math>-\!\!\!<</math>” for implication. In one place he explains “<math>-\!\!\!<</math>” as a variant of the sign “<math>\le</math>” for ''less than or equal to''; in another place he suggests that <math>A \,-\!\!\!< B</math> is an iconic way of representing a state of affairs where <math>A,\!</math> in every way that it can be, is <math>B.\!</math> ==Graphical proof== Under the existential interpretation of Peirce's [[logical graphs]], Peirce's law is represented by means of the following formal equivalence or logical equation. {| align="center" border="0" cellpadding="10" cellspacing="0" | [[Image:Peirce's Law 1.0 Splash Page.png|500px]] || (1) |} '''Proof.''' Using the axiom set given in the entry for [[logical graphs]], Peirce's law may be proved in the following manner. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Peirce's Law 1.0 Marquee Title.png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 1.png|500px]] |- | [[Image:Equational Inference Band Collect p.png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 2.png|500px]] |- | [[Image:Equational Inference Band Quit ((q)).png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 3.png|500px]] |- | [[Image:Equational Inference Band Cancel (( )).png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 4.png|500px]] |- | [[Image:Equational Inference Band Delete p.png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 5.png|500px]] |- | [[Image:Equational Inference Band Cancel (( )).png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 6.png|500px]] |- | [[Image:Equational Inference Marquee QED.png|500px]] |} | (2) |} The following animation replays the steps of the proof. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Peirce's Law 2.0 Animation.gif]] |} | (3) |} ==Equational form== A stronger form of Peirce's law also holds, in which the final implication is observed to be reversible: {| align="center" cellpadding="10" | <math>((p \Rightarrow q) \Rightarrow p) \Leftrightarrow p</math> |} ===Proof 1=== Given what precedes, it remains to show that: {| align="center" cellpadding="10" | <math>p \Rightarrow ((p \Rightarrow q) \Rightarrow p)</math> |} But this is immediate, since <math>p \Rightarrow (r \Rightarrow p)</math> for any proposition <math>r.\!</math> ===Proof 2=== Representing propositions as logical graphs under the existential interpretation, the strong form of Peirce's law is expressed by the following equation: {| align="center" border="0" cellpadding="10" cellspacing="0" | [[Image:Peirce's Law Strong Form 1.0 Splash Page.png|500px]] || (4) |} Using the axioms and theorems listed in the article on [[logical graphs]], the equational form of Peirce's law may be proved in the following manner: {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Peirce's Law Strong Form 1.0 Marquee Title.png|500px]] |- | [[Image:Peirce's Law Strong Form 1.0 Storyboard 1.png|500px]] |- | [[Image:Equational Inference Rule Collect p.png|500px]] |- | [[Image:Peirce's Law Strong Form 1.0 Storyboard 2.png|500px]] |- | [[Image:Equational Inference Rule Quit ((q)).png|500px]] |- | [[Image:Peirce's Law Strong Form 1.0 Storyboard 3.png|500px]] |- | [[Image:Equational Inference Rule Cancel (( )).png|500px]] |- | [[Image:Peirce's Law Strong Form 1.0 Storyboard 4.png|500px]] |- | [[Image:Equational Inference Marquee QED.png|500px]] |} | (5) |} The following animation replays the steps of the proof. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Peirce's Law Strong Form 2.0 Animation.gif]] |} | (6) |} ==Bibliography== * [[Charles Sanders Peirce|Peirce, Charles Sanders]] (1885), "On the Algebra of Logic : A Contribution to the Philosophy of Notation", ''American Journal of Mathematics'' 7 (1885), 180&ndash;202. Reprinted (CP&nbsp;3.359&ndash;403), (CE&nbsp;5, 162&ndash;190). * Peirce, Charles Sanders (1931&ndash;1935, 1958), ''Collected Papers of Charles Sanders Peirce'', vols. 1&ndash;6, Charles Hartshorne and Paul Weiss (eds.), vols. 7&ndash;8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA. Cited as (CP&nbsp;volume.paragraph). * Peirce, Charles Sanders (1981&ndash;), ''Writings of Charles S. Peirce : A Chronological Edition'', Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN. Cited as (CE&nbsp;volume,&nbsp;page). ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Peirce's_law Peirce's Law @ InterSciWiki] * [http://mywikibiz.com/Peirce's_law Peirce's Law @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Peirce's_law Peirce's Law @ Subject Wikis] * [http://en.wikiversity.org/wiki/Peirce's_law Peirce's Law @ Wikiversity] * [http://beta.wikiversity.org/wiki/Peirce's_law Peirce's Law @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Peirce's_law Peirce's Law], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Peirce's_law Peirce's Law], [http://mywikibiz.com/ MyWikiBiz] * [http://planetmath.org/PeircesLaw Peirce's Law], [http://planetmath.org/ PlanetMath] * [http://wikinfo.org/w/index.php/Peirce's_law Peirce's Law], [http://www.wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Peirce's_law Peirce's Law], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Peirce's_law Peirce's Law], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Peirce%27s_law&oldid=60606482 Peirce's Law], [http://en.wikipedia.org/ Wikipedia] [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Artificial Intelligence]] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Cybernetics]] [[Category:Equational Reasoning]] [[Category:Formal Languages]] [[Category:Formal Systems]] [[Category:Graph Theory]] [[Category:History of Logic]] [[Category:History of Mathematics]] [[Category:Knowledge Representation]] [[Category:Logic]] [[Category:Logical Graphs]] [[Category:Mathematics]] [[Category:Philosophy]] [[Category:Semiotics]] [[Category:Visualization]] cc45f3ccc1aa2cde618365930770378e91e1903c 0 374 729 2015-11-07T16:02:26Z Jon Awbrey 3 add article wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''propositional calculus''' (or a '''sentential calculus''') is a formal system that represents the materials and the principles of ''propositional logic'' (or ''sentential logic''). Propositional logic is a domain of formal subject matter that is, up to somorphism, constituted by the structural relationships of mathematical objects called ''propositions''. In general terms, a calculus is a formal system that consists of a set of syntactic expressions (''well-formed formulas'' or ''wffs''), a distinguished subset of these expressions, plus a set of transformation rules that define a binary relation on the space of expressions. When the expressions are interpreted for mathematical purposes, the transformation rules are typically intended to preserve some type of semantic equivalence relation among the expressions. In particular, when the expressions are interpreted as a logical system, the semantic equivalence is typically intended to be logical equivalence. In this setting, the transformation rules can be used to derive logically equivalent expressions from any given expression. These derivations include as special cases (1) the problem of ''simplifying'' expressions and (2) the problem of deciding whether a given expression is equivalent to an expression in the distinguished subset, typically interpreted as the subset of logical ''axioms''. The set of axioms may be empty, a nonempty finite set, a countably infinite set, or given by axiom schemata. A formal grammar recursively defines the expressions and well-formed formulas (wffs) of the language. In addition a semantics is given which defines truth and valuations (or interpretations). It allows us to determine which wffs are valid, that is, are theorems. The language of a propositional calculus consists of (1) a set of primitive symbols, variously referred to as ''atomic formulas'', ''placeholders'', ''proposition letters'', or ''variables'', and (2) a set of operator symbols, variously interpreted as ''logical operators'' or ''logical connectives''. A ''well-formed formula'' (''wff'') is any atomic formula or any formula that can be built up from atomic formulas by means of operator symbols. The following outlines a standard propositional calculus. Many different formulations exist which are all more or less equivalent but differ in (1) their language, that is, the particular collection of primitive symbols and operator symbols, (2) the set of axioms, or distingushed formulas, and (3) the set of transformation rules that are available. ==Abstraction and application== Although it is possible to construct an abstract formal calculus that has no immediate practical use and next to nothing in the way of obvious applications, the very name ''calculus'' indicates that this species of formal system owes its origin to the utility of its prototypical members in practical calculation. Generally speaking, any mathematical calculus is designed with the intention of representing a given domain of formal objects, and typically with the aim of facilitating the computations and inferences that need to be carried out in this representation. Thus some idea of the intended denotation, the formal objects that the formulas of the calculus are intended to denote, is given in advance of developing the calculus itself. Viewed over the course of its historical development, a formal calculus for any given subject matter normally arises through a process of gradual abstraction, stepwise refinement, and trial-and-error synthesis from the array of informal notational systems that inform prior use, each of which covers the object domain only in part or from a particular angle. ==Generic description of a propositional calculus== A '''propositional calculus''' is a formal system <math>\mathcal{L} = \mathcal{L}\ (\Alpha,\ \Omega,\ \Zeta,\ \Iota)</math>, whose formulas are constructed in the following manner: * The ''alpha set'' <math>\Alpha\!</math> is a finite set of elements called ''proposition symbols'' or ''propositional variables''. Syntactically speaking, these are the most basic elements of the formal language <math>\mathcal{L},</math> otherwise referred to as ''atomic formulas'' or ''terminal elements''. In the examples to follow, the elements of <math>\Alpha\!</math> are typically the letters ''p'', ''q'', ''r'', and so on. * The ''omega set'' <math>\Omega\!</math> is a finite set of elements called ''operator symbols'' or ''logical connectives''. The set <math>\Omega\!</math> is partitioned into disjoint subsets as follows: ::: <math>\Omega = \Omega_0 \cup \Omega_1 \cup \ldots \cup \Omega_j \cup \ldots \cup \Omega_m.\!</math> : In this partition, <math>\Omega_j\!</math> is the set of operator symbols of ''arity'' <math>j.\!</math> : In the more familiar propositional calculi, <math>\Omega\!</math> is typically partitioned as follows: ::: <math>\Omega_1 = \{ \lnot \},\!</math> ::: <math>\Omega_2 \subseteq \{ \land, \lor, \rightarrow, \leftrightarrow \}.\!</math> : A frequently adopted option treats the constant logical values as operators of arity zero, thus: ::: <math>\Omega_0 = \{ 0, 1 \}.\!</math> : Some writers use the tilde (~) instead of (¬) and some use the ampersand (&) instead of (&#8743;). Notation varies even more for the set of logical values, with symbols like {false, true}, {F, T}, {0, 1}, and {<math>\bot</math>, <math>\top</math>} all being seen in various contexts. * Depending on the precise formal grammar or the grammar formalism that is being used, syntactic auxiliaries like the left parenthesis, "(", and the right parentheses, ")", may be necessary to complete the construction of formulas. The ''language'' of <math>\mathcal{L}</math>, also known as its set of ''formulas'', ''well-formed formulas'' or ''wffs'', is inductively or recursively defined by the following rules: # Base. Any element of the alpha set <math>\Alpha\!</math> is a formula of <math>\mathcal{L}</math>. # Step (a). If ''p'' is a formula, then ¬''p'' is a formula. # Step (b). If ''p'' and ''q'' are formulas, then (''p'' &#8743; ''q''), (''p'' &#8744; ''q''), (''p'' &#8594; ''q''), and (''p'' &#8596; ''q'') are formulas. # Close. Nothing else is a formula of <math>\mathcal{L}</math>. Repeated applications of these rules permits the construction of complex formulas. For example: # By rule 1, ''p'' is a formula. # By rule 2, ¬''p'' is a formula. # By rule 1, ''q'' is a formula. # By rule 3, (¬''p'' &#8744; ''q'') is a formula. * The ''zeta set'' <math>\Zeta\!</math> is a finite set of ''transformation rules'' that are called ''inference rules'' when they acquire logical applications. * The ''iota set'' <math>\Iota\!</math> is a finite set of ''initial points'' that are called ''axioms'' when they receive logical interpretations. ==Example 1. Simple axiom system== Let <math>\mathcal{L}_1 = \mathcal{L}\ (\Alpha,\ \Omega,\ \Zeta,\ \Iota)</math>, where <math>\Alpha,\ \Omega,\ \Zeta,\ \Iota</math> are defined as follows: * The alpha set <math>\Alpha \!</math> is a finite set of symbols that is large enough to supply the needs of a given discussion, for example: ::: <math>\Alpha = \{p, q, r, s, t, u \} \,.</math> Of the three connectives for conjunction, disjunction, and implication (&#8743;, &#8744;, and &#8594;), one can be taken as primitive and the other two can be defined in terms of it and negation (¬). Indeed, all of the logical connectives can be defined in terms of a [[sole sufficient operator]]. The biconditional (&#8596;) can of course be defined in terms of conjunction and implication, with a &#8596; b defined as (a &#8594; b) &#8743; (b &#8594; a). Adopting negation and implication as the two primitive operations of a propositional calculus is tantamount to having the omega set <math>\Omega = \Omega_1 \cup \Omega_2</math> partition as follows: ::: <math>\Omega_1 = \{ \lnot \} \,,</math> ::: <math>\Omega_2 = \{ \Rightarrow \} \,.</math> An axiom system discovered by [[Jan &#321;ukasiewicz|Jan Lukasiewicz]] formulates a propositional calculus in this language as follows: ::* <math>p \Rightarrow (q \Rightarrow p)</math> ::* <math>(p \Rightarrow (q \Rightarrow r)) \Rightarrow ((p \Rightarrow q) \Rightarrow (p \Rightarrow r))</math> ::* <math>(\neg p \Rightarrow \neg q) \Rightarrow (q \Rightarrow p)</math> The inference rule is ''[[modus ponens]]'': ::* From ''p'', (''p'' &rArr; ''q''), infer ''q''. Then we have the following definitions: ::* ''p'' &or; ''q'' is defined as &not;''p'' &rArr; ''q''. ::* ''p'' &and; ''q'' is defined as &not;(''p'' &rArr; &not;''q''). ==Example 2. Natural deduction system== Let <math>\mathcal{L}_2 = \mathcal{L}\ (\Alpha,\ \Omega,\ \Zeta,\ \Iota)</math>, where <math>\Alpha,\ \Omega,\ \Zeta,\ \Iota</math> are defined as follows: * The alpha set <math>\Alpha \!</math> is a finite set of symbols that is large enough to supply the needs of a given discussion, for example: ::: <math>\Alpha = \{p, q, r, s, t, u \} \,.</math> * The omega set <math>\Omega = \Omega_1 \cup \Omega_2</math> partitions as follows: ::: <math>\Omega_1 = \{ \lnot \} \,,</math> ::: <math>\Omega_2 = \{ \land, \lor, \rightarrow, \leftrightarrow \} \,.</math> In the following example of a propositional calculus, the transformation rules are intended to be interpreted as the inference rules of a so-called ''[[natural deduction system]]''. The particular system presented here has no initial points, which means that its interpretation for logical applications derives its [[theorem]]s from an empty axiom set. * The set of initial points is empty, that is, <math>\Iota = \varnothing \,.</math> * The set of transformation rules, <math>\Zeta ,\!</math>, is described as follows: ==Graphical calculi== : ''Main article'' : [[Logical graph]] It is possible to generalize the definition of a formal language from a set of finite sequences over a finite basis to include many other sets of mathematical structures, so long as they are built up by finitary means from finite materials. What's more, many of these families of formal structures are especially well-suited for use in logic. For example, there are many families of graphs that are close enough analogues of formal languages that the concept of a calculus is quite easily and naturally extended to them. Indeed, many species of graphs arise as ''parse graphs'' in the syntactic analysis of the corresponding families of text structures. The exigencies of practical computation on formal languages frequently demand that text strings be converted into pointer structure renditions of parse graphs, simply as a matter of checking whether strings are wffs or not. Once this is done, there are many advantages to be gained from developing the graphical analogue of the calculus on strings. The mapping from strings to parse graphs is called ''parsing'' and the inverse mapping from parse graphs to strings is achieved by an operation that is called ''traversing'' the graph. ==References== * Brown, Frank Markham (2003), ''Boolean Reasoning: The Logic of Boolean Equations'', 1st edition, Kluwer Academic Publishers, Norwell, MA. 2nd edition, Dover Publications, Mineola, NY, 2003. * Chang, C.C., and Keisler, H.J. (1973), ''Model Theory'', North-Holland, Amsterdam, Netherlands. * Kohavi, Zvi (1978), ''Switching and Finite Automata Theory'', 1st edition, McGraw–Hill, 1970. 2nd edition, McGraw–Hill, 1978. * Korfhage, Robert R. (1974), ''Discrete Computational Structures'', Academic Press, New York, NY. * Lambek, J., and Scott, P.J. (1986), ''Introduction to Higher Order Categorical Logic'', Cambridge University Press, Cambridge, UK. * Mendelson, Elliot (1964), ''Introduction to Mathematical Logic'', D. Van Nostrand Company. ==Resources== * Klement, Kevin C. (2006), &ldquo;Propositional Logic&rdquo;, in James Fieser and Bradley Dowden (eds.), ''Internet Encyclopedia of Philosophy''. [http://www.iep.utm.edu/p/prop-log.htm Online]. * Magnus, P.D. ''[http://www.fecundity.com/logic/ Forall x : An Introduction to Formal Logic]''. * [http://www.visualstatistics.net/Scaling/Propositional%20Calculus/Elements%20of%20Propositional%20Calculus.htm Elements of Propositional Calculus]. * [http://www.ltn.lv/~podnieks/mlog/ml2.htm Introduction to Mathematical Logic]. ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_calculus Propositional Calculus @ InterSciWiki] * [http://mywikibiz.com/Propositional_calculus Propositional Calculus @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Propositional_calculus Propositional Calculus @ Subject Wikis] * [http://en.wikiversity.org/wiki/Propositional_calculus Propositional Calculus @ Wikiversity] * [http://beta.wikiversity.org/wiki/Propositional_calculus Propositional Calculus @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_calculus Propositional Calculus], [http://intersci.ss.uci.edu/wiki/index.php/Main_Page InterSciWiki] * [http://mywikibiz.com/Propositional_calculus Propositional Calculus], [http://mywikibiz.com/ MyWikiBiz] * [http://planetmath.org/PropositionalCalculus Propositional Calculus], [http://planetmath.org/ PlanetMath] * [http://wikinfo.org/w/index.php/Propositional_calculus Propositional Calculus], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Propositional_calculus Propositional Calculus], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Propositional_calculus Propositional Calculus], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Propositional_calculus&oldid=77110794 Propositional Calculus], [http://en.wikipedia.org/ Wikipedia] [[Category:Charles Sanders Peirce]] [[Category:Computer Science]] [[Category:Formal Grammars]] [[Category:Formal Languages]] [[Category:Formal Sciences]] [[Category:Formal Systems]] [[Category:Inquiry]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Model Theory]] [[Category:Normative Sciences]] [[Category:Pragmatics]] [[Category:Proof Theory]] [[Category:Semantics]] [[Category:Semiotics]] [[Category:Syntax]] 8c317b4925e5ff9cc0a1739f354f8d74a24b2cd7 0 340 730 562 2015-11-07T18:10:08Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''sole sufficient operator''' is an operator that is sufficient by itself to generate every operator in a specified class of operators.&nbsp; In the context of [[logic]], it is a logical operator that suffices to generate every [[boolean-valued function]], <math>f : X \to \mathbb{B},\!</math> where <math>X\!</math> is an arbitrary set and where <math>\mathbb{B}\!</math> is a generic two-element set, typically <math>\mathbb{B} = \{ 0, 1 \} = \{ \mathrm{false}, \mathrm{true} \},\!</math> in particular, to generate every finitary [[boolean function]], <math>f : \mathbb{B}^k \to \mathbb{B}.\!</math> ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Sole_sufficient_operator Sole Sufficient Operator @ InterSciWiki] * [http://mywikibiz.com/Sole_sufficient_operator Sole Sufficient Operator @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Sole_sufficient_operator Sole Sufficient Operator @ Subject Wikis] * [http://en.wikiversity.org/wiki/Sole_sufficient_operator Sole Sufficient Operator @ Wikiversity] * [http://beta.wikiversity.org/wiki/Sole_sufficient_operator Sole Sufficient Operator @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Sole_sufficient_operator Sole Sufficient Operator], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Sole_sufficient_operator Sole Sufficient Operator], [http://mywikibiz.com/ MyWikiBiz] * [http://http://planetmath.org/SoleSufficientOperator Sole Sufficient Operator], [http://planetmath.org/ PlanetMath] * [http://semanticweb.org/wiki/Sole_sufficient_operator Sole Sufficient Operator], [http://semanticweb.org/ SemanticWeb] * [http://wikinfo.org/w/index.php/Sole_sufficient_operator Sole Sufficient Operator], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Sole_sufficient_operator Sole Sufficient Operator], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Sole_sufficient_operator Sole Sufficient Operator], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Sole_sufficient_operator&oldid=156136346 Sole Sufficient Operator], [http://en.wikipedia.org/ Wikipedia] [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Semiotics]] c1222e181b55fe66e12439915fe2208eb5f1777a 0 338 731 560 2015-11-07T18:31:02Z Jon Awbrey 3 wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''truth table''' is a tabular array that illustrates the computation of a ''logical function'', that is, a function of the form <math>f : \mathbb{A}^k \to \mathbb{A},</math> where <math>k\!</math> is a non-negative integer and <math>\mathbb{A}</math> is the domain of logical values <math>\{ \operatorname{false}, \operatorname{true} \}.</math> The names of the logical values, or ''truth values'', are commonly abbreviated in accord with the equations <math>\operatorname{F} = \operatorname{false}</math> and <math>\operatorname{T} = \operatorname{true}.</math> In many applications it is usual to represent a truth function by a [[boolean function]], that is, a function of the form <math>f : \mathbb{B}^k \to \mathbb{B},</math> where <math>k\!</math> is a non-negative integer and <math>\mathbb{B}</math> is the [[boolean domain]] <math>\{ 0, 1 \}.\!</math> In most applications <math>\operatorname{false}</math> is represented by <math>0\!</math> and <math>\operatorname{true}</math> is represented by <math>1\!</math> but the opposite representation is also possible, depending on the overall representation of truth functions as boolean functions. The remainder of this article assumes the usual representation, taking the equations <math>\operatorname{F} = 0</math> and <math>\operatorname{T} = 1</math> for granted. ==Logical negation== '''[[Logical negation]]''' is an operation on one logical value, typically the value of a proposition, that produces a value of ''true'' when its operand is false and a value of ''false'' when its operand is true. The truth table of <math>\operatorname{NOT}~ p,</math> also written <math>\lnot p,\!</math> appears below: <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:45%" |+ style="height:30px" | <math>\text{Logical Negation}\!</math> |- style="height:40px; background:#f0f0ff" | style="width:50%" | <math>p\!</math> | style="width:50%" | <math>\lnot p\!</math> |- | <math>\operatorname{F}</math> || <math>\operatorname{T}</math> |- | <math>\operatorname{T}</math> || <math>\operatorname{F}</math> |} <br> The negation of a proposition <math>p\!</math> may be found notated in various ways in various contexts of application, often merely for typographical convenience. Among these variants are the following: <br> {| align="center" border="1" cellpadding="8" cellspacing="0" width="45%" |+ style="height:30px" | <math>\text{Variant Notations}\!</math> |- style="height:40px; background:#f0f0ff" | width="50%" align="center" | <math>\text{Notation}\!</math> | width="50%" | <math>\text{Vocalization}\!</math> |- | align="center" | <math>\bar{p}\!</math> | <math>p\!</math> bar |- | align="center" | <math>\tilde{p}\!</math> | <math>p\!</math> tilde |- | align="center" | <math>p'\!</math> | <math>p\!</math> prime<br> <math>p\!</math> complement |- | align="center" | <math>!p\!</math> | bang <math>p\!</math> |} <br> ==Logical conjunction== '''[[Logical conjunction]]''' is an operation on two logical values, typically the values of two propositions, that produces a value of ''true'' if and only if both of its operands are true. The truth table of <math>p ~\operatorname{AND}~ q,</math> also written <math>p \land q\!</math> or <math>p \cdot q,\!</math> appears below: <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:45%" |+ style="height:30px" | <math>\text{Logical Conjunction}\!</math> |- style="height:40px; background:#f0f0ff" | style="width:33%" | <math>p\!</math> | style="width:33%" | <math>q\!</math> | style="width:33%" | <math>p \land q</math> |- | <math>\operatorname{F}</math> || <math>\operatorname{F}</math> || <math>\operatorname{F}</math> |- | <math>\operatorname{F}</math> || <math>\operatorname{T}</math> || <math>\operatorname{F}</math> |- | <math>\operatorname{T}</math> || <math>\operatorname{F}</math> || <math>\operatorname{F}</math> |- | <math>\operatorname{T}</math> || <math>\operatorname{T}</math> || <math>\operatorname{T}</math> |} <br> ==Logical disjunction== '''[[Logical disjunction]]''', also called '''logical alternation''', is an operation on two logical values, typically the values of two propositions, that produces a value of ''false'' if and only if both of its operands are false. The truth table of <math>p ~\operatorname{OR}~ q,</math> also written <math>p \lor q,\!</math> appears below: <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:45%" |+ style="height:30px" | <math>\text{Logical Disjunction}\!</math> |- style="height:40px; background:#f0f0ff" | style="width:33%" | <math>p\!</math> | style="width:33%" | <math>q\!</math> | style="width:33%" | <math>p \lor q</math> |- | <math>\operatorname{F}</math> || <math>\operatorname{F}</math> || <math>\operatorname{F}</math> |- | <math>\operatorname{F}</math> || <math>\operatorname{T}</math> || <math>\operatorname{T}</math> |- | <math>\operatorname{T}</math> || <math>\operatorname{F}</math> || <math>\operatorname{T}</math> |- | <math>\operatorname{T}</math> || <math>\operatorname{T}</math> || <math>\operatorname{T}</math> |} <br> ==Logical equality== '''[[Logical equality]]''' is an operation on two logical values, typically the values of two propositions, that produces a value of ''true'' if and only if both operands are false or both operands are true. The truth table of <math>p ~\operatorname{EQ}~ q,</math> also written <math>p = q,\!</math> <math>p \Leftrightarrow q,\!</math> or <math>p \equiv q,\!</math> appears below: <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:45%" |+ style="height:30px" | <math>\text{Logical Equality}\!</math> |- style="height:40px; background:#f0f0ff" | style="width:33%" | <math>p\!</math> | style="width:33%" | <math>q\!</math> | style="width:33%" | <math>p = q\!</math> |- | <math>\operatorname{F}</math> || <math>\operatorname{F}</math> || <math>\operatorname{T}</math> |- | <math>\operatorname{F}</math> || <math>\operatorname{T}</math> || <math>\operatorname{F}</math> |- | <math>\operatorname{T}</math> || <math>\operatorname{F}</math> || <math>\operatorname{F}</math> |- | <math>\operatorname{T}</math> || <math>\operatorname{T}</math> || <math>\operatorname{T}</math> |} <br> ==Exclusive disjunction== '''[[Exclusive disjunction]]''', also known as '''logical inequality''' or '''symmetric difference''', is an operation on two logical values, typically the values of two propositions, that produces a value of ''true'' just in case exactly one of its operands is true. The truth table of <math>p ~\operatorname{XOR}~ q,</math> also written <math>p + q\!</math> or <math>p \ne q,\!</math> appears below: <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:45%" |+ style="height:30px" | <math>\text{Exclusive Disjunction}\!</math> |- style="height:40px; background:#f0f0ff" | style="width:33%" | <math>p\!</math> | style="width:33%" | <math>q\!</math> | style="width:33%" | <math>p ~\operatorname{XOR}~ q</math> |- | <math>\operatorname{F}</math> || <math>\operatorname{F}</math> || <math>\operatorname{F}</math> |- | <math>\operatorname{F}</math> || <math>\operatorname{T}</math> || <math>\operatorname{T}</math> |- | <math>\operatorname{T}</math> || <math>\operatorname{F}</math> || <math>\operatorname{T}</math> |- | <math>\operatorname{T}</math> || <math>\operatorname{T}</math> || <math>\operatorname{F}</math> |} <br> The following equivalents may then be deduced: {| align="center" cellspacing="10" width="90%" | <math>\begin{matrix} p + q & = & (p \land \lnot q) & \lor & (\lnot p \land q) \\[6pt] & = & (p \lor q) & \land & (\lnot p \lor \lnot q) \\[6pt] & = & (p \lor q) & \land & \lnot (p \land q) \end{matrix}</math> |} ==Logical implication== The '''[[logical implication]]''' relation and the '''material conditional''' function are both associated with an operation on two logical values, typically the values of two propositions, that produces a value of ''false'' if and only if the first operand is true and the second operand is false. The truth table associated with the material conditional <math>\text{if}~ p ~\text{then}~ q,\!</math> symbolized <math>p \rightarrow q,\!</math> and the logical implication <math>p ~\text{implies}~ q,\!</math> symbolized <math>p \Rightarrow q,\!</math> appears below: <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:45%" |+ style="height:30px" | <math>\text{Logical Implication}\!</math> |- style="height:40px; background:#f0f0ff" | style="width:33%" | <math>p\!</math> | style="width:33%" | <math>q\!</math> | style="width:33%" | <math>p \Rightarrow q\!</math> |- | <math>\operatorname{F}</math> || <math>\operatorname{F}</math> || <math>\operatorname{T}</math> |- | <math>\operatorname{F}</math> || <math>\operatorname{T}</math> || <math>\operatorname{T}</math> |- | <math>\operatorname{T}</math> || <math>\operatorname{F}</math> || <math>\operatorname{F}</math> |- | <math>\operatorname{T}</math> || <math>\operatorname{T}</math> || <math>\operatorname{T}</math> |} <br> ==Logical NAND== The '''[[logical NAND]]''' is an operation on two logical values, typically the values of two propositions, that produces a value of ''false'' if and only if both of its operands are true. In other words, it produces a value of ''true'' if and only if at least one of its operands is false. The truth table of <math>p ~\operatorname{NAND}~ q,</math> also written <math>p \stackrel{\circ}{\curlywedge} q\!</math> or <math>p \barwedge q,\!</math> appears below: <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:45%" |+ style="height:30px" | <math>\text{Logical NAND}\!</math> |- style="height:40px; background:#f0f0ff" | style="width:33%" | <math>p\!</math> | style="width:33%" | <math>q\!</math> | style="width:33%" | <math>p \stackrel{\circ}{\curlywedge} q\!</math> |- | <math>\operatorname{F}</math> || <math>\operatorname{F}</math> || <math>\operatorname{T}</math> |- | <math>\operatorname{F}</math> || <math>\operatorname{T}</math> || <math>\operatorname{T}</math> |- | <math>\operatorname{T}</math> || <math>\operatorname{F}</math> || <math>\operatorname{T}</math> |- | <math>\operatorname{T}</math> || <math>\operatorname{T}</math> || <math>\operatorname{F}</math> |} <br> ==Logical NNOR== The '''[[logical NNOR]]''' (&ldquo;Neither Nor&rdquo;) is an operation on two logical values, typically the values of two propositions, that produces a value of ''true'' if and only if both of its operands are false. In other words, it produces a value of ''false'' if and only if at least one of its operands is true. The truth table of <math>p ~\operatorname{NNOR}~ q,</math> also written <math>p \curlywedge q,\!</math> appears below: <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:45%" |+ style="height:30px" | <math>\text{Logical NNOR}\!</math> |- style="height:40px; background:#f0f0ff" | style="width:33%" | <math>p\!</math> | style="width:33%" | <math>q\!</math> | style="width:33%" | <math>p \curlywedge q\!</math> |- | <math>\operatorname{F}</math> || <math>\operatorname{F}</math> || <math>\operatorname{T}</math> |- | <math>\operatorname{F}</math> || <math>\operatorname{T}</math> || <math>\operatorname{F}</math> |- | <math>\operatorname{T}</math> || <math>\operatorname{F}</math> || <math>\operatorname{F}</math> |- | <math>\operatorname{T}</math> || <math>\operatorname{T}</math> || <math>\operatorname{F}</math> |} <br> ==Translations== * [http://zh.wikipedia.org/wiki/%E7%9C%9F%E5%80%BC%E8%A1%A8 &#20013;&#25991; : &#30495;&#20540;&#34920;] ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Truth_table Truth Table @ InterSciWiki] * [http://mywikibiz.com/Truth_table Truth Table @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Truth_table Truth Table @ Subject Wikis] * [http://en.wikiversity.org/wiki/Truth_table Truth Table @ Wikiversity] * [http://beta.wikiversity.org/wiki/Truth_table Truth Table @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Truth_table Truth Table], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Truth_table Truth Table], [http://mywikibiz.com/ MyWikiBiz] * [http://semanticweb.org/wiki/Truth_table Truth Table], [http://semanticweb.org/ SemanticWeb] * [http://wikinfo.org/w/index.php/Truth_table Truth Table], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Truth_table Truth Table], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Truth_table Truth Table], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Truth_table&oldid=77110085 Truth Table], [http://en.wikipedia.org/ Wikipedia] [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Charles Sanders Peirce]] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Discrete Mathematics]] [[Category:Formal Languages]] [[Category:Formal Sciences]] [[Category:Formal Systems]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Philosophy]] [[Category:Semiotics]] 1d7765993e339aa5ae838e3b985daa4f12eb68f6 0 339 732 561 2015-11-07T19:14:55Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. The term '''''universe of discourse''''' is generally attributed to Augustus De&nbsp;Morgan (1846).&nbsp; George Boole (1854) defines it in the following manner: {| align="center" cellpadding="4" width="90%" | <p>In every discourse, whether of the mind conversing with its own thoughts, or of the individual in his intercourse with others, there is an assumed or expressed limit within which the subjects of its operation are confined. &hellip; Now, whatever may be the extent of the field within which all the objects of our discourse are found, that field may properly be termed the universe of discourse. (Boole 1854/1958, p.&nbsp;42).</p> |} ==References== * Boole, George (1854/1958), ''An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities'', Macmillan Publishers, 1854. Reprinted with corrections, Dover Publications, New York, NY, 1958. * De Morgan, Augustus (1846), ''Cambridge Philosophical Transactions'', ''viii'', p. 380. ==Resources== * [http://www.helsinki.fi/science/commens/dictionary.html Commens Dictionary of Peirce's Terms] ** [http://www.helsinki.fi/science/commens/terms/logicaluniv.html Logical Universe] ** [http://www.helsinki.fi/science/commens/terms/universedisc.html Universe of Discourse] ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Universe_of_discourse Universe of Discourse @ InterSciWiki] * [http://mywikibiz.com/Universe_of_discourse Universe of Discourse @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Universe_of_discourse Universe of Discourse @ Subject Wikis] * [http://en.wikiversity.org/wiki/Universe_of_discourse Universe of Discourse @ Wikiversity] * [http://beta.wikiversity.org/wiki/Universe_of_discourse Universe of Discourse @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Universe_of_discourse Universe of Discourse], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Universe_of_discourse Universe of Discourse], [http://mywikibiz.com/ MyWikiBiz] * [http://planetmath.org/UniverseOfDiscourse Universe of Discourse], [http://planetmath.org/ PlanetMath] * [http://semanticweb.org/wiki/Universe_of_discourse Universe of Discourse], [http://semanticweb.org/ SemanticWeb] * [http://en.wikiversity.org/wiki/Universe_of_discourse Universe of Discourse], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Universe_of_discourse Universe of Discourse], [http://beta.wikiversity.org/ Wikiversity Beta] [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Computer Science]] [[Category:Formal Grammars]] [[Category:Formal Languages]] [[Category:Formal Sciences]] [[Category:Formal Systems]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Model Theory]] [[Category:Normative Sciences]] [[Category:Pragmatics]] [[Category:Proof Theory]] [[Category:Semantics]] [[Category:Semiotics]] [[Category:Syntax]] ef6fd172e16643ac57d92c7a02028de4f8d1d3a1 0 341 733 563 2015-11-08T03:44:42Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. '''''Zeroth order logic''''' is an informal term that is sometimes used to indicate the common principles underlying the algebra of sets, boolean algebra, [[boolean functions]], logical connectives, monadic predicate calculus, [[propositional calculus]], and sentential logic.&nbsp; The term serves to mark a level of abstraction in which the more inessential differences among these subjects can be subsumed under the appropriate isomorphisms. ==Propositional forms on two variables== By way of initial orientation, Table 1 lists equivalent expressions for the sixteen functions of concrete type <math>X \times Y \to \mathbb{B}</math> and abstract type <math>\mathbb{B} \times \mathbb{B} \to \mathbb{B}</math> in a number of different languages for zeroth order logic. <br> {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%" |+ '''Table 1. Propositional Forms on Two Variables''' |- style="background:#e6e6ff" ! style="width:15%" | L<sub>1</sub> ! style="width:15%" | L<sub>2</sub> ! style="width:15%" | L<sub>3</sub> ! style="width:15%" | L<sub>4</sub> ! style="width:15%" | L<sub>5</sub> ! style="width:15%" | L<sub>6</sub> |- style="background:#e6e6ff" | &nbsp; | align="right" | x : | 1 1 0 0 | &nbsp; | &nbsp; | &nbsp; |- style="background:#e6e6ff" | &nbsp; | align="right" | y : | 1 0 1 0 | &nbsp; | &nbsp; | &nbsp; |- | f<sub>0</sub> || f<sub>0000</sub> || 0 0 0 0 || ( ) || false || 0 |- | f<sub>1</sub> || f<sub>0001</sub> || 0 0 0 1 || (x)(y) || neither x nor y || ¬x &and; ¬y |- | f<sub>2</sub> || f<sub>0010</sub> || 0 0 1 0 || (x) y || y and not x || ¬x &and; y |- | f<sub>3</sub> || f<sub>0011</sub> || 0 0 1 1 || (x) || not x || ¬x |- | f<sub>4</sub> || f<sub>0100</sub> || 0 1 0 0 || x (y) || x and not y || x &and; ¬y |- | f<sub>5</sub> || f<sub>0101</sub> || 0 1 0 1 || (y) || not y || ¬y |- | f<sub>6</sub> || f<sub>0110</sub> || 0 1 1 0 || (x, y) || x not equal to y || x &ne; y |- | f<sub>7</sub> || f<sub>0111</sub> || 0 1 1 1 || (x y) || not both x and y || ¬x &or; ¬y |- | f<sub>8</sub> || f<sub>1000</sub> || 1 0 0 0 || x y || x and y || x &and; y |- | f<sub>9</sub> || f<sub>1001</sub> || 1 0 0 1 || ((x, y)) || x equal to y || x = y |- | f<sub>10</sub> || f<sub>1010</sub> || 1 0 1 0 || y || y || y |- | f<sub>11</sub> || f<sub>1011</sub> || 1 0 1 1 || (x (y)) || not x without y || x &rarr; y |- | f<sub>12</sub> || f<sub>1100</sub> || 1 1 0 0 || x || x || x |- | f<sub>13</sub> || f<sub>1101</sub> || 1 1 0 1 || ((x) y) || not y without x || x &larr; y |- | f<sub>14</sub> || f<sub>1110</sub> || 1 1 1 0 || ((x)(y)) || x or y || x &or; y |- | f<sub>15</sub> || f<sub>1111</sub> || 1 1 1 1 || (( )) || true || 1 |} <br> These six languages for the sixteen boolean functions are conveniently described in the following order: * Language '''L<sub>3</sub>''' describes each boolean function ''f'' : '''B'''<sup>2</sup> &#8594; '''B''' by means of the sequence of four boolean values (''f''(1,1), ''f''(1,0), ''f''(0,1), ''f''(0,0)). Such a sequence, perhaps in another order, and perhaps with the logical values ''F'' and ''T'' instead of the boolean values 0 and 1, respectively, would normally be displayed as a column in a [[truth table]]. * Language '''L<sub>2</sub>''' lists the sixteen functions in the form '''f<sub>i</sub>''', where the index '''i''' is a [[bit string]] formed from the sequence of boolean values in '''L<sub>3</sub>'''. * Language '''L<sub>1</sub>''' notates the boolean functions '''f<sub>i</sub>''' with an index '''i''' that is the decimal equivalent of the binary numeral index in '''L<sub>2</sub>'''. * Language '''L<sub>4</sub>''' expresses the sixteen functions in terms of logical [[conjunction]], indicated by concatenating function names or proposition expressions in the manner of products, plus the family of ''[[minimal negation operator]]s'', the first few of which are given in the following variant notations: : <math>\begin{matrix} (\ ) & = & 0 & = & \mbox{false} \\ (x) & = & \tilde{x} & = & x' \\ (x, y) & = & \tilde{x}y \lor x\tilde{y} & = & x'y \lor xy' \\ (x, y, z) & = & \tilde{x}yz \lor x\tilde{y}z \lor xy\tilde{z} & = & x'yz \lor xy'z \lor xyz' \end{matrix}</math> It may also be noted that <math>(x, y)\!</math> is the same function as <math>x + y\!</math> and <math>x \ne y</math>, and that the inclusive disjunctions indicated for <math>(x, y)\!</math> and for <math>(x, y, z)\!</math> may be replaced with exclusive disjunctions without affecting the meaning, because the terms disjoined are already disjoint. However, the function <math>(x, y, z)\!</math> is not the same thing as the function <math>x + y + z\!</math>. * Language '''L<sub>5</sub>''' lists ordinary language expressions for the sixteen functions. Many other paraphrases are possible, but these afford a sample of the simplest equivalents. * Language '''L<sub>6</sub>''' expresses the sixteen functions in one of several notations that are commonly used in formal logic. ==Translations== * [http://zh.wikipedia.org/wiki/%E9%9B%B6%E9%98%B6%E9%80%BB%E8%BE%91 &#20013;&#25991; : &#38646;&#38454;&#36923;&#36753;] ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Zeroth_order_logic Zeroth Order Logic @ InterSciWiki] * [http://mywikibiz.com/Zeroth_order_logic Zeroth Order Logic @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Zeroth_order_logic Zeroth Order Logic @ Subject Wikis] * [http://en.wikiversity.org/wiki/Zeroth_order_logic Zeroth Order Logic @ Wikiversity] * [http://beta.wikiversity.org/wiki/Zeroth_order_logic Zeroth Order Logic @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Zeroth_order_logic Zeroth Order Logic], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Zeroth_order_logic Zeroth Order Logic], [http://mywikibiz.com/ MyWikiBiz] * [http://planetmath.org/ZerothOrderLogic Zeroth Order Logic], [http://planetmath.org/ PlanetMath] * [http://wikinfo.org/w/index.php/Zeroth_order_logic Zeroth Order Logic], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Zeroth_order_logic Zeroth Order Logic], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Zeroth_order_logic Zeroth Order Logic], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Zeroth-order_logic&oldid=77109225 Zeroth Order Logic], [http://en.wikipedia.org/ Wikipedia] * [http://web.archive.org/web/20050323065233/http://www.altheim.com/cs/zol.html Zeroth Order Logic], [http://web.archive.org/web/20070305032442/http://www.altheim.com/cs/ Altheim.com] [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Computer Science]] [[Category:Formal Languages]] [[Category:Formal Sciences]] [[Category:Formal Systems]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Normative Sciences]] d66e3ae28a939cb088aa6afd22309eb8b51cae28 734 733 2015-11-08T15:50:18Z Jon Awbrey 3 update with TeX table wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. '''''Zeroth order logic''''' is an informal term that is sometimes used to indicate the common principles underlying the algebra of sets, boolean algebra, [[boolean functions]], logical connectives, monadic predicate calculus, [[propositional calculus]], and sentential logic.&nbsp; The term serves to mark a level of abstraction in which the more inessential differences among these subjects can be subsumed under the appropriate isomorphisms. ==Propositional forms on two variables== By way of initial orientation, Table 1 lists equivalent expressions for the sixteen functions of concrete type <math>X \times Y \to \mathbb{B}</math> and abstract type <math>\mathbb{B} \times \mathbb{B} \to \mathbb{B}</math> in a number of different languages for zeroth order logic. <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%" |+ style="height:30px" | <math>\text{Table 1.} ~~ \text{Propositional Forms on Two Variables}\!</math> |- style="height:40px; background:ghostwhite" | style="width:15%" | <math>\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}\!</math> | style="width:15%" | <math>\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}\!</math> | style="width:15%" | <math>\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}\!</math> | style="width:15%" | <math>\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}\!</math> | style="width:25%" | <math>\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}\!</math> | style="width:15%" | <math>\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}\!</math> |- style="background:ghostwhite" | &nbsp; | align="right" | <math>x\colon\!</math> | <math>1~1~0~0\!</math> | &nbsp; | &nbsp; | &nbsp; |- style="background:ghostwhite" | &nbsp; | align="right" | <math>y\colon\!</math> | <math>1~0~1~0\!</math> | &nbsp; | &nbsp; | &nbsp; |- | valign="bottom" | <math>\begin{matrix} f_{0} \\[4pt] f_{1} \\[4pt] f_{2} \\[4pt] f_{3} \\[4pt] f_{4} \\[4pt] f_{5} \\[4pt] f_{6} \\[4pt] f_{7} \end{matrix}\!</math> | valign="bottom" | <math>\begin{matrix} f_{0000} \\[4pt] f_{0001} \\[4pt] f_{0010} \\[4pt] f_{0011} \\[4pt] f_{0100} \\[4pt] f_{0101} \\[4pt] f_{0110} \\[4pt] f_{0111} \end{matrix}\!</math> | valign="bottom" | <math>\begin{matrix} 0~0~0~0 \\[4pt] 0~0~0~1 \\[4pt] 0~0~1~0 \\[4pt] 0~0~1~1 \\[4pt] 0~1~0~0 \\[4pt] 0~1~0~1 \\[4pt] 0~1~1~0 \\[4pt] 0~1~1~1 \end{matrix}\!</math> | valign="bottom" | <math>\begin{matrix} \texttt{(~)} \\[4pt] \texttt{(} x \texttt{)(} y \texttt{)} \\[4pt] \texttt{(} x \texttt{)} ~ y ~ \\[4pt] \texttt{(} x \texttt{)} \\[4pt] ~ x ~ \texttt{(} y \texttt{)} \\[4pt] \texttt{(} y \texttt{)} \\[4pt] \texttt{(} x \texttt{,} ~ y \texttt{)} \\[4pt] \texttt{(} x ~ y \texttt{)} \end{matrix}\!</math> | valign="bottom" | <math>\begin{matrix} \text{false} \\[4pt] \text{neither}~ x ~\text{nor}~ y \\[4pt] y ~\text{without}~ x \\[4pt] \text{not}~ x \\[4pt] x ~\text{without}~ y \\[4pt] \text{not}~ y \\[4pt] x ~\text{not equal to}~ y \\[4pt] \text{not both}~ x ~\text{and}~ y \end{matrix}\!</math> | valign="bottom" | <math>\begin{matrix} 0 \\[4pt] \lnot x \land \lnot y \\[4pt] \lnot x \land y \\[4pt] \lnot x \\[4pt] x \land \lnot y \\[4pt] \lnot y \\[4pt] x \ne y \\[4pt] \lnot x \lor \lnot y \end{matrix}\!</math> |- | valign="bottom" | <math>\begin{matrix} f_{8} \\[4pt] f_{9} \\[4pt] f_{10} \\[4pt] f_{11} \\[4pt] f_{12} \\[4pt] f_{13} \\[4pt] f_{14} \\[4pt] f_{15} \end{matrix}\!</math> | valign="bottom" | <math>\begin{matrix} f_{1000} \\[4pt] f_{1001} \\[4pt] f_{1010} \\[4pt] f_{1011} \\[4pt] f_{1100} \\[4pt] f_{1101} \\[4pt] f_{1110} \\[4pt] f_{1111} \end{matrix}\!</math> | valign="bottom" | <math>\begin{matrix} 1~0~0~0 \\[4pt] 1~0~0~1 \\[4pt] 1~0~1~0 \\[4pt] 1~0~1~1 \\[4pt] 1~1~0~0 \\[4pt] 1~1~0~1 \\[4pt] 1~1~1~0 \\[4pt] 1~1~1~1 \end{matrix}\!</math> | valign="bottom" | <math>\begin{matrix} x ~ y \\[4pt] \texttt{((} x \texttt{,} ~ y \texttt{))} \\[4pt] y \\[4pt] \texttt{(} x ~ \texttt{(} y \texttt{))} \\[4pt] x \\[4pt] \texttt{((} x \texttt{)} ~ y \texttt{)} \\[4pt] \texttt{((} x \texttt{)(} y \texttt{))} \\[4pt] \texttt{((~))} \end{matrix}\!</math> | valign="bottom" | <math>\begin{matrix} x ~\text{and}~ y \\[4pt] x ~\text{equal to}~ y \\[4pt] y \\[4pt] \text{not}~ x ~\text{without}~ y \\[4pt] x \\[4pt] \text{not}~ y ~\text{without}~ x \\[4pt] x ~\text{or}~ y \\[4pt] \text{true} \end{matrix}\!</math> | valign="bottom" | <math>\begin{matrix} x \land y \\[4pt] x = y \\[4pt] y \\[4pt] x \Rightarrow y \\[4pt] x \\[4pt] x \Leftarrow y \\[4pt] x \lor y \\[4pt] 1 \end{matrix}\!</math> |} <br> These six languages for the sixteen boolean functions are conveniently described in the following order: * Language '''L<sub>3</sub>''' describes each boolean function ''f'' : '''B'''<sup>2</sup> &#8594; '''B''' by means of the sequence of four boolean values (''f''(1,1), ''f''(1,0), ''f''(0,1), ''f''(0,0)). Such a sequence, perhaps in another order, and perhaps with the logical values ''F'' and ''T'' instead of the boolean values 0 and 1, respectively, would normally be displayed as a column in a [[truth table]]. * Language '''L<sub>2</sub>''' lists the sixteen functions in the form '''f<sub>i</sub>''', where the index '''i''' is a [[bit string]] formed from the sequence of boolean values in '''L<sub>3</sub>'''. * Language '''L<sub>1</sub>''' notates the boolean functions '''f<sub>i</sub>''' with an index '''i''' that is the decimal equivalent of the binary numeral index in '''L<sub>2</sub>'''. * Language '''L<sub>4</sub>''' expresses the sixteen functions in terms of logical [[conjunction]], indicated by concatenating function names or proposition expressions in the manner of products, plus the family of ''[[minimal negation operator]]s'', the first few of which are given in the following variant notations: : <math>\begin{matrix} (\ ) & = & 0 & = & \mbox{false} \\ (x) & = & \tilde{x} & = & x' \\ (x, y) & = & \tilde{x}y \lor x\tilde{y} & = & x'y \lor xy' \\ (x, y, z) & = & \tilde{x}yz \lor x\tilde{y}z \lor xy\tilde{z} & = & x'yz \lor xy'z \lor xyz' \end{matrix}</math> It may also be noted that <math>(x, y)\!</math> is the same function as <math>x + y\!</math> and <math>x \ne y</math>, and that the inclusive disjunctions indicated for <math>(x, y)\!</math> and for <math>(x, y, z)\!</math> may be replaced with exclusive disjunctions without affecting the meaning, because the terms disjoined are already disjoint. However, the function <math>(x, y, z)\!</math> is not the same thing as the function <math>x + y + z\!</math>. * Language '''L<sub>5</sub>''' lists ordinary language expressions for the sixteen functions. Many other paraphrases are possible, but these afford a sample of the simplest equivalents. * Language '''L<sub>6</sub>''' expresses the sixteen functions in one of several notations that are commonly used in formal logic. ==Translations== * [http://zh.wikipedia.org/wiki/%E9%9B%B6%E9%98%B6%E9%80%BB%E8%BE%91 &#20013;&#25991; : &#38646;&#38454;&#36923;&#36753;] ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Zeroth_order_logic Zeroth Order Logic @ InterSciWiki] * [http://mywikibiz.com/Zeroth_order_logic Zeroth Order Logic @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Zeroth_order_logic Zeroth Order Logic @ Subject Wikis] * [http://en.wikiversity.org/wiki/Zeroth_order_logic Zeroth Order Logic @ Wikiversity] * [http://beta.wikiversity.org/wiki/Zeroth_order_logic Zeroth Order Logic @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Zeroth_order_logic Zeroth Order Logic], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Zeroth_order_logic Zeroth Order Logic], [http://mywikibiz.com/ MyWikiBiz] * [http://planetmath.org/ZerothOrderLogic Zeroth Order Logic], [http://planetmath.org/ PlanetMath] * [http://wikinfo.org/w/index.php/Zeroth_order_logic Zeroth Order Logic], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Zeroth_order_logic Zeroth Order Logic], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Zeroth_order_logic Zeroth Order Logic], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Zeroth-order_logic&oldid=77109225 Zeroth Order Logic], [http://en.wikipedia.org/ Wikipedia] * [http://web.archive.org/web/20050323065233/http://www.altheim.com/cs/zol.html Zeroth Order Logic], [http://web.archive.org/web/20070305032442/http://www.altheim.com/cs/ Altheim.com] [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Computer Science]] [[Category:Formal Languages]] [[Category:Formal Sciences]] [[Category:Formal Systems]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Normative Sciences]] [[Category:Semiotics]] aa2c8698de50b8d50345027809e5c4ebb31d84b4 735 734 2015-11-09T03:08:28Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. '''''Zeroth order logic''''' is an informal term that is sometimes used to indicate the common principles underlying the algebra of sets, boolean algebra, [[boolean functions]], logical connectives, monadic predicate calculus, [[propositional calculus]], and sentential logic.&nbsp; The term serves to mark a level of abstraction in which the more inessential differences among these subjects can be subsumed under the appropriate isomorphisms. ==Propositional forms on two variables== By way of initial orientation, Table&nbsp;1 lists equivalent expressions for the sixteen functions of concrete type <math>X \times Y \to \mathbb{B}\!</math> and abstract type <math>\mathbb{B} \times \mathbb{B} \to \mathbb{B}\!</math> in a number of different languages for zeroth order logic. <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%" |+ style="height:30px" | <math>\text{Table 1.} ~~ \text{Propositional Forms on Two Variables}\!</math> |- style="height:40px; background:ghostwhite" | style="width:15%" | <math>\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}\!</math> | style="width:15%" | <math>\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}\!</math> | style="width:15%" | <math>\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}\!</math> | style="width:15%" | <math>\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}\!</math> | style="width:25%" | <math>\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}\!</math> | style="width:15%" | <math>\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}\!</math> |- style="background:ghostwhite" | &nbsp; | align="right" | <math>x\colon\!</math> | <math>1~1~0~0\!</math> | &nbsp; | &nbsp; | &nbsp; |- style="background:ghostwhite" | &nbsp; | align="right" | <math>y\colon\!</math> | <math>1~0~1~0\!</math> | &nbsp; | &nbsp; | &nbsp; |- | valign="bottom" | <math>\begin{matrix} f_{0} \\[4pt] f_{1} \\[4pt] f_{2} \\[4pt] f_{3} \\[4pt] f_{4} \\[4pt] f_{5} \\[4pt] f_{6} \\[4pt] f_{7} \end{matrix}\!</math> | valign="bottom" | <math>\begin{matrix} f_{0000} \\[4pt] f_{0001} \\[4pt] f_{0010} \\[4pt] f_{0011} \\[4pt] f_{0100} \\[4pt] f_{0101} \\[4pt] f_{0110} \\[4pt] f_{0111} \end{matrix}\!</math> | valign="bottom" | <math>\begin{matrix} 0~0~0~0 \\[4pt] 0~0~0~1 \\[4pt] 0~0~1~0 \\[4pt] 0~0~1~1 \\[4pt] 0~1~0~0 \\[4pt] 0~1~0~1 \\[4pt] 0~1~1~0 \\[4pt] 0~1~1~1 \end{matrix}\!</math> | valign="bottom" | <math>\begin{matrix} \texttt{(~)} \\[4pt] \texttt{(} x \texttt{)(} y \texttt{)} \\[4pt] \texttt{(} x \texttt{)} ~ y ~ \\[4pt] \texttt{(} x \texttt{)} \\[4pt] ~ x ~ \texttt{(} y \texttt{)} \\[4pt] \texttt{(} y \texttt{)} \\[4pt] \texttt{(} x \texttt{,} ~ y \texttt{)} \\[4pt] \texttt{(} x ~ y \texttt{)} \end{matrix}\!</math> | valign="bottom" | <math>\begin{matrix} \text{false} \\[4pt] \text{neither}~ x ~\text{nor}~ y \\[4pt] y ~\text{without}~ x \\[4pt] \text{not}~ x \\[4pt] x ~\text{without}~ y \\[4pt] \text{not}~ y \\[4pt] x ~\text{not equal to}~ y \\[4pt] \text{not both}~ x ~\text{and}~ y \end{matrix}\!</math> | valign="bottom" | <math>\begin{matrix} 0 \\[4pt] \lnot x \land \lnot y \\[4pt] \lnot x \land y \\[4pt] \lnot x \\[4pt] x \land \lnot y \\[4pt] \lnot y \\[4pt] x \ne y \\[4pt] \lnot x \lor \lnot y \end{matrix}\!</math> |- | valign="bottom" | <math>\begin{matrix} f_{8} \\[4pt] f_{9} \\[4pt] f_{10} \\[4pt] f_{11} \\[4pt] f_{12} \\[4pt] f_{13} \\[4pt] f_{14} \\[4pt] f_{15} \end{matrix}\!</math> | valign="bottom" | <math>\begin{matrix} f_{1000} \\[4pt] f_{1001} \\[4pt] f_{1010} \\[4pt] f_{1011} \\[4pt] f_{1100} \\[4pt] f_{1101} \\[4pt] f_{1110} \\[4pt] f_{1111} \end{matrix}\!</math> | valign="bottom" | <math>\begin{matrix} 1~0~0~0 \\[4pt] 1~0~0~1 \\[4pt] 1~0~1~0 \\[4pt] 1~0~1~1 \\[4pt] 1~1~0~0 \\[4pt] 1~1~0~1 \\[4pt] 1~1~1~0 \\[4pt] 1~1~1~1 \end{matrix}\!</math> | valign="bottom" | <math>\begin{matrix} x ~ y \\[4pt] \texttt{((} x \texttt{,} ~ y \texttt{))} \\[4pt] y \\[4pt] \texttt{(} x ~ \texttt{(} y \texttt{))} \\[4pt] x \\[4pt] \texttt{((} x \texttt{)} ~ y \texttt{)} \\[4pt] \texttt{((} x \texttt{)(} y \texttt{))} \\[4pt] \texttt{((~))} \end{matrix}\!</math> | valign="bottom" | <math>\begin{matrix} x ~\text{and}~ y \\[4pt] x ~\text{equal to}~ y \\[4pt] y \\[4pt] \text{not}~ x ~\text{without}~ y \\[4pt] x \\[4pt] \text{not}~ y ~\text{without}~ x \\[4pt] x ~\text{or}~ y \\[4pt] \text{true} \end{matrix}\!</math> | valign="bottom" | <math>\begin{matrix} x \land y \\[4pt] x = y \\[4pt] y \\[4pt] x \Rightarrow y \\[4pt] x \\[4pt] x \Leftarrow y \\[4pt] x \lor y \\[4pt] 1 \end{matrix}\!</math> |} <br> These six languages for the sixteen boolean functions are conveniently described in the following order: * Language <math>\mathcal{L}_{3}\!</math> describes each boolean function <math>f : \mathbb{B}^2 \to \mathbb{B}\!</math> by means of the sequence of four boolean values, <math>f(1,1),\!</math> <math>f(1,0),\!</math> <math>f(0,1),\!</math> <math>f(0,0).\!</math>&nbsp; Such a sequence, perhaps in another order, and perhaps with the logical values <math>\mathrm{F}\!</math> and <math>\mathrm{T}\!</math> instead of the boolean values <math>0\!</math> and <math>1,\!</math> respectively, would normally be displayed as a column in a [[truth table]]. * Language <math>\mathcal{L}_{2}\!</math> lists the sixteen functions in the form <math>f_i,\!</math> where the index <math>i\!</math> is a bit string formed from the sequence of boolean values in <math>\mathcal{L}_{3}.\!</math> * Language <math>\mathcal{L}_{1}\!</math> notates the boolean functions <math>f_i\!</math> with an index <math>i\!</math> that is the decimal equivalent of the binary numeral index in <math>\mathcal{L}_{2}.\!</math> * Language <math>\mathcal{L}_{4}\!</math> expresses the sixteen functions in terms of [[logical conjunction]], indicated by concatenating function names or proposition expressions in the manner of products, plus the family of ''[[minimal negation operator]]s'', the first few of which are given in the following variant notations: {| align="center" cellpadding="8" style="text-align:center" | <math>\begin{matrix} \texttt{()} & = & 0 & = & \mathrm{false} \\[6pt] \texttt{(} x \texttt{)} & = & \tilde{x} & = & x^\prime \\[6pt] \texttt{(} x \texttt{,} y \texttt{)} & = & \tilde{x}y \lor x\tilde{y} & = & x^\prime y \lor x y^\prime \\[6pt] \texttt{(} x \texttt{,} y \texttt{,} z \texttt{)} & = & \tilde{x}yz \lor x\tilde{y}z \lor xy\tilde{z} & = & x^\prime y z \lor x y^\prime z \lor x y z^\prime \end{matrix}\!</math> |} It may be noted that <math>\texttt{(} x \texttt{,} y \texttt{)}\!</math> is the same function as <math>x + y\!</math> and <math>x \ne y.</math> The inclusive disjunctions indicated for <math>\texttt{(} x \texttt{,} y \texttt{)}\!</math> and for <math>\texttt{(} x \texttt{,} y \texttt{,} z \texttt{)}\!</math> may be replaced with exclusive disjunctions without affecting the meaning, since the terms disjoined are already disjoint. However, the function <math>\texttt{(} x \texttt{,} y \texttt{,} z \texttt{)}\!</math> is not the same thing as the function <math>x + y + z.\!</math> * Language <math>\mathcal{L}_{5}\!</math> lists ordinary language expressions for the sixteen functions. Many other paraphrases are possible, but these afford a sample of the simplest equivalents. * Language <math>\mathcal{L}_{6}\!</math> expresses the sixteen functions in one of several notations that are commonly used in formal logic. ==Translations== * [http://zh.wikipedia.org/wiki/%E9%9B%B6%E9%98%B6%E9%80%BB%E8%BE%91 &#20013;&#25991; : &#38646;&#38454;&#36923;&#36753;] ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Zeroth_order_logic Zeroth Order Logic @ InterSciWiki] * [http://mywikibiz.com/Zeroth_order_logic Zeroth Order Logic @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Zeroth_order_logic Zeroth Order Logic @ Subject Wikis] * [http://en.wikiversity.org/wiki/Zeroth_order_logic Zeroth Order Logic @ Wikiversity] * [http://beta.wikiversity.org/wiki/Zeroth_order_logic Zeroth Order Logic @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Zeroth_order_logic Zeroth Order Logic], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Zeroth_order_logic Zeroth Order Logic], [http://mywikibiz.com/ MyWikiBiz] * [http://planetmath.org/ZerothOrderLogic Zeroth Order Logic], [http://planetmath.org/ PlanetMath] * [http://wikinfo.org/w/index.php/Zeroth_order_logic Zeroth Order Logic], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Zeroth_order_logic Zeroth Order Logic], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Zeroth_order_logic Zeroth Order Logic], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Zeroth-order_logic&oldid=77109225 Zeroth Order Logic], [http://en.wikipedia.org/ Wikipedia] * [http://web.archive.org/web/20050323065233/http://www.altheim.com/cs/zol.html Zeroth Order Logic], [http://web.archive.org/web/20070305032442/http://www.altheim.com/cs/ Altheim.com] [[Category:Inquiry]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] [[Category:Computer Science]] [[Category:Formal Languages]] [[Category:Formal Sciences]] [[Category:Formal Systems]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Normative Sciences]] [[Category:Semiotics]] 2119a1cf735c412ca3e8900f32e0958668b45144 0 326 736 547 2015-11-09T15:14:01Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''continuous predicate''', as described by Charles Sanders Peirce, is a special type of [[relation (mathematics)|relational predicate]] that arises as the limit of an iterated process of [[hypostatic abstraction]]. Here is one of Peirce's definitive discussions of the concept: <div style="margin-left:5em; margin-right:20em"> When we have analyzed a proposition so as to throw into the subject everything that can be removed from the predicate, all that it remains for the predicate to represent is the form of connection between the different subjects as expressed in the propositional ''form''.&nbsp; What I mean by &ldquo;everything that can be removed from the predicate&rdquo; is best explained by giving an example of something not so removable. But first take something removable.&nbsp; &ldquo;Cain kills Abel.&rdquo;&nbsp; Here the predicate appears as &ldquo;&mdash; kills &mdash;.&rdquo;&nbsp; But&nbsp;we can remove killing from the predicate and make the latter &ldquo;&mdash; stands in the relation &mdash; to &mdash;.&rdquo;&nbsp; Suppose we attempt to remove more from the predicate and put the last into the form &ldquo;&mdash; exercises the function of relate of the relation &mdash; to &mdash;&rdquo; and then putting &ldquo;the function of relate to the relation&rdquo; into another subject leave as predicate &ldquo;&mdash; exercises &mdash; in respect to &mdash; to &mdash;.&rdquo;&nbsp; But&nbsp;this &ldquo;exercises&rdquo; expresses &ldquo;exercises the function&rdquo;.&nbsp; Nay more, it expresses &ldquo;exercises the function of relate&rdquo;, so that we find that though we may put this into a separate subject, it continues in the predicate just the same. Stating this in another form, to say that &ldquo;A is in the relation R to B&rdquo; is to say that A is in a certain relation to R.&nbsp; Let us separate this out thus:&nbsp; &ldquo;A is in the relation R<sup>1</sup> (where R<sup>1</sup> is the relation of a relate to the relation of which it is the relate) to R to B&rdquo;.&nbsp; But&nbsp;A is here said to be in a certain relation to the relation R<sup>1</sup>.&nbsp; So that we can express the same fact by saying, &ldquo;A is in the relation R<sup>1</sup> to the relation R<sup>1</sup> to the relation R to B&rdquo;, and so on ''ad&nbsp;infinitum''. A predicate which can thus be analyzed into parts all homogeneous with the whole I call a ''continuous predicate''.&nbsp; It is very important in logical analysis, because a continuous predicate obviously cannot be a ''compound'' except of continuous predicates, and thus when we have carried analysis so far as to leave only a continuous predicate, we have carried it to its ultimate elements. (C.S. Peirce, &ldquo;Letters to Lady Welby&rdquo; (14 December 1908), ''Selected Writings'', pp.&nbsp;396&ndash;397). </div> ==References== * [[Charles Sanders Peirce|Peirce, C.S.]], &ldquo;Letters to Lady Welby&rdquo;, pp.&nbsp;380&ndash;432 in ''Charles S. Peirce : Selected Writings (Values in a Universe of Chance)'', Philip P. Wiener (ed.), Dover&nbsp;Publications, New&nbsp;York, NY, 1966. ==Resources== * [http://vectors.usc.edu/thoughtmesh/publish/147.php Continuous Predicate &rarr; ThoughtMesh] ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Continuous_predicate Continuous Predicate @ InterSciWiki] * [http://mywikibiz.com/Continuous_predicate Continuous Predicate @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Continuous_predicate Continuous Predicate @ Subject Wikis] * [http://en.wikiversity.org/wiki/Continuous_predicate Continuous Predicate @ Wikiversity] * [http://beta.wikiversity.org/wiki/Continuous_predicate Continuous Predicate @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Continuous_predicate Continuous Predicate], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Continuous_predicate Continuous Predicate], [http://mywikibiz.com/ MyWikiBiz] * [http://planetmath.org/ContinuousPredicate Continuous Predicate], [http://planetmath.org/ PlanetMath] * [http://semanticweb.org/wiki/Continuous_predicate Continuous Predicate], [http://semanticweb.org/ SemanticWeb] * [http://vectors.usc.edu/thoughtmesh/publish/147.php Continuous Predicate], [http://vectors.usc.edu/thoughtmesh/ ThoughtMesh] * [http://wikinfo.org/w/index.php?title=Continuous_predicate Continuous Predicate], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Continuous_predicate Continuous Predicate], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Continuous_predicate Continuous Predicate], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Continuous_predicate&oldid=96870273 Continuous Predicate], [http://en.wikipedia.org/ Wikipedia] [[Category:Charles Sanders Peirce]] [[Category:Inquiry]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Philosophy]] [[Category:Semiotics]] cade0050ad27cc78e58dd28aaadb69447619e55d 738 736 2015-11-10T04:10:56Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''continuous predicate''', as described by Charles Sanders Peirce, is a special type of [[relation (mathematics)|relational predicate]] that arises as the limit of an iterated process of [[hypostatic abstraction]]. Here is one of Peirce's definitive discussions of the concept: <div style="margin-left:5em; margin-right:20em"> When we have analyzed a proposition so as to throw into the subject everything that can be removed from the predicate, all that it remains for the predicate to represent is the form of connection between the different subjects as expressed in the propositional ''form''.&nbsp; What I mean by &ldquo;everything that can be removed from the predicate&rdquo; is best explained by giving an example of something not so removable. But first take something removable.&nbsp; &ldquo;Cain kills Abel.&rdquo;&nbsp; Here the predicate appears as &ldquo;&mdash; kills &mdash;.&rdquo;&nbsp; But&nbsp;we can remove killing from the predicate and make the latter &ldquo;&mdash; stands in the relation &mdash; to &mdash;.&rdquo;&nbsp; Suppose we attempt to remove more from the predicate and put the last into the form &ldquo;&mdash; exercises the function of relate of the relation &mdash; to &mdash;&rdquo; and then putting &ldquo;the function of relate to the relation&rdquo; into another subject leave as predicate &ldquo;&mdash; exercises &mdash; in respect to &mdash; to &mdash;.&rdquo;&nbsp; But&nbsp;this &ldquo;exercises&rdquo; expresses &ldquo;exercises the function&rdquo;.&nbsp; Nay more, it expresses &ldquo;exercises the function of relate&rdquo;, so that we find that though we may put this into a separate subject, it continues in the predicate just the same. Stating this in another form, to say that &ldquo;A is in the relation R to B&rdquo; is to say that A is in a certain relation to R.&nbsp; Let us separate this out thus:&nbsp; &ldquo;A is in the relation R<sup>1</sup> (where R<sup>1</sup> is the relation of a relate to the relation of which it is the relate) to R to B&rdquo;.&nbsp; But&nbsp;A is here said to be in a certain relation to the relation R<sup>1</sup>.&nbsp; So that we can express the same fact by saying, &ldquo;A is in the relation R<sup>1</sup> to the relation R<sup>1</sup> to the relation R to B&rdquo;, and so on ''ad&nbsp;infinitum''. A predicate which can thus be analyzed into parts all homogeneous with the whole I call a ''continuous predicate''.&nbsp; It is very important in logical analysis, because a continuous predicate obviously cannot be a ''compound'' except of continuous predicates, and thus when we have carried analysis so far as to leave only a continuous predicate, we have carried it to its ultimate elements. (C.S. Peirce, &ldquo;Letters to Lady Welby&rdquo; (14 December 1908), ''Selected Writings'', pp.&nbsp;396&ndash;397). </div> ==References== * [[Charles Sanders Peirce|Peirce, C.S.]], &ldquo;Letters to Lady Welby&rdquo;, pp.&nbsp;380&ndash;432 in ''Charles S. Peirce : Selected Writings (Values in a Universe of Chance)'', Philip P. Wiener (ed.), Dover&nbsp;Publications, New&nbsp;York, NY, 1966. ==Resources== * [http://vectors.usc.edu/thoughtmesh/publish/147.php Continuous Predicate &rarr; ThoughtMesh] ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Continuous_predicate Continuous Predicate @ InterSciWiki] * [http://mywikibiz.com/Continuous_predicate Continuous Predicate @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Continuous_predicate Continuous Predicate @ Subject Wikis] * [http://en.wikiversity.org/wiki/Continuous_predicate Continuous Predicate @ Wikiversity] * [http://beta.wikiversity.org/wiki/Continuous_predicate Continuous Predicate @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Continuous_predicate Continuous Predicate], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Continuous_predicate Continuous Predicate], [http://mywikibiz.com/ MyWikiBiz] * [http://planetmath.org/ContinuousPredicate Continuous Predicate], [http://planetmath.org/ PlanetMath] * [http://semanticweb.org/wiki/Continuous_predicate Continuous Predicate], [http://semanticweb.org/ SemanticWeb] * [http://vectors.usc.edu/thoughtmesh/publish/147.php Continuous Predicate], [http://vectors.usc.edu/thoughtmesh/ ThoughtMesh] * [http://wikinfo.org/w/index.php?title=Continuous_predicate Continuous Predicate], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Continuous_predicate Continuous Predicate], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Continuous_predicate Continuous Predicate], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Continuous_predicate&oldid=96870273 Continuous Predicate], [http://en.wikipedia.org/ Wikipedia] [[Category:Charles Sanders Peirce]] [[Category:Inquiry]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Ontology]] [[Category:Philosophy]] [[Category:Pragmatism]] [[Category:Semiotics]] bf8f4b29fd6467ada1cc2b223dd5dc6bf1dff61c 0 327 737 548 2015-11-09T22:30:46Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. '''Hypostatic abstraction''' is a formal operation that takes an element of information, as expressed in a proposition <math>X ~\text{is}~ Y,\!</math> and conceives its information to consist in the relation between that subject and another subject, as expressed in the proposition <math>X ~\text{has}~ Y\!\text{-ness}.\!</math>&nbsp; The existence of the abstract subject <math>Y\!\text{-ness}\!</math> consists solely in the truth of those propositions that contain the concrete predicate <math>Y.\!</math>&nbsp; Hypostatic abstraction is known by many names, for example, ''hypostasis'', ''objectification'', ''reification'', and ''subjectal abstraction''.&nbsp; The object of discussion or thought thus introduced is termed a ''[[hypostatic object]]''. The above definition is adapted from the one given by introduced [[Charles Sanders Peirce]] (CP&nbsp;4.235, &ldquo;[[The Simplest Mathematics]]&rdquo; (1902), in ''Collected Papers'', CP&nbsp;4.227&ndash;323). The way that Peirce describes it, the main thing about the formal operation of hypostatic abstraction, insofar as it can be observed to operate on formal linguistic expressions, is that it converts an adjective or some part of a predicate into an extra subject, upping the ''arity'', also called the ''adicity'', of the main predicate in the process. For example, a typical case of hypostatic abstraction occurs in the transformation from &ldquo;honey is sweet&rdquo; to &ldquo;honey possesses sweetness&rdquo;, which transformation can be viewed in the following variety of ways: <br> <p>[[Image:Hypostatic Abstraction Figure 1.png|center]]</p><br> <p>[[Image:Hypostatic Abstraction Figure 2.png|center]]</p><br> <p>[[Image:Hypostatic Abstraction Figure 3.png|center]]</p><br> <p>[[Image:Hypostatic Abstraction Figure 4.png|center]]</p><br> The grammatical trace of this hypostatic transformation tells of a process that abstracts the adjective &ldquo;sweet&rdquo; from the main predicate &ldquo;is sweet&rdquo;, thus arriving at a new, increased-arity predicate &ldquo;possesses&rdquo;, and as a by-product of the reaction, as it were, precipitating out the substantive &ldquo;sweetness&rdquo; as a new second subject of the new predicate, &ldquo;possesses&rdquo;. ==References== * [[Charles Sanders Peirce|Peirce, C.S.]], ''Collected Papers of Charles Sanders Peirce'', vols. 1&ndash;6, Charles Hartshorne and Paul Weiss (eds.), vols. 7&ndash;8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931&ndash;1935, 1958. ==Resources== * [http://vectors.usc.edu/thoughtmesh/publish/146.php Hypostatic Abstraction &rarr; ThoughtMesh] * [http://web.clas.ufl.edu/users/jzeman/peirce_on_abstraction.htm J. Jay Zeman, ''Peirce on Abstraction''] ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Hypostatic_abstraction Hypostatic Abstraction @ InterSciWiki] * [http://mywikibiz.com/Hypostatic_abstraction Hypostatic Abstraction @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Hypostatic_abstraction Hypostatic Abstraction @ Subject Wikis] * [http://en.wikiversity.org/wiki/Hypostatic_abstraction Hypostatic Abstraction @ Wikiversity] * [http://beta.wikiversity.org/wiki/Hypostatic_abstraction Hypostatic Abstraction @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Hypostatic_abstraction Hypostatic Abstraction], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Hypostatic_abstraction Hypostatic Abstraction], [http://mywikibiz.com/ MyWikiBiz] * [http://planetmath.org/HypostaticAbstraction Hypostatic Abstraction], [http://planetmath.org/ PlanetMath] * [http://vectors.usc.edu/thoughtmesh/publish/146.php Hypostatic Abstraction], [http://vectors.usc.edu/thoughtmesh/ ThoughtMesh] * [http://wikinfo.org/w/index.php?title=Hypostatic_abstraction Hypostatic Abstraction], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Hypostatic_abstraction Hypostatic Abstraction], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Hypostatic_abstraction Hypostatic Abstraction], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Hypostatic_abstraction&oldid=69736615 Hypostatic Abstraction], [http://en.wikipedia.org/ Wikipedia] [[Category:Charles Sanders Peirce]] [[Category:Inquiry]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Ontology]] [[Category:Philosophy]] [[Category:Pragmatism]] [[Category:Semiotics]] 48739c50b41e66a0d876e652a2381a1517014775 740 737 2015-11-10T16:10:09Z Jon Awbrey 3 + ==See also== wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. '''Hypostatic abstraction''' is a formal operation that takes an element of information, as expressed in a proposition <math>X ~\text{is}~ Y,\!</math> and conceives its information to consist in the relation between that subject and another subject, as expressed in the proposition <math>X ~\text{has}~ Y\!\text{-ness}.\!</math>&nbsp; The existence of the abstract subject <math>Y\!\text{-ness}\!</math> consists solely in the truth of those propositions that contain the concrete predicate <math>Y.\!</math>&nbsp; Hypostatic abstraction is known by many names, for example, ''hypostasis'', ''objectification'', ''reification'', and ''subjectal abstraction''.&nbsp; The object of discussion or thought thus introduced is termed a ''[[hypostatic object]]''. The above definition is adapted from the one given by introduced [[Charles Sanders Peirce]] (CP&nbsp;4.235, &ldquo;[[The Simplest Mathematics]]&rdquo; (1902), in ''Collected Papers'', CP&nbsp;4.227&ndash;323). The way that Peirce describes it, the main thing about the formal operation of hypostatic abstraction, insofar as it can be observed to operate on formal linguistic expressions, is that it converts an adjective or some part of a predicate into an extra subject, upping the ''arity'', also called the ''adicity'', of the main predicate in the process. For example, a typical case of hypostatic abstraction occurs in the transformation from &ldquo;honey is sweet&rdquo; to &ldquo;honey possesses sweetness&rdquo;, which transformation can be viewed in the following variety of ways: <br> <p>[[Image:Hypostatic Abstraction Figure 1.png|center]]</p><br> <p>[[Image:Hypostatic Abstraction Figure 2.png|center]]</p><br> <p>[[Image:Hypostatic Abstraction Figure 3.png|center]]</p><br> <p>[[Image:Hypostatic Abstraction Figure 4.png|center]]</p><br> The grammatical trace of this hypostatic transformation tells of a process that abstracts the adjective &ldquo;sweet&rdquo; from the main predicate &ldquo;is sweet&rdquo;, thus arriving at a new, increased-arity predicate &ldquo;possesses&rdquo;, and as a by-product of the reaction, as it were, precipitating out the substantive &ldquo;sweetness&rdquo; as a new second subject of the new predicate, &ldquo;possesses&rdquo;. ==See also== * [[Hypostatic object]] * [[Prescisive abstraction]] * [[The Simplest Mathematics]] ==References== * [[Charles Sanders Peirce|Peirce, C.S.]], ''Collected Papers of Charles Sanders Peirce'', vols. 1&ndash;6, Charles Hartshorne and Paul Weiss (eds.), vols. 7&ndash;8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931&ndash;1935, 1958. ==Resources== * [http://vectors.usc.edu/thoughtmesh/publish/146.php Hypostatic Abstraction &rarr; ThoughtMesh] * [http://web.clas.ufl.edu/users/jzeman/peirce_on_abstraction.htm J. Jay Zeman, ''Peirce on Abstraction''] ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Hypostatic_abstraction Hypostatic Abstraction @ InterSciWiki] * [http://mywikibiz.com/Hypostatic_abstraction Hypostatic Abstraction @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Hypostatic_abstraction Hypostatic Abstraction @ Subject Wikis] * [http://en.wikiversity.org/wiki/Hypostatic_abstraction Hypostatic Abstraction @ Wikiversity] * [http://beta.wikiversity.org/wiki/Hypostatic_abstraction Hypostatic Abstraction @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Hypostatic_abstraction Hypostatic Abstraction], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Hypostatic_abstraction Hypostatic Abstraction], [http://mywikibiz.com/ MyWikiBiz] * [http://planetmath.org/HypostaticAbstraction Hypostatic Abstraction], [http://planetmath.org/ PlanetMath] * [http://vectors.usc.edu/thoughtmesh/publish/146.php Hypostatic Abstraction], [http://vectors.usc.edu/thoughtmesh/ ThoughtMesh] * [http://wikinfo.org/w/index.php?title=Hypostatic_abstraction Hypostatic Abstraction], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Hypostatic_abstraction Hypostatic Abstraction], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Hypostatic_abstraction Hypostatic Abstraction], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Hypostatic_abstraction&oldid=69736615 Hypostatic Abstraction], [http://en.wikipedia.org/ Wikipedia] [[Category:Charles Sanders Peirce]] [[Category:Inquiry]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Ontology]] [[Category:Philosophy]] [[Category:Pragmatism]] [[Category:Semiotics]] 7474e2b5f729479b3a14638e7d72506e8a376c58 742 740 2015-11-10T18:58:54Z Jon Awbrey 3 wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. '''Hypostatic abstraction''' is a formal operation that takes an element of information, as expressed in a proposition <math>X ~\text{is}~ Y,\!</math> and conceives its information to consist in the relation between that subject and another subject, as expressed in the proposition <math>X ~\text{has}~ Y\!\text{-ness}.\!</math>&nbsp; The existence of the abstract subject <math>Y\!\text{-ness}\!</math> consists solely in the truth of those propositions that contain the concrete predicate <math>Y.\!</math>&nbsp; Hypostatic abstraction is known by many names, for example, ''hypostasis'', ''objectification'', ''reification'', and ''subjectal abstraction''.&nbsp; The object of discussion or thought thus introduced is termed a ''[[hypostatic object]]''. The above definition is adapted from the one given by introduced [[Charles Sanders Peirce]] (CP&nbsp;4.235, &ldquo;[[The Simplest Mathematics]]&rdquo; (1902), in ''Collected Papers'', CP&nbsp;4.227&ndash;323). The way that Peirce describes it, the main thing about the formal operation of hypostatic abstraction, insofar as it can be observed to operate on formal linguistic expressions, is that it converts an adjective or some part of a predicate into an extra subject, upping the ''arity'', also called the ''adicity'', of the main predicate in the process. For example, a typical case of hypostatic abstraction occurs in the transformation from &ldquo;honey is sweet&rdquo; to &ldquo;honey possesses sweetness&rdquo;, which transformation can be viewed in the following variety of ways: <br> <p>[[Image:Hypostatic Abstraction Figure 1.png|center]]</p><br> <p>[[Image:Hypostatic Abstraction Figure 2.png|center]]</p><br> <p>[[Image:Hypostatic Abstraction Figure 3.png|center]]</p><br> <p>[[Image:Hypostatic Abstraction Figure 4.png|center]]</p><br> The grammatical trace of this hypostatic transformation tells of a process that abstracts the adjective &ldquo;sweet&rdquo; from the main predicate &ldquo;is sweet&rdquo;, thus arriving at a new, increased-arity predicate &ldquo;possesses&rdquo;, and as a by-product of the reaction, as it were, precipitating out the substantive &ldquo;sweetness&rdquo; as a new second subject of the new predicate, &ldquo;possesses&rdquo;. ==See also== * [[Hypostatic object]] * [[Prescisive abstraction]] ==References== * [[Charles Sanders Peirce|Peirce, C.S.]], ''Collected Papers of Charles Sanders Peirce'', vols. 1&ndash;6, Charles Hartshorne and Paul Weiss (eds.), vols. 7&ndash;8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931&ndash;1935, 1958. ==Resources== * [http://vectors.usc.edu/thoughtmesh/publish/146.php Hypostatic Abstraction &rarr; ThoughtMesh] * [http://web.clas.ufl.edu/users/jzeman/peirce_on_abstraction.htm J. Jay Zeman, ''Peirce on Abstraction''] ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Hypostatic_abstraction Hypostatic Abstraction @ InterSciWiki] * [http://mywikibiz.com/Hypostatic_abstraction Hypostatic Abstraction @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Hypostatic_abstraction Hypostatic Abstraction @ Subject Wikis] * [http://en.wikiversity.org/wiki/Hypostatic_abstraction Hypostatic Abstraction @ Wikiversity] * [http://beta.wikiversity.org/wiki/Hypostatic_abstraction Hypostatic Abstraction @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Hypostatic_abstraction Hypostatic Abstraction], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Hypostatic_abstraction Hypostatic Abstraction], [http://mywikibiz.com/ MyWikiBiz] * [http://planetmath.org/HypostaticAbstraction Hypostatic Abstraction], [http://planetmath.org/ PlanetMath] * [http://vectors.usc.edu/thoughtmesh/publish/146.php Hypostatic Abstraction], [http://vectors.usc.edu/thoughtmesh/ ThoughtMesh] * [http://wikinfo.org/w/index.php?title=Hypostatic_abstraction Hypostatic Abstraction], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Hypostatic_abstraction Hypostatic Abstraction], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Hypostatic_abstraction Hypostatic Abstraction], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Hypostatic_abstraction&oldid=69736615 Hypostatic Abstraction], [http://en.wikipedia.org/ Wikipedia] [[Category:Charles Sanders Peirce]] [[Category:Inquiry]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Ontology]] [[Category:Philosophy]] [[Category:Pragmatism]] [[Category:Semiotics]] 31c05a8a6b0f5e4f4ebd61cf2211826441945f38 0 375 739 2015-11-10T15:14:47Z Jon Awbrey 3 add article wikitext text/x-wiki '''Prescisive abstraction''' or '''prescision''', variously spelled as '''precisive abstraction''' or '''prescission''', is a formal operation that marks, selects, or singles out one feature of a concrete experience to the disregard of others. The above definition is adapted from the one given by [[Charles Sanders Peirce]] (CP&nbsp;4.235, &ldquo;[[The Simplest Mathematics]]&rdquo; (1902), in ''Collected Papers'', CP&nbsp;4.227&ndash;393). ==References== * [[Charles Sanders Peirce|Peirce, C.S.]], ''Collected Papers of Charles Sanders Peirce'', vols. 1&ndash;6, Charles Hartshorne and Paul Weiss (eds.), vols. 7&ndash;8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931&ndash;1935, 1958. ==See also== * [[Hypostatic abstraction]] * [[Hypostatic object]] [[Category:Charles Sanders Peirce]] [[Category:Inquiry]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Ontology]] [[Category:Philosophy]] [[Category:Pragmatism]] [[Category:Semiotics]] 4475677fd825f6476842c7bb339a7da9a11ea77c 743 739 2015-11-10T19:12:07Z Jon Awbrey 3 wikitext text/x-wiki '''Prescisive abstraction''' or '''prescision''', variously spelled as '''precisive abstraction''' or '''prescission''', is a formal operation that marks, selects, or singles out one feature of a concrete experience to the disregard of others. The above definition is adapted from the one given by [[Charles Sanders Peirce]] (CP&nbsp;4.235, &ldquo;[[The Simplest Mathematics]]&rdquo; (1902), in ''Collected Papers'', CP&nbsp;4.227&ndash;393). ==References== * [[Charles Sanders Peirce|Peirce, C.S.]], ''Collected Papers of Charles Sanders Peirce'', vols. 1&ndash;6, Charles Hartshorne and Paul Weiss (eds.), vols. 7&ndash;8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931&ndash;1935, 1958. ==See also== * [[Charles Sanders Peirce]] * [[Hypostatic abstraction]] * [[Hypostatic object]] [[Category:Charles Sanders Peirce]] [[Category:Inquiry]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Ontology]] [[Category:Philosophy]] [[Category:Pragmatism]] [[Category:Semiotics]] 734bf99c9a898d10aec1c4be319da986c68a31bc 0 376 741 2015-11-10T18:45:04Z Jon Awbrey 3 add article wikitext text/x-wiki A '''hypostatic object''', also known in certain senses as an '''abstract object''' or a '''formal object''', is an object of discussion or thought that results as the normal product of a process of ''[[hypostatic abstraction]]''. ==See also== * [[Charles Sanders Peirce]] * [[Hypostatic abstraction]] * [[Prescisive abstraction]] [[Category:Charles Sanders Peirce]] [[Category:Inquiry]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Ontology]] [[Category:Philosophy]] [[Category:Pragmatism]] [[Category:Semiotics]] 7cb2cffde65a9c5887e6807c614e461f1346b990 0 377 744 2015-11-10T19:28:07Z Jon Awbrey 3 add article wikitext text/x-wiki '''''The Simplest Mathematics''''' is the title of a paper by [[Charles Sanders Peirce]], intended as Chapter&nbsp;3 of his unfinished magnum opus ''The Minute Logic''. The paper is dated January&ndash;February 1902 but was not published until the appearance of his ''Collected Papers, Volume&nbsp;4'' in 1933. Peirce introduces the subject of the paper as &ldquo;certain extremely simple branches of mathematics which, owing to their utility in logic, have to be treated in considerable detail, although to the mathematician they are hardly worth consideration&rdquo; (CP&nbsp;4.227). ==References== * Peirce, Benjamin (1870), &ldquo;Linear Associative Algebra&rdquo;, §&nbsp;1. See ''American Journal of Mathematics'' 4 (1881). * [[Charles Sanders Peirce (Bibliography)|Peirce, C.S., Bibliography]]. * Peirce, C.S. (1902), &ldquo;The Simplest Mathematics&rdquo;, MS dated January&ndash;February 1902, intended as Chapter&nbsp;3 of the ''Minute Logic'', CP&nbsp;4.227&ndash;323 in ''Collected Papers''. * Peirce, C.S., ''Collected Papers of Charles Sanders Peirce'', vols. 1&ndash;6, Charles Hartshorne and Paul Weiss (eds.), vols. 7&ndash;8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931&ndash;1935, 1958. Cited as CP&nbsp;volume.paragraph. ==See also== * [[Foundations of mathematics]] * [[Kaina Stoicheia]] * [[Philosophy of mathematics]] [[Category:Charles Sanders Peirce]] [[Category:Inquiry]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Ontology]] [[Category:Philosophy]] [[Category:Pragmatism]] [[Category:Semiotics]] 00153783b1b1003a896415ed019e06ea3c4c3b93 746 744 2015-11-11T15:22:01Z Jon Awbrey 3 update wikitext text/x-wiki '''''The Simplest Mathematics''''' is the title of a paper by [[Charles Sanders Peirce]], intended as Chapter&nbsp;3 of his unfinished magnum opus ''The Minute Logic''. The paper is dated January&ndash;February 1902 but was not published until the appearance of his ''Collected Papers, Volume&nbsp;4'' in 1933. Peirce introduces the subject of the paper as &ldquo;certain extremely simple branches of mathematics which, owing to their utility in logic, have to be treated in considerable detail, although to the mathematician they are hardly worth consideration&rdquo; (CP&nbsp;4.227). ==References== * [[Charles Sanders Peirce (Bibliography)]]. * Peirce, C.S. (1902), &ldquo;The Simplest Mathematics&rdquo;, MS dated January&ndash;February 1902, intended as Chapter&nbsp;3 of the ''Minute Logic'', CP&nbsp;4.227&ndash;323 in ''Collected Papers''. * Peirce, C.S., ''Collected Papers of Charles Sanders Peirce'', vols. 1&ndash;6, Charles Hartshorne and Paul Weiss (eds.), vols. 7&ndash;8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931&ndash;1935, 1958. Cited as CP&nbsp;volume.paragraph. ==See also== * [[Kaina Stoicheia]] [[Category:Charles Sanders Peirce]] [[Category:Inquiry]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Ontology]] [[Category:Philosophy]] [[Category:Pragmatism]] [[Category:Semiotics]] 2fbb646ca6acd3f364d338cf197db59933d19078 747 746 2015-11-11T15:36:55Z Jon Awbrey 3 wikitext text/x-wiki '''''The Simplest Mathematics''''' is the title of a paper by [[Charles Sanders Peirce]], intended as Chapter&nbsp;3 of his unfinished magnum opus ''The Minute Logic''. The paper is dated January&ndash;February 1902 but was not published until the appearance of his ''Collected Papers, Volume&nbsp;4'' in 1933. Peirce introduces the subject of the paper as &ldquo;certain extremely simple branches of mathematics which, owing to their utility in logic, have to be treated in considerable detail, although to the mathematician they are hardly worth consideration&rdquo; (CP&nbsp;4.227). ==Related topic== * [[Kaina Stoicheia]] ==References== * [[Charles Sanders Peirce (Bibliography)]]. * Peirce, C.S. (1902), &ldquo;The Simplest Mathematics&rdquo;, MS dated January&ndash;February 1902, intended as Chapter&nbsp;3 of the ''Minute Logic'', CP&nbsp;4.227&ndash;323 in ''Collected Papers''. * Peirce, C.S., ''Collected Papers of Charles Sanders Peirce'', vols. 1&ndash;6, Charles Hartshorne and Paul Weiss (eds.), vols. 7&ndash;8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931&ndash;1935, 1958. Cited as CP&nbsp;volume.paragraph. [[Category:Charles Sanders Peirce]] [[Category:Inquiry]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Ontology]] [[Category:Philosophy]] [[Category:Pragmatism]] [[Category:Semiotics]] a8941e6122416e51e06926e0f905aff8ea45c435 0 332 745 554 2015-11-10T22:54:22Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. The '''logic of relatives''', more precisely, the '''logic of relative terms''', is the study of [[relation (mathematics)|relation]]s as represented in symbolic forms known as ''rhemes'', ''rhemata'', or ''relative terms''. The treatment of relations by way of their corresponding relative terms affords a distinctive perspective on the subject, even though all angles of approach must ultimately converge on the same formal subject matter. The consideration of ''[[relative term]]s'' has its roots in antiquity, but it entered a radically new phase of development with the work of [[Charles Sanders Peirce]], beginning with his 1870 paper &ldquo;[[Logic of Relatives (1870)|Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic]]&rdquo;. ==See also== * [[Logic of Relatives (1870)]] * [[Logic of Relatives (1883)]] ==References== * [[Charles Sanders Peirce|Peirce, C.S.]], &ldquo;Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic&rdquo;, ''Memoirs of the American Academy of Arts and Sciences'' 9, 317&ndash;378, 1870. Reprinted, ''Collected Papers'' CP&nbsp;3.45&ndash;149, ''Chronological Edition'' CE&nbsp;2, 359&ndash;429. * [http://intersci.ss.uci.edu/wiki/index.php/Peirce%27s_1870_Logic_Of_Relatives Awbrey, J.L., &ldquo;Peirce's 1870 Logic of Relatives&rdquo;] ==Bibliography== * Aristotle, &ldquo;The Categories&rdquo;, Harold P. Cooke (trans.), pp.&nbsp;1&ndash;109 in ''Aristotle, Volume&nbsp;1'', Loeb Classical Library, William Heinemann, London, UK, 1938. * Aristotle, &ldquo;On Interpretation&rdquo;, Harold P. Cooke (trans.), pp.&nbsp;111&ndash;179 in ''Aristotle, Volume&nbsp;1'', Loeb Classical Library, William Heinemann, London, UK, 1938. * Aristotle, &ldquo;Prior Analytics&rdquo;, Hugh Tredennick (trans.), pp.&nbsp;181&ndash;531 in ''Aristotle, Volume&nbsp;1'', Loeb Classical Library, William Heinemann, London, UK, 1938. * Boole, George, ''An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities'', Macmillan Publishers, 1854. Reprinted with corrections, Dover Publications, New&nbsp;York, NY, 1958. * Maddux, Roger D., ''Relation Algebras'', vol. 150 in &lsquo;Studies in Logic and the Foundations of Mathematics&rsquo;, Elsevier Science, 2006. * Peirce, C.S., ''Collected Papers of Charles Sanders Peirce'', vols. 1&ndash;6, Charles Hartshorne and Paul Weiss (eds.), vols. 7&ndash;8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931&ndash;1935, 1958. Cited as CP&nbsp;volume.paragraph. * Peirce, C.S., ''Writings of Charles S. Peirce : A Chronological Edition, Volume 2, 1867&ndash;1871'', Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1984. Cited as CE&nbsp;2. ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Logic_of_relatives Logic of Relatives @ InterSciWiki] * [http://mywikibiz.com/Logic_of_relatives Logic of Relatives @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Logic_of_relatives Logic of Relatives @ Subject Wikis] * [http://en.wikiversity.org/wiki/Logic_of_relatives Logic of Relatives @ Wikiversity] * [http://beta.wikiversity.org/wiki/Logic_of_relatives Logic of Relatives @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Logic_of_relatives Logic of Relatives], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Logic_of_relatives Logic of Relatives], [http://mywikibiz.com/ MyWikiBiz] * [http://semanticweb.org/wiki/Logic_of_Relatives Logic of Relatives], [http://semanticweb.org/ Semantic Web] * [http://wikinfo.org/w/index.php/Logic_of_relatives Logic of Relatives], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Logic_of_relatives Logic of Relatives], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Logic_of_relatives Logic of Relatives], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Logic_of_relatives&oldid=43501411 Logic of Relatives], [http://en.wikipedia.org/ Wikipedia] [[Category:Charles Sanders Peirce]] [[Category:Formal Languages]] [[Category:Formal Sciences]] [[Category:Formal Systems]] [[Category:Inquiry]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Philosophy]] [[Category:Science]] [[Category:Semiotics]] 67191d3e585a5fdef1710d6631db3b66d1e6204a 0 333 748 638 2015-11-11T16:16:49Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''logical matrix''', in the finite dimensional case, is a <math>k\!</math>-dimensional array with entries from the [[boolean domain]] <math>\mathbb{B} = \{ 0,1 \}.</math> Such a matrix affords a matrix representation of a <math>k\!</math>-adic [[relation (mathematics)|relation]]. ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Logical_matrix Logical Matrix @ InterSciWiki] * [http://mywikibiz.com/Logical_matrix Logical Matrix @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Logical_matrix Logical Matrix @ Subject Wikis] * [http://en.wikiversity.org/wiki/Logical_matrix Logical Matrix @ Wikiversity] * [http://beta.wikiversity.org/wiki/Logical_matrix Logical Matrix @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Logical_matrix Logical Matrix], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Logical_matrix Logical Matrix], [http://mywikibiz.com/ MyWikiBiz] * [http://planetmath.org/LogicalMatrix Logical Matrix], [http://planetmath.org/ PlanetMath] * [http://wikinfo.org/w/index.php/Logical_matrix Logical Matrix], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Logical_matrix Logical Matrix], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Logical_matrix Logical Matrix], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Logical_matrix&oldid=43606082 Logical Matrix], [http://en.wikipedia.org/ Wikipedia] [[Category:Charles Sanders Peirce]] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Discrete Mathematics]] [[Category:Inquiry]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Semiotics]] 9542e5de322d7b08529f9bed8fa14662b277cdbd 0 334 749 556 2015-11-12T21:14:57Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. In mathematics, a '''finitary relation''' is defined by one of the formal definitions given below. * The basic idea is to generalize the concept of a two-place relation, such as the relation of ''equality'' denoted by the sign &ldquo;<math>=\!</math>&rdquo; in a statement like <math>5 + 7 = 12\!</math> or the relation of ''order'' denoted by the sign &ldquo;<math>{<}\!</math>&rdquo; in a statement like <math>5 < 12.\!</math>&nbsp; Relations that involve two ''places'' or ''roles'' are called ''binary relations'' by some and ''dyadic relations'' by others, the latter being historically prior but also useful when necessary to avoid confusion with ''binary (base 2) numerals''. * The concept of a two-place relation is generalized by considering relations with increasing but still finite numbers of places or roles.&nbsp; These are called ''finite-place'' or ''finitary'' relations.&nbsp; A&nbsp;finitary relation involving <math>k\!</math> places is variously called a ''<math>k\!</math>-ary'', a ''<math>k\!</math>-adic'', or a ''<math>k\!</math>-dimensional'' relation.&nbsp; The number <math>k\!</math> is then called the ''arity'', the ''adicity'', or the ''dimension'' of the relation, respectively. ==Informal introduction== The definition of ''relation'' given in the next section formally captures a concept that is actually quite familiar from everyday life.&nbsp; For example, consider the relationship, involving three roles that people might play, expressed in a statement of the form <math>X ~\text{suspects that}~ Y ~\text{likes}~ Z.\!</math>&nbsp; The facts of a concrete situation could be organized in the form of a Table like the one below: <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:60%" |+ style="height:30px" | <math>\text{Relation}~ S ~:~ X ~\text{suspects that}~ Y ~\text{likes}~ Z\!</math> |- style="height:40px; background:ghostwhite" | <math>\text{Person}~ X\!</math> | <math>\text{Person}~ Y\!</math> | <math>\text{Person}~ Z\!</math> |- | <math>\text{Alice}\!</math> | <math>\text{Bob}\!</math> | <math>\text{Denise}\!</math> |- | <math>\text{Charles}\!</math> | <math>\text{Alice}\!</math> | <math>\text{Bob}\!</math> |- | <math>\text{Charles}\!</math> | <math>\text{Charles}\!</math> | <math>\text{Alice}\!</math> |- | <math>\text{Denise}\!</math> | <math>\text{Denise}\!</math> | <math>\text{Denise}\!</math> |} <br> Each row of the Table records a fact or makes an assertion of the form <math>X ~\text{suspects that}~ Y ~\text{likes}~ Z.\!</math>&nbsp; For instance, the first row says, in effect, <math>\text{Alice suspects that Bob likes Denise.}\!</math>&nbsp; The Table represents a relation <math>S\!</math> over the set <math>P\!</math> of people under discussion: {| align="center" cellpadding="10" | <math>P ~=~ \{ \text{Alice}, \text{Bob}, \text{Charles}, \text{Denise} \}\!</math> |} The data of the Table are equivalent to the following set of ordered triples: {| align="center" cellpadding="10" | <math>\begin{smallmatrix} S & = & \{ & \text{(Alice, Bob, Denise)}, & \text{(Charles, Alice, Bob)}, & \text{(Charles, Charles, Alice)}, & \text{(Denise, Denise, Denise)} & \} \end{smallmatrix}\!</math> |} By a slight overuse of notation, it is usual to write <math>S (\text{Alice}, \text{Bob}, \text{Denise})\!</math> to say the same thing as the first row of the Table.&nbsp; The relation <math>S\!</math> is a ''triadic'' or ''ternary'' relation, since there are ''three'' items involved in each row.&nbsp; The relation itself is a mathematical object, defined in terms of concepts from set theory, that carries all the information from the Table in one neat package. The Table for relation <math>S\!</math> is an extremely simple example of a relational database.&nbsp; The theoretical aspects of databases are the specialty of one branch of computer science, while their practical impacts have become all too familiar in our everyday lives.&nbsp; Computer scientists, logicians, and mathematicians, however, tend to see different things when they look at these concrete examples and samples of the more general concept of a relation. For one thing, databases are designed to deal with empirical data, and experience is always finite, whereas mathematics is nothing if not concerned with infinity, at the very least, potential infinity. This difference in perspective brings up a number of ideas that are usefully introduced at this point, if by no means covered in depth. ==Example: Divisibility== A more typical example of a two-place relation in mathematics is the relation of ''divisibility'' between two positive integers <math>n\!</math> and <math>m\!</math> that is expressed in statements like <math>{}^{\backprime\backprime} n ~\text{divides}~ m {}^{\prime\prime}\!</math> or <math>{}^{\backprime\backprime} n ~\text{goes into}~ m {}^{\prime\prime}.\!</math>&nbsp; This is a relation that comes up so often that a special symbol <math>{}^{\backprime\backprime} | {}^{\prime\prime}\!</math> is reserved to express it, allowing one to write <math>{}^{\backprime\backprime} n|m {}^{\prime\prime}\!</math> for <math>{}^{\backprime\backprime} n ~\text{divides}~ m {}^{\prime\prime}.\!</math> To express the binary relation of divisibility in terms of sets, we have the set <math>P\!</math> of positive integers, <math>P = \{ 1, 2, 3, \ldots \},\!</math> and we have the binary relation <math>D\!</math> on <math>P\!</math> such that the ordered pair <math>(n, m)\!</math> is in the relation <math>D\!</math> just in case <math>n|m.\!</math>&nbsp; In other turns of phrase that are frequently used, one says that the number <math>n\!</math> is related by <math>D\!</math> to the number <math>m\!</math> just in case <math>n\!</math> is a factor of <math>m,\!</math> that is, just in case <math>n\!</math> divides <math>m\!</math> with no remainder.&nbsp; The relation <math>D,\!</math> regarded as a set of ordered pairs, consists of all pairs of numbers <math>(n, m)\!</math> such that <math>n\!</math> divides <math>m.\!</math> For example, <math>2\!</math> is a factor of <math>4,\!</math> and <math>6\!</math> is a factor of <math>72,\!</math> which two facts can be written either as <math>2|4\!</math> and <math>6|72\!</math> or as <math>D(2, 4)\!</math> and <math>D(6, 72).\!</math> ==Formal definitions== There are two definitions of k-place relations that are commonly encountered in mathematics. In order of simplicity, the first of these definitions is as follows: '''Definition 1.''' A '''relation''' ''L'' over the [[set]]s ''X''<sub>1</sub>, …, ''X''<sub>''k''</sub> is a [[subset]] of their [[cartesian product]], written ''L'' &sube; ''X''<sub>1</sub> &times; … &times; ''X''<sub>''k''</sub>. Under this definition, then, a ''k''-ary relation is simply a set of ''k''-[[tuple]]s. The second definition makes use of an idiom that is common in mathematics, stipulating that "such and such is an ''n''-tuple" in order to ensure that such and such a mathematical object is determined by the specification of ''n'' component mathematical objects. In the case of a relation ''L'' over ''k'' sets, there are ''k'' + 1 things to specify, namely, the ''k'' sets plus a subset of their cartesian product. In the idiom, this is expressed by saying that ''L'' is a (''k''+1)-tuple. '''Definition 2.''' A '''relation''' ''L'' over the sets ''X''<sub>1</sub>, …, ''X''<sub>''k''</sub> is a (''k''+1)-tuple ''L'' = (''X''<sub>1</sub>, …, ''X''<sub>''k''</sub>, ''G''(''L'')), where ''G''(''L'') is a subset of the cartesian product ''X''<sub>1</sub> &times; … &times; ''X''<sub>''k''</sub>. ''G''(''L'') is called the ''[[graph]]'' of ''L''. Elements of a relation are more briefly denoted by using boldface characters, for example, the constant element <math>\mathbf{a}</math> = (a<sub>1</sub>,&nbsp;…,&nbsp;a<sub>''k''</sub>) or the variable element <math>\mathbf{x}</math> = (''x''<sub>1</sub>,&nbsp;…,&nbsp;''x''<sub>''k''</sub>). A statement of the form "<math>\mathbf{a}</math> is in the relation ''L''&nbsp;" is taken to mean that <math>\mathbf{a}</math> is in ''L'' under the first definition and that <math>\mathbf{a}</math> is in ''G''(''L'') under the second definition. The following considerations apply under either definition: :* The sets ''X''<sub>''j''</sub> for ''j'' = 1 to ''k'' are called the ''[[domain]]s'' of the relation. In the case of the first definition, the relation itself does not uniquely determine a given sequence of domains. :* If all of the domains ''X''<sub>''j''</sub> are the same set ''X'', then ''L'' is more simply referred to as a ''k''-ary relation over ''X''. :* If any of the domains ''X''<sub>''j''</sub> is empty, then the cartesian product is empty, and the only relation over such a sequence of domains is the empty relation ''L'' = <math>\varnothing</math>. As a result, naturally occurring applications of the relation concept typically involve a stipulation that all of the domains be nonempty. As a rule, whatever definition best fits the application at hand will be chosen for that purpose, and anything that falls under it will be called a 'relation' for the duration of that discussion. If it becomes necessary to distinguish the two alternatives, the latter type of object can be referred to as an ''embedded'' or ''included'' relation. If ''L'' is a relation over the domains ''X''<sub>1</sub>, …, ''X''<sub>''k''</sub>, it is conventional to consider a sequence of terms called ''variables'', ''x''<sub>1</sub>, …, ''x''<sub>''k''</sub>, that are said to ''range over'' the respective domains. A ''[[boolean domain]]'' '''B''' is a generic 2-element set, say, '''B''' = {0, 1}, whose elements are interpreted as logical values, typically 0 = false and 1 = true. The ''[[characteristic function]]'' of the relation ''L'', written ''f''<sub>''L''</sub> or &chi;(''L''), is the [[boolean-valued function]] ''f''<sub>''L''</sub>&nbsp;:&nbsp;''X''<sub>1</sub>&nbsp;&times;&nbsp;…&nbsp;&times;&nbsp;''X''<sub>''k''</sub>&nbsp;→&nbsp;'''B''', defined in such a way that ''f''<sub>''L''</sub>(<math>\mathbf{x}</math>) = 1 just in case the ''k''-tuple <math>\mathbf{x}</math> is in the relation ''L''. The characteristic function of a relation may also be called its ''[[indicator function]]'', especially in probability and statistics. It is conventional in applied mathematics, computer science, and statistics to refer to a boolean-valued function like ''f''<sub>''L''</sub> as a ''k''-place ''[[predicate]]''. From the more abstract viewpoints of [[formal logic]] and [[model theory]], the relation ''L'' is seen as constituting a ''logical model'' or a ''relational structure'' that serves as one of many possible [[interpretation]]s of a corresponding ''k''-place ''predicate symbol'', as that term is used in ''[[first-order logic|predicate calculus]]''. Due to the convergence of many different styles of study on the same areas of interest, the reader will find much variation in usage here. The variation presented in this article treats a relation as the [[set theory|set-theoretic]] ''[[extension (semantics)|extension]]'' of a relational concept or term. Another variation reserves the term 'relation' to the corresponding logical entity, either the ''[[comprehension (logic)|logical comprehension]]'', which is the totality of ''[[intension]]s'' or abstract ''[[property (philosophy)|properties]]'' that all of the elements of the relation in extension have in common, or else the symbols that are taken to denote these elements and intensions. Further, but hardly finally, some writers of the latter persuasion introduce terms with more concrete connotations, like 'relational structure', for the set-theoretic extension of a given relational concept. ==Example: coplanarity== For lines ''L'' in three-dimensional space, there is a ternary relation picking out the triples of lines that are [[coplanar]]. This ''does not'' reduce to the binary [[symmetric relation]] of coplanarity of pairs of lines. In other words, writing ''P''(''L'', ''M'', ''N'') when the lines ''L'', ''M'', and ''N'' lie in a plane, and ''Q''(''L'', ''M'') for the binary relation, it is not true that ''Q''(''L'', ''M''), ''Q''(''M'', ''N'') and ''Q''(''N'', ''L'') together imply ''P''(''L'', ''M'', ''N''); although the converse is certainly true (any pair out of three coplanar lines is coplanar, ''a fortiori''). There are two geometrical reasons for this. In one case, for example taking the ''x''-axis, ''y''-axis and ''z''-axis, the three lines are concurrent, i.e. intersect at a single point. In another case, ''L'', ''M'', and ''N'' can be three edges of an infinite [[triangular prism]]. What is true is that if each pair of lines intersects, and the points of intersection are distinct, then pairwise coplanarity implies coplanarity of the triple. ==Remarks== Relations are classified according to the number of sets in the cartesian product, in other words the number of terms in the expression: :* Unary relation or [[property (philosophy)|property]]: ''L''(''u'') :* Binary relation: ''L''(''u'', ''v'') or ''u'' ''L'' ''v'' :* Ternary relation: ''L''(''u'', ''v'', ''w'') :* Quaternary relation: ''L''(''u'', ''v'', ''w'', ''x'') Relations with more than four terms are usually referred to as ''k''-ary, for example, "a 5-ary relation". ==References== * [[Charles Sanders Peirce|Peirce, C.S.]] (1870), &ldquo;Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic&rdquo;, ''Memoirs of the American Academy of Arts and Sciences'' 9, 317&ndash;378, 1870. Reprinted, ''Collected Papers'' CP&nbsp;3.45&ndash;149, ''Chronological Edition'' CE&nbsp;2, 359&ndash;429. * Ulam, S.M., and Bednarek, A.R. (1990), &ldquo;On the Theory of Relational Structures and Schemata for Parallel Computation&rdquo;, pp. 477&ndash;508 in A.R. Bednarek and Françoise Ulam (eds.), ''Analogies Between Analogies : The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators'', University of California Press, Berkeley, CA. ==Bibliography== * [[Nicolas Bourbaki|Bourbaki, N.]] (1994), ''Elements of the History of Mathematics'', John Meldrum (trans.), Springer-Verlag, Berlin, Germany. * [[Paul Richard Halmos|Halmos, P.R.]] (1960), ''Naive Set Theory'', D. Van Nostrand Company, Princeton, NJ. * [[Francis William Lawvere|Lawvere, F.W.]], and [[Robert Rosebrugh|Rosebrugh, R.]] (2003), ''Sets for Mathematics'', Cambridge University Press, Cambridge, UK. * Maddux, R.D. (2006), ''Relation Algebras'', vol.&nbsp;150 in 'Studies in Logic and the Foundations of Mathematics', Elsevier Science. * Mili, A., Desharnais, J., Mili, F., with Frappier, M. (1994), ''Computer Program Construction'', Oxford University Press, New York, NY. * [[Marvin L. Minsky|Minsky, M.L.]], and [[Seymour A. Papert|Papert, S.A.]] (1969/1988), ''[[Perceptron]]s, An Introduction to Computational Geometry'', MIT Press, Cambridge, MA, 1969. Expanded edition, 1988. * [[Charles Sanders Peirce|Peirce, C.S.]] (1984), ''Writings of Charles S. Peirce : A Chronological Edition, Volume 2, 1867-1871'', Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN. * [[Josiah Royce|Royce, J.]] (1961), ''The Principles of Logic'', Philosophical Library, New York, NY. * [[Alfred Tarski|Tarski, A.]] (1956/1983), ''Logic, Semantics, Metamathematics, Papers from 1923 to 1938'', J.H. Woodger (trans.), 1st edition, Oxford University Press, 1956. 2nd edition, J. Corcoran (ed.), Hackett Publishing, Indianapolis, IN, 1983. * [[Stanisław Marcin Ulam|Ulam, S.M.]] (1990), ''Analogies Between Analogies : The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators'', A.R. Bednarek and Françoise Ulam (eds.), University of California Press, Berkeley, CA. * [[Paulus Venetus|Venetus, P.]] (1984), ''Logica Parva, Translation of the 1472 Edition with Introduction and Notes'', Alan R. Perreiah (trans.), Philosophia Verlag, Munich, Germany. ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Relation_(mathematics) Relation @ InterSciWiki] * [http://mywikibiz.com/Relation_(mathematics) Relation @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Relation_(mathematics) Relation @ Subject Wikis] * [http://en.wikiversity.org/wiki/Relation_(mathematics) Relation @ Wikiversity] * [http://beta.wikiversity.org/wiki/Relation_(mathematics) Relation @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Relation_(mathematics) Relation], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Relation_(mathematics) Relation], [http://mywikibiz.com/ MyWikiBiz] * [http://wikinfo.org/w/index.php/Relation_(mathematics) Relation], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Relation_(mathematics) Relation], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Relation_(mathematics) Relation], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Relation_(mathematics)&oldid=73324659 Relation], [http://en.wikipedia.org/ Wikipedia] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Database Theory]] [[Category:Discrete Mathematics]] [[Category:Inquiry]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Relation Theory]] [[Category:Semiotics]] [[Category:Set Theory]] f69b98dc52859a511281fff55476961fea6e7802 750 749 2015-11-13T18:14:24Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. In mathematics, a '''finitary relation''' is defined by one of the formal definitions given below. * The basic idea is to generalize the concept of a two-place relation, such as the relation of ''equality'' denoted by the sign &ldquo;<math>=\!</math>&rdquo; in a statement like <math>5 + 7 = 12\!</math> or the relation of ''order'' denoted by the sign &ldquo;<math>{<}\!</math>&rdquo; in a statement like <math>5 < 12.\!</math>&nbsp; Relations that involve two ''places'' or ''roles'' are called ''binary relations'' by some and ''dyadic relations'' by others, the latter being historically prior but also useful when necessary to avoid confusion with ''binary (base 2) numerals''. * The concept of a two-place relation is generalized by considering relations with increasing but still finite numbers of places or roles.&nbsp; These are called ''finite-place'' or ''finitary'' relations.&nbsp; A&nbsp;finitary relation involving <math>k\!</math> places is variously called a ''<math>k\!</math>-ary'', a ''<math>k\!</math>-adic'', or a ''<math>k\!</math>-dimensional'' relation.&nbsp; The number <math>k\!</math> is then called the ''arity'', the ''adicity'', or the ''dimension'' of the relation, respectively. ==Informal introduction== The definition of ''relation'' given in the next section formally captures a concept that is actually quite familiar from everyday life.&nbsp; For example, consider the relationship, involving three roles that people might play, expressed in a statement of the form <math>X ~\text{suspects that}~ Y ~\text{likes}~ Z.\!</math>&nbsp; The facts of a concrete situation could be organized in the form of a Table like the one below: <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:60%" |+ style="height:30px" | <math>\text{Relation}~ S ~:~ X ~\text{suspects that}~ Y ~\text{likes}~ Z\!</math> |- style="height:40px; background:ghostwhite" | <math>\text{Person}~ X\!</math> | <math>\text{Person}~ Y\!</math> | <math>\text{Person}~ Z\!</math> |- | <math>\text{Alice}\!</math> | <math>\text{Bob}\!</math> | <math>\text{Denise}\!</math> |- | <math>\text{Charles}\!</math> | <math>\text{Alice}\!</math> | <math>\text{Bob}\!</math> |- | <math>\text{Charles}\!</math> | <math>\text{Charles}\!</math> | <math>\text{Alice}\!</math> |- | <math>\text{Denise}\!</math> | <math>\text{Denise}\!</math> | <math>\text{Denise}\!</math> |} <br> Each row of the Table records a fact or makes an assertion of the form <math>X ~\text{suspects that}~ Y ~\text{likes}~ Z.\!</math>&nbsp; For instance, the first row says, in effect, <math>\text{Alice suspects that Bob likes Denise.}\!</math>&nbsp; The Table represents a relation <math>S\!</math> over the set <math>P\!</math> of people under discussion: {| align="center" cellpadding="10" | <math>P ~=~ \{ \text{Alice}, \text{Bob}, \text{Charles}, \text{Denise} \}\!</math> |} The data of the Table are equivalent to the following set of ordered triples: {| align="center" cellpadding="10" | <math>\begin{smallmatrix} S & = & \{ & \text{(Alice, Bob, Denise)}, & \text{(Charles, Alice, Bob)}, & \text{(Charles, Charles, Alice)}, & \text{(Denise, Denise, Denise)} & \} \end{smallmatrix}\!</math> |} By a slight overuse of notation, it is usual to write <math>S (\text{Alice}, \text{Bob}, \text{Denise})\!</math> to say the same thing as the first row of the Table.&nbsp; The relation <math>S\!</math> is a ''triadic'' or ''ternary'' relation, since there are ''three'' items involved in each row.&nbsp; The relation itself is a mathematical object, defined in terms of concepts from set theory, that carries all the information from the Table in one neat package. The Table for relation <math>S\!</math> is an extremely simple example of a relational database.&nbsp; The theoretical aspects of databases are the specialty of one branch of computer science, while their practical impacts have become all too familiar in our everyday lives.&nbsp; Computer scientists, logicians, and mathematicians, however, tend to see different things when they look at these concrete examples and samples of the more general concept of a relation. For one thing, databases are designed to deal with empirical data, and experience is always finite, whereas mathematics is nothing if not concerned with infinity, at the very least, potential infinity. This difference in perspective brings up a number of ideas that are usefully introduced at this point, if by no means covered in depth. ==Example 1. Divisibility== A more typical example of a two-place relation in mathematics is the relation of ''divisibility'' between two positive integers <math>n\!</math> and <math>m\!</math> that is expressed in statements like <math>{}^{\backprime\backprime} n ~\text{divides}~ m {}^{\prime\prime}\!</math> or <math>{}^{\backprime\backprime} n ~\text{goes into}~ m {}^{\prime\prime}.\!</math>&nbsp; This is a relation that comes up so often that a special symbol <math>{}^{\backprime\backprime} | {}^{\prime\prime}\!</math> is reserved to express it, allowing one to write <math>{}^{\backprime\backprime} n|m {}^{\prime\prime}\!</math> for <math>{}^{\backprime\backprime} n ~\text{divides}~ m {}^{\prime\prime}.\!</math> To express the binary relation of divisibility in terms of sets, we have the set <math>P\!</math> of positive integers, <math>P = \{ 1, 2, 3, \ldots \},\!</math> and we have the binary relation <math>D\!</math> on <math>P\!</math> such that the ordered pair <math>(n, m)\!</math> is in the relation <math>D\!</math> just in case <math>n|m.\!</math>&nbsp; In other turns of phrase that are frequently used, one says that the number <math>n\!</math> is related by <math>D\!</math> to the number <math>m\!</math> just in case <math>n\!</math> is a factor of <math>m,\!</math> that is, just in case <math>n\!</math> divides <math>m\!</math> with no remainder.&nbsp; The relation <math>D,\!</math> regarded as a set of ordered pairs, consists of all pairs of numbers <math>(n, m)\!</math> such that <math>n\!</math> divides <math>m.\!</math> For example, <math>2\!</math> is a factor of <math>4,\!</math> and <math>6\!</math> is a factor of <math>72,\!</math> which two facts can be written either as <math>2|4\!</math> and <math>6|72\!</math> or as <math>D(2, 4)\!</math> and <math>D(6, 72).\!</math> ==Formal definitions== There are two definitions of <math>k\!</math>-place relations that are commonly encountered in mathematics.&nbsp; In order of simplicity, the first of these definitions is as follows: '''Definition 1.''' &nbsp; A '''relation''' <math>L\!</math> over the sets <math>X_1, \ldots, X_k\!</math> is a subset of their cartesian product, written <math>L \subseteq X_1 \times \ldots \times X_k.\!</math>&nbsp; Under this definition, then, a <math>k\!</math>-ary relation is simply a set of <math>k\!</math>-tuples. The second definition makes use of an idiom that is common in mathematics, saying that &ldquo;such and such is an <math>n\!</math>-tuple&rdquo; to mean that the mathematical object being defined is determined by the specification of <math>n\!</math> component mathematical objects.&nbsp; In the case of a relation <math>L\!</math> over <math>k\!</math> sets, there are <math>k + 1\!</math> things to specify, namely, the <math>k\!</math> sets plus a subset of their cartesian product.&nbsp; In the idiom, this is expressed by saying that <math>L\!</math> is a <math>(k+1)\!</math>-tuple. '''Definition 2.''' &nbsp; A '''relation''' <math>L\!</math> over the sets <math>X_1, \ldots, X_k\!</math> is a <math>(k+1)\!</math>-tuple <math>L = (X_1, \ldots, X_k, \mathrm{graph}(L)),\!</math> where <math>\mathrm{graph}(L)\!</math> is a subset of the cartesian product <math>X_1 \times \ldots \times X_k~\!</math> called the ''graph'' of <math>L.\!</math> Elements of a relation are sometimes denoted by using boldface characters, for example, the constant element <math>\mathbf{a} = (a_1, \ldots, a_k)\!</math> or the variable element <math>\mathbf{x} = (x_1, \ldots, x_k).\!</math> A statement of the form &ldquo;<math>\mathbf{a}\!</math> is in the relation <math>L\!</math>&rdquo; is taken to mean that <math>\mathbf{a}\!</math> is in <math>L\!</math> under the first definition and that <math>\mathbf{a}\!</math> is in <math>\mathrm{graph}(L)\!</math> under the second definition. The following considerations apply under either definition: :* The sets ''X''<sub>''j''</sub> for ''j'' = 1 to ''k'' are called the ''[[domain]]s'' of the relation. In the case of the first definition, the relation itself does not uniquely determine a given sequence of domains. :* If all of the domains ''X''<sub>''j''</sub> are the same set ''X'', then ''L'' is more simply referred to as a ''k''-ary relation over ''X''. :* If any of the domains ''X''<sub>''j''</sub> is empty, then the cartesian product is empty, and the only relation over such a sequence of domains is the empty relation ''L'' = <math>\varnothing</math>. As a result, naturally occurring applications of the relation concept typically involve a stipulation that all of the domains be nonempty. As a rule, whatever definition best fits the application at hand will be chosen for that purpose, and anything that falls under it will be called a 'relation' for the duration of that discussion. If it becomes necessary to distinguish the two alternatives, the latter type of object can be referred to as an ''embedded'' or ''included'' relation. If ''L'' is a relation over the domains ''X''<sub>1</sub>, …, ''X''<sub>''k''</sub>, it is conventional to consider a sequence of terms called ''variables'', ''x''<sub>1</sub>, …, ''x''<sub>''k''</sub>, that are said to ''range over'' the respective domains. A ''[[boolean domain]]'' '''B''' is a generic 2-element set, say, '''B''' = {0, 1}, whose elements are interpreted as logical values, typically 0 = false and 1 = true. The ''[[characteristic function]]'' of the relation ''L'', written ''f''<sub>''L''</sub> or &chi;(''L''), is the [[boolean-valued function]] ''f''<sub>''L''</sub>&nbsp;:&nbsp;''X''<sub>1</sub>&nbsp;&times;&nbsp;…&nbsp;&times;&nbsp;''X''<sub>''k''</sub>&nbsp;→&nbsp;'''B''', defined in such a way that ''f''<sub>''L''</sub>(<math>\mathbf{x}</math>) = 1 just in case the ''k''-tuple <math>\mathbf{x}</math> is in the relation ''L''. The characteristic function of a relation may also be called its ''[[indicator function]]'', especially in probability and statistics. It is conventional in applied mathematics, computer science, and statistics to refer to a boolean-valued function like ''f''<sub>''L''</sub> as a ''k''-place ''[[predicate]]''. From the more abstract viewpoints of [[formal logic]] and [[model theory]], the relation ''L'' is seen as constituting a ''logical model'' or a ''relational structure'' that serves as one of many possible [[interpretation]]s of a corresponding ''k''-place ''predicate symbol'', as that term is used in ''[[first-order logic|predicate calculus]]''. Due to the convergence of many different styles of study on the same areas of interest, the reader will find much variation in usage here. The variation presented in this article treats a relation as the [[set theory|set-theoretic]] ''[[extension (semantics)|extension]]'' of a relational concept or term. Another variation reserves the term 'relation' to the corresponding logical entity, either the ''[[comprehension (logic)|logical comprehension]]'', which is the totality of ''[[intension]]s'' or abstract ''[[property (philosophy)|properties]]'' that all of the elements of the relation in extension have in common, or else the symbols that are taken to denote these elements and intensions. Further, but hardly finally, some writers of the latter persuasion introduce terms with more concrete connotations, like 'relational structure', for the set-theoretic extension of a given relational concept. ==Example 2. Coplanarity== For lines ''L'' in three-dimensional space, there is a ternary relation picking out the triples of lines that are [[coplanar]]. This ''does not'' reduce to the binary [[symmetric relation]] of coplanarity of pairs of lines. In other words, writing ''P''(''L'', ''M'', ''N'') when the lines ''L'', ''M'', and ''N'' lie in a plane, and ''Q''(''L'', ''M'') for the binary relation, it is not true that ''Q''(''L'', ''M''), ''Q''(''M'', ''N'') and ''Q''(''N'', ''L'') together imply ''P''(''L'', ''M'', ''N''); although the converse is certainly true (any pair out of three coplanar lines is coplanar, ''a fortiori''). There are two geometrical reasons for this. In one case, for example taking the ''x''-axis, ''y''-axis and ''z''-axis, the three lines are concurrent, i.e. intersect at a single point. In another case, ''L'', ''M'', and ''N'' can be three edges of an infinite [[triangular prism]]. What is true is that if each pair of lines intersects, and the points of intersection are distinct, then pairwise coplanarity implies coplanarity of the triple. ==Remarks== Relations are classified according to the number of sets in the cartesian product, in other words the number of terms in the expression: :* Unary relation or [[property (philosophy)|property]]: ''L''(''u'') :* Binary relation: ''L''(''u'', ''v'') or ''u'' ''L'' ''v'' :* Ternary relation: ''L''(''u'', ''v'', ''w'') :* Quaternary relation: ''L''(''u'', ''v'', ''w'', ''x'') Relations with more than four terms are usually referred to as ''k''-ary, for example, "a 5-ary relation". ==References== * [[Charles Sanders Peirce|Peirce, C.S.]] (1870), &ldquo;Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic&rdquo;, ''Memoirs of the American Academy of Arts and Sciences'' 9, 317&ndash;378, 1870. Reprinted, ''Collected Papers'' CP&nbsp;3.45&ndash;149, ''Chronological Edition'' CE&nbsp;2, 359&ndash;429. * Ulam, S.M., and Bednarek, A.R. (1990), &ldquo;On the Theory of Relational Structures and Schemata for Parallel Computation&rdquo;, pp. 477&ndash;508 in A.R. Bednarek and Françoise Ulam (eds.), ''Analogies Between Analogies : The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators'', University of California Press, Berkeley, CA. ==Bibliography== * Bourbaki, N. (1994), ''Elements of the History of Mathematics'', John Meldrum (trans.), Springer-Verlag, Berlin, Germany. * Halmos, P.R. (1960), ''Naive Set Theory'', D. Van Nostrand Company, Princeton, NJ. * Lawvere, F.W., and Rosebrugh, R. (2003), ''Sets for Mathematics'', Cambridge University Press, Cambridge, UK. * Maddux, R.D. (2006), ''Relation Algebras'', vol.&nbsp;150 in Studies in Logic and the Foundations of Mathematics, Elsevier Science. * Mili, A., Desharnais, J., Mili, F., with Frappier, M. (1994), ''Computer Program Construction'', Oxford University Press, New&nbsp;York, NY. * Minsky, M.L., and Papert, S.A. (1969/1988), ''Perceptrons, An Introduction to Computational Geometry'', MIT Press, Cambridge, MA, 1969. Expanded edition, 1988. * [[Charles Sanders Peirce|Peirce, C.S.]] (1984), ''Writings of Charles S. Peirce : A Chronological Edition, Volume&nbsp;2, 1867&ndash;1871'', Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN. * Royce, J. (1961), ''The Principles of Logic'', Philosophical Library, New&nbsp;York, NY. * Tarski, A. (1956/1983), ''Logic, Semantics, Metamathematics, Papers from 1923 to 1938'', J.H. Woodger (trans.), 1st&nbsp;edition, Oxford University Press, 1956. 2nd&nbsp;edition, J. Corcoran (ed.), Hackett Publishing, Indianapolis, IN, 1983. * Ulam, S.M. (1990), ''Analogies Between Analogies : The Mathematical Reports of S.M. Ulam and His Los&nbsp;Alamos Collaborators'', A.R. Bednarek and Françoise Ulam (eds.), University of California Press, Berkeley, CA. * Venetus, P. (1984), ''Logica Parva, Translation of the 1472 Edition with Introduction and Notes'', Alan R. Perreiah (trans.), Philosophia Verlag, Munich, Germany. ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Relation_(mathematics) Relation @ InterSciWiki] * [http://mywikibiz.com/Relation_(mathematics) Relation @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Relation_(mathematics) Relation @ Subject Wikis] * [http://en.wikiversity.org/wiki/Relation_(mathematics) Relation @ Wikiversity] * [http://beta.wikiversity.org/wiki/Relation_(mathematics) Relation @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Relation_(mathematics) Relation], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Relation_(mathematics) Relation], [http://mywikibiz.com/ MyWikiBiz] * [http://wikinfo.org/w/index.php/Relation_(mathematics) Relation], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Relation_(mathematics) Relation], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Relation_(mathematics) Relation], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Relation_(mathematics)&oldid=73324659 Relation], [http://en.wikipedia.org/ Wikipedia] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Database Theory]] [[Category:Discrete Mathematics]] [[Category:Inquiry]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Relation Theory]] [[Category:Semiotics]] [[Category:Set Theory]] 763f47fb9547772659b216f5e36c75057072097a 751 750 2015-11-14T17:10:44Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. In mathematics, a '''finitary relation''' is defined by one of the formal definitions given below. * The basic idea is to generalize the concept of a two-place relation, such as the relation of ''equality'' denoted by the sign &ldquo;<math>=\!</math>&rdquo; in a statement like <math>5 + 7 = 12\!</math> or the relation of ''order'' denoted by the sign &ldquo;<math>{<}\!</math>&rdquo; in a statement like <math>5 < 12.\!</math>&nbsp; Relations that involve two ''places'' or ''roles'' are called ''binary relations'' by some and ''dyadic relations'' by others, the latter being historically prior but also useful when necessary to avoid confusion with ''binary (base 2) numerals''. * The concept of a two-place relation is generalized by considering relations with increasing but still finite numbers of places or roles.&nbsp; These are called ''finite-place'' or ''finitary'' relations.&nbsp; A&nbsp;finitary relation involving <math>k\!</math> places is variously called a ''<math>k\!</math>-ary'', ''<math>k\!</math>-adic'', or ''<math>k\!</math>-dimensional'' relation.&nbsp; The number <math>k\!</math> is then called the ''arity'', the ''adicity'', or the ''dimension'' of the relation, respectively. ==Informal introduction== The definition of ''relation'' given in the next section formally captures a concept that is actually quite familiar from everyday life.&nbsp; For example, consider the relationship, involving three roles that people might play, expressed in a statement of the form <math>X ~\text{suspects that}~ Y ~\text{likes}~ Z.\!</math>&nbsp; The facts of a concrete situation could be organized in the form of a Table like the one below: <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:60%" |+ style="height:30px" | <math>\text{Relation}~ S ~:~ X ~\text{suspects that}~ Y ~\text{likes}~ Z\!</math> |- style="height:40px; background:ghostwhite" | <math>\text{Person}~ X\!</math> | <math>\text{Person}~ Y\!</math> | <math>\text{Person}~ Z\!</math> |- | <math>\text{Alice}\!</math> | <math>\text{Bob}\!</math> | <math>\text{Denise}\!</math> |- | <math>\text{Charles}\!</math> | <math>\text{Alice}\!</math> | <math>\text{Bob}\!</math> |- | <math>\text{Charles}\!</math> | <math>\text{Charles}\!</math> | <math>\text{Alice}\!</math> |- | <math>\text{Denise}\!</math> | <math>\text{Denise}\!</math> | <math>\text{Denise}\!</math> |} <br> Each row of the Table records a fact or makes an assertion of the form <math>X ~\text{suspects that}~ Y ~\text{likes}~ Z.\!</math>&nbsp; For instance, the first row says, in effect, <math>\text{Alice suspects that Bob likes Denise.}\!</math>&nbsp; The Table represents a relation <math>S\!</math> over the set <math>P\!</math> of people under discussion: {| align="center" cellpadding="10" | <math>P ~=~ \{ \text{Alice}, \text{Bob}, \text{Charles}, \text{Denise} \}\!</math> |} The data of the Table are equivalent to the following set of ordered triples: {| align="center" cellpadding="10" | <math>\begin{smallmatrix} S & = & \{ & \text{(Alice, Bob, Denise)}, & \text{(Charles, Alice, Bob)}, & \text{(Charles, Charles, Alice)}, & \text{(Denise, Denise, Denise)} & \} \end{smallmatrix}\!</math> |} By a slight overuse of notation, it is usual to write <math>S (\text{Alice}, \text{Bob}, \text{Denise})\!</math> to say the same thing as the first row of the Table.&nbsp; The relation <math>S\!</math> is a ''triadic'' or ''ternary'' relation, since there are ''three'' items involved in each row.&nbsp; The relation itself is a mathematical object, defined in terms of concepts from set theory, that carries all the information from the Table in one neat package. The Table for relation <math>S\!</math> is an extremely simple example of a relational database.&nbsp; The theoretical aspects of databases are the specialty of one branch of computer science, while their practical impacts have become all too familiar in our everyday lives.&nbsp; Computer scientists, logicians, and mathematicians, however, tend to see different things when they look at these concrete examples and samples of the more general concept of a relation. For one thing, databases are designed to deal with empirical data, and experience is always finite, whereas mathematics is nothing if not concerned with infinity, at the very least, potential infinity. This difference in perspective brings up a number of ideas that are usefully introduced at this point, if by no means covered in depth. ==Example 1. Divisibility== A more typical example of a two-place relation in mathematics is the relation of ''divisibility'' between two positive integers <math>n\!</math> and <math>m\!</math> that is expressed in statements like <math>{}^{\backprime\backprime} n ~\text{divides}~ m {}^{\prime\prime}\!</math> or <math>{}^{\backprime\backprime} n ~\text{goes into}~ m {}^{\prime\prime}.\!</math>&nbsp; This is a relation that comes up so often that a special symbol <math>{}^{\backprime\backprime} | {}^{\prime\prime}\!</math> is reserved to express it, allowing one to write <math>{}^{\backprime\backprime} n|m {}^{\prime\prime}\!</math> for <math>{}^{\backprime\backprime} n ~\text{divides}~ m {}^{\prime\prime}.\!</math> To express the binary relation of divisibility in terms of sets, we have the set <math>P\!</math> of positive integers, <math>P = \{ 1, 2, 3, \ldots \},\!</math> and we have the binary relation <math>D\!</math> on <math>P\!</math> such that the ordered pair <math>(n, m)\!</math> is in the relation <math>D\!</math> just in case <math>n|m.\!</math>&nbsp; In other turns of phrase that are frequently used, one says that the number <math>n\!</math> is related by <math>D\!</math> to the number <math>m\!</math> just in case <math>n\!</math> is a factor of <math>m,\!</math> that is, just in case <math>n\!</math> divides <math>m\!</math> with no remainder.&nbsp; The relation <math>D,\!</math> regarded as a set of ordered pairs, consists of all pairs of numbers <math>(n, m)\!</math> such that <math>n\!</math> divides <math>m.\!</math> For example, <math>2\!</math> is a factor of <math>4,\!</math> and <math>6\!</math> is a factor of <math>72,\!</math> which two facts can be written either as <math>2|4\!</math> and <math>6|72\!</math> or as <math>D(2, 4)\!</math> and <math>D(6, 72).\!</math> ==Formal definitions== There are two definitions of <math>k\!</math>-place relations that are commonly encountered in mathematics.&nbsp; In order of simplicity, the first of these definitions is as follows: '''Definition 1.''' &nbsp; A '''relation''' <math>L\!</math> over the sets <math>X_1, \ldots, X_k\!</math> is a subset of their cartesian product, written <math>L \subseteq X_1 \times \ldots \times X_k.\!</math>&nbsp; Under this definition, then, a <math>k\!</math>-ary relation is simply a set of <math>k\!</math>-tuples. The second definition makes use of an idiom that is common in mathematics, saying that &ldquo;such and such is an <math>n\!</math>-tuple&rdquo; to mean that the mathematical object being defined is determined by the specification of <math>n\!</math> component mathematical objects.&nbsp; In the case of a relation <math>L\!</math> over <math>k\!</math> sets, there are <math>k + 1\!</math> things to specify, namely, the <math>k\!</math> sets plus a subset of their cartesian product.&nbsp; In the idiom, this is expressed by saying that <math>L\!</math> is a <math>(k+1)\!</math>-tuple. '''Definition 2.''' &nbsp; A '''relation''' <math>L\!</math> over the sets <math>X_1, \ldots, X_k\!</math> is a <math>(k+1)\!</math>-tuple <math>L = (X_1, \ldots, X_k, \mathrm{graph}(L)),\!</math> where <math>\mathrm{graph}(L)\!</math> is a subset of the cartesian product <math>X_1 \times \ldots \times X_k~\!</math> called the ''graph'' of <math>L.\!</math> Elements of a relation are sometimes denoted by using boldface characters, for example, the constant element <math>\mathbf{a} = (a_1, \ldots, a_k)\!</math> or the variable element <math>\mathbf{x} = (x_1, \ldots, x_k).\!</math> A statement of the form &ldquo;<math>\mathbf{a}\!</math> is in the relation <math>L\!</math>&rdquo; is taken to mean that <math>\mathbf{a}\!</math> is in <math>L\!</math> under the first definition and that <math>\mathbf{a}\!</math> is in <math>\mathrm{graph}(L)\!</math> under the second definition. The following considerations apply under either definition: :* The sets <math>X_j~\!</math> for <math>j = 1 ~\text{to}~ k\!</math> are called the ''domains'' of the relation.&nbsp; In the case of the first definition, the relation itself does not uniquely determine a given sequence of domains. :* If all the domains <math>X_j~\!</math> are the same set <math>X,\!</math> then <math>L\!</math> is more simply referred to as a <math>k\!</math>-ary relation over <math>X.\!</math> :* If any domain <math>X_j~\!</math> is empty then the cartesian product is empty and the only relation over such a sequence of domains is the empty relation <math>L = \varnothing.\!</math>&nbsp; Most applications of the relation concept will set aside this trivial case and assume that all domains are nonempty. If <math>L\!</math> is a relation over the domains <math>X_1, \ldots, X_k,\!</math> it is conventional to consider a sequence of terms called ''variables'', <math>x_1, \ldots, x_k,\!</math> that are said to ''range over'' the respective domains. A ''[[boolean domain]]'' <math>\mathbb{B}\!</math> is a generic 2-element set, say, <math>\mathbb{B} = \{ 0, 1 \},\!</math> whose elements are interpreted as logical values, typically <math>0 = \mathrm{false}\!</math> and <math>1 = \mathrm{true}.\!</math> The ''characteristic function'' of the relation <math>L,\!</math> written <math>f_L\!</math> or <math>\chi(L),\!</math> is the [[boolean-valued function]] <math>f_L : X_1 \times \ldots \times X_k \to \mathbb{B},\!</math> defined in such a way that <math>f_L (\mathbf{x}) = 1\!</math> just in case the <math>k\!</math>-tuple <math>\mathbf{x} = (x_1, \ldots, x_k)\!</math> is in the relation <math>L.\!</math>&nbsp; The characteristic function of a relation may also be called its ''indicator function'', especially in probabilistic and statistical contexts. It is conventional in applied mathematics, computer science, and statistics to refer to a boolean-valued function like <math>f_L\!</math> as a <math>k\!</math>-place ''predicate''.&nbsp; From the more abstract viewpoints of formal logic and model theory, the relation <math>L\!</math> is seen as constituting a ''logical model'' or a ''relational structure'' that serves as one of many possible interpretations of a corresponding <math>k\!</math>-place ''predicate symbol'', as that term is used in ''predicate calculus''. Due to the convergence of many traditions of study, there are wide variations in the language used to describe relations.&nbsp; The ''extensional'' approach presented in this article treats a relation as the set-theoretic ''extension'' of a relational concept or term.&nbsp; An alternative, ''intensional approach'' reserves the term ''relation'' to the corresponding logical entity, either the ''logical comprehension'', which is the totality of ''intensions'' or abstract ''properties'' that all the elements of the extensional relation have in common, or else the symbols that are taken to denote those elements and intensions. ==Example 2. Coplanarity== For lines <math>\ell\!</math> in three-dimensional space, there is a triadic relation picking out the triples of lines that are coplanar.&nbsp; This does not reduce to the dyadic relation of coplanarity between pairs of lines. In other words, writing <math>P(\ell, m, n)\!</math> when the lines <math>\ell, m, n\!</math> lie in a plane, and <math>Q(\ell, m)\!</math> for the binary relation, it is not true that <math>Q(\ell, m),\!</math> <math>Q(m, n),\!</math> and <math>Q(n, \ell)\!</math> together imply <math>P(\ell, m, n),\!</math> although the converse is certainly true:&nbsp; any two of three coplanar lines are necessarily coplanar.&nbsp; There are two geometrical reasons for this. In one case, for example taking the <math>x\!</math>-axis, <math>y\!</math>-axis, and <math>z\!</math>-axis, the three lines are concurrent, that is, they intersect at a single point.&nbsp; In another case, <math>\ell, m, n\!</math> can be three edges of an infinite triangular prism. What is true is that if each pair of lines intersects, and the points of intersection are distinct, then pairwise coplanarity implies coplanarity of the triple. ==Remarks== Relations are classified by the number of sets in the cartesian product, in other words, the number of places or terms in the relational expression: {| align="center" cellspacing="6" width="90%" | width="18%" | <math>L(a)\!</math> | Monadic or unary relation, in other words, a property or set |- | <math>L(a, b) ~\text{or}~ a L b\!</math> | Dyadic or binary relation |- | <math>L(a, b, c)\!</math> | Triadic or ternary relation |- | <math>L(a, b, c, d)\!</math> | Tetradic or quaternary relation |- | <math>L(a, b, c, d, e)\!</math> | Pentadic or quinary relation |} Relations with more than five terms are usually referred to as <math>k\!</math>-adic or <math>k\!</math>-ary, for example, a 6-adic, 6-ary, or hexadic relation. ==References== * [[Charles Sanders Peirce|Peirce, C.S.]] (1870), &ldquo;Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic&rdquo;, ''Memoirs of the American Academy of Arts and Sciences'' 9, 317&ndash;378, 1870. Reprinted, ''Collected Papers'' CP&nbsp;3.45&ndash;149, ''Chronological Edition'' CE&nbsp;2, 359&ndash;429. * Ulam, S.M., and Bednarek, A.R. (1990), &ldquo;On the Theory of Relational Structures and Schemata for Parallel Computation&rdquo;, pp. 477&ndash;508 in A.R. Bednarek and Françoise Ulam (eds.), ''Analogies Between Analogies : The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators'', University of California Press, Berkeley, CA. ==Bibliography== * Bourbaki, N. (1994), ''Elements of the History of Mathematics'', John Meldrum (trans.), Springer-Verlag, Berlin, Germany. * Halmos, P.R. (1960), ''Naive Set Theory'', D. Van Nostrand Company, Princeton, NJ. * Lawvere, F.W., and Rosebrugh, R. (2003), ''Sets for Mathematics'', Cambridge University Press, Cambridge, UK. * Maddux, R.D. (2006), ''Relation Algebras'', vol.&nbsp;150 in Studies in Logic and the Foundations of Mathematics, Elsevier Science. * Mili, A., Desharnais, J., Mili, F., with Frappier, M. (1994), ''Computer Program Construction'', Oxford University Press, New&nbsp;York, NY. * Minsky, M.L., and Papert, S.A. (1969/1988), ''Perceptrons, An Introduction to Computational Geometry'', MIT Press, Cambridge, MA, 1969. Expanded edition, 1988. * [[Charles Sanders Peirce|Peirce, C.S.]] (1984), ''Writings of Charles S. Peirce : A Chronological Edition, Volume&nbsp;2, 1867&ndash;1871'', Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN. * Royce, J. (1961), ''The Principles of Logic'', Philosophical Library, New&nbsp;York, NY. * Tarski, A. (1956/1983), ''Logic, Semantics, Metamathematics, Papers from 1923 to 1938'', J.H. Woodger (trans.), 1st&nbsp;edition, Oxford University Press, 1956. 2nd&nbsp;edition, J. Corcoran (ed.), Hackett Publishing, Indianapolis, IN, 1983. * Ulam, S.M. (1990), ''Analogies Between Analogies : The Mathematical Reports of S.M. Ulam and His Los&nbsp;Alamos Collaborators'', A.R. Bednarek and Françoise Ulam (eds.), University of California Press, Berkeley, CA. * Venetus, P. (1984), ''Logica Parva, Translation of the 1472 Edition with Introduction and Notes'', Alan R. Perreiah (trans.), Philosophia Verlag, Munich, Germany. ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Relation_(mathematics) Relation @ InterSciWiki] * [http://mywikibiz.com/Relation_(mathematics) Relation @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Relation_(mathematics) Relation @ Subject Wikis] * [http://en.wikiversity.org/wiki/Relation_(mathematics) Relation @ Wikiversity] * [http://beta.wikiversity.org/wiki/Relation_(mathematics) Relation @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Relation_(mathematics) Relation], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Relation_(mathematics) Relation], [http://mywikibiz.com/ MyWikiBiz] * [http://wikinfo.org/w/index.php/Relation_(mathematics) Relation], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Relation_(mathematics) Relation], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Relation_(mathematics) Relation], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Relation_(mathematics)&oldid=73324659 Relation], [http://en.wikipedia.org/ Wikipedia] [[Category:Charles Sanders Peirce]] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Database Theory]] [[Category:Discrete Mathematics]] [[Category:Inquiry]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Relation Theory]] [[Category:Semiotics]] [[Category:Set Theory]] 0e66a314fbde04abcb79fd75696b1b131d31670a 0 335 752 724 2015-11-14T18:18:41Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. '''Relation composition''', or the composition of [[relation (mathematics)|relations]], is the generalization of function composition, or the composition of functions.&nbsp; The following treatment of relation composition takes the &ldquo;strongly typed&rdquo; approach to relations that is outlined in the article on [[relation theory]]. ==Preliminaries== There are several ways to formalize the subject matter of relations. Relations and their combinations may be described in the logic of relative terms, in set theories of various kinds, and through a broadening of category theory from functions to relations in general. The first order of business is to define the operation on [[relation (mathematics)|relations]] that is variously known as the ''composition of relations'', ''relational composition'', or ''relative multiplication''. In approaching the more general constructions, it pays to begin with the composition of dyadic and triadic relations. As an incidental observation on usage, there are many different conventions of syntax for denoting the application and composition of relations, with perhaps even more options in general use than are common for the application and composition of functions. In this case there is little chance of standardization, since the convenience of conventions is relative to the context of use, and the same writers use different styles of syntax in different settings, depending on the ease of analysis and computation. {| align="center" cellpadding="4" width="90%" | <p>The first dimension of variation in syntax has to do with the correspondence between the order of operation and the linear order of terms on the page.</p> |- | <p>The second dimension of variation in syntax has to do with the automatic assumptions in place about the associations of terms in the absence of associations marked by parentheses. This becomes a significant factor with relations in general because the usual property of associativity is lost as both the complexities of compositions and the dimensions of relations increase.</p> |} These two factors together generate the following four styles of syntax: {| align="center" cellpadding="4" width="90%" | LALA = left application, left association. |- | LARA = left application, right association. |- | RALA = right application, left association. |- | RARA = right application, right association. |} ==Definition== A notion of relational composition is to be defined that generalizes the usual notion of functional composition: {| align="center" cellpadding="4" width="90%" | <p>Composing ''on the right'', <math>f : X \to Y</math> followed by <math>g : Y \to Z</math></p> <p>results in a ''composite function'' formulated as <math>fg : X \to Z.</math></p> |- | <p>Composing ''on the left'', <math>f : X \to Y</math> followed by <math>g : Y \to Z</math></p> <p>results in a ''composite function'' formulated as <math>gf : X \to Z.</math></p> |} Note on notation. The ordinary symbol for functional composition is the ''composition sign'', a small circle "<math>\circ</math>" written between the names of the functions being composed, as <math>f \circ g,</math> but the sign is often omitted if there is no risk of confusing the composition of functions with their algebraic product. In contexts where both compositions and products occur, either the composition is marked on each occasion or else the product is marked by means of a ''center dot'' &ldquo;<math>\cdot</math>&rdquo;, as <math>f \cdot g.</math> Generalizing the paradigm along parallel lines, the ''composition'' of a pair of dyadic relations is formulated in the following two ways: {| align="center" cellpadding="4" width="90%" | <p>Composing ''on the right'', <math>P \subseteq X \times Y</math> followed by <math>Q \subseteq Y \times Z</math></p> <p>results in a ''composite relation'' formulated as <math>PQ \subseteq X \times Z.</math></p> |- | <p>Composing ''on the left'', <math>P \subseteq X \times Y</math> followed by <math>Q \subseteq Y \times Z</math></p> <p>results in a ''composite relation'' formulated as <math>QP \subseteq X \times Z.</math></p> |} ==Geometric construction== There is a neat way of defining relational compositions in geometric terms, not only showing their relationship to the projection operations that come with any cartesian product, but also suggesting natural directions for generalizing relational compositions beyond the dyadic case, and even beyond relations that have any fixed arity, in effect, to the general case of formal languages as generalized relations. This way of looking at relational compositions is sometimes referred to as Tarski's Trick, on account of his having put it to especially good use in his work (Ulam and Bednarek, 1977). It supplies the imagination with a geometric way of visualizing the relational composition of a pair of dyadic relations, doing this by attaching concrete imagery to the basic set-theoretic operations, namely, intersections, projections, and a certain class of operations inverse to projections, here called ''tacit extensions''. The stage is set for Tarski's trick by highlighting the links between two topics that are likely to appear wholly unrelated at first, namely: :* The use of [[logical conjunction]], as denoted by the symbol <math>\land,\!</math> in expressions of the form <math>F(x, y, z) = G(x, y) \land H(y, z),\!</math> to define a triadic relation <math>F\!</math> in terms of a pair of dyadic relations <math>G\!</math> and <math>H.\!</math> :* The concepts of dyadic ''projection'' and ''projective determination'', that are invoked in the &ldquo;weak&rdquo; notion of ''projective reducibility''. The relational composition <math>G \circ H\!</math> of a pair of dyadic relations <math>G\!</math> and <math>H\!</math> will be constructed in three stages, first, by taking the tacit extensions of <math>G\!</math> and <math>H\!</math> to triadic relations that reside in the same space, next, by taking the intersection of these extensions, tantamount to the maximal triadic relation that is consistent with the ''prima facie'' dyadic relation data, finally, by projecting this intersection on a suitable plane to form a third dyadic relation, constituting in fact the relational composition <math>G \circ H\!</math> of the relations <math>G\!</math> and <math>H.\!</math> The construction of a relational composition in a specifically mathematical setting normally begins with [[relation (mathematics)|mathematical relations]] at a higher level of abstraction than the corresponding objects in linguistic or logical settings. This is due to the fact that mathematical objects are typically specified only ''up to isomorphism'' as the conventional saying goes, that is, any objects that have the &ldquo;same form&rdquo; are generally regarded as the being the same thing, for most all intents and mathematical purposes. Thus the mathematical construction of a relational composition begins by default with a pair of dyadic relations that reside, without loss of generality, in the same plane, say, <math>G, H \subseteq X \times Y,\!</math> as shown in Figure&nbsp;1. {| align="center" border="0" cellpadding="10" | <pre> o-------------------------------------------------o | | | o o | | |\ |\ | | | \ | \ | | | \ | \ | | | \ | \ | | | \ | \ | | | \ | \ | | | * \ | * \ | | X * Y X * Y | | \ * | \ * | | | \ G | \ H | | | \ | \ | | | \ | \ | | | \ | \ | | | \ | \ | | | \| \| | | o o | | | o-------------------------------------------------o Figure 1. Dyadic Relations G, H c X x Y </pre> |} The dyadic relations <math>G\!</math> and <math>H\!</math> cannot be composed at all at this point, not without additional information or further stipulation. In order for their relational composition to be possible, one of two types of cases has to happen: :* The first type of case occurs when <math>X = Y.\!</math> In this case, both of the compositions <math>G \circ H\!</math> and <math>H \circ G\!</math> are defined. :* The second type of case occurs when <math>X\!</math> and <math>Y\!</math> are distinct, but when it nevertheless makes sense to speak of a dyadic relation <math>\hat{H}\!</math> that is isomorphic to <math>H,\!</math> but living in the plane <math>YZ,\!</math> that is, in the space of the cartesian product <math>Y \times Z,\!</math> for some set <math>Z.\!</math> Whether you view isomorphic things to be the same things or not, you still have to specify the exact isomorphisms that are needed to transform any given representation of a thing into a required representation of the same thing. Let us imagine that we have done this, and say how later: {| align="center" border="0" cellpadding="10" | <pre> o-------------------------------------------------o | | | o o | | |\ /| | | | \ / | | | | \ / | | | | \ / | | | | \ / | | | | \ / | | | | * \ / * | | | X * Y Y * Z | | \ * | | * / | | \ G | | &#292; / | | \ | | / | | \ | | / | | \ | | / | | \ | | / | | \| |/ | | o o | | | o-------------------------------------------------o Figure 2. Dyadic Relations G c X x Y and &#292; c Y x Z </pre> |} With the required spaces carefully swept out, the stage is set for the presentation of Tarski's trick, and the invocation of the following symbolic formula, claimed to be a definition of the relational composition <math>P \circ Q\!</math> of a pair of dyadic relations <math>P, Q \subseteq X \times X.\!</math> : '''Definition.''' <math>P \circ Q = \mathrm{proj}_{13} (P \times X ~\cap~ X \times Q).\!</math> To get this drift of this definition one needs to understand that it comes from a point of view that regards all dyadic relations as covered well enough by subsets of a suitable cartesian square and thus of the form <math>L \subseteq X \times X.\!</math> So, if one has started out with a dyadic relation of the shape <math>L \subseteq U \times V,\!</math> one merely lets <math>X = U \cup V,\!</math> trading in the initial <math>L\!</math> for a new <math>L \subseteq X \times X\!</math> as need be. The projection <math>\mathrm{proj}_{13}\!</math> is just the projection of the cartesian cube <math>X \times X \times X\!</math> on the space of shape <math>X \times X\!</math> that is spanned by the first and the third domains, but since they now have the same names and the same contents it is necessary to distinguish them by numbering their relational places. Finally, the notation of the cartesian product sign &ldquo;<math>\times\!</math>&rdquo; is extended to signify two other products with respect to a dyadic relation <math>L \subseteq X \times X\!</math> and a subset <math>W \subseteq X,\!</math> as follows: : '''Definition.''' <math>L \times W ~=~ \{ (x, y, z) \in X^3 ~:~ (x, y) \in L ~\mathrm{and}~ z \in W \}.\!</math> : '''Definition.''' <math>W \times L ~=~ \{ (x, y, z) \in X^3 ~:~ x \in W ~\mathrm{and}~ (y, z) \in L \}.\!</math> Applying these definitions to the case <math>P, Q \subseteq X \times X,\!</math> the two dyadic relations whose relational composition <math>P \circ Q \subseteq X \times X\!</math> is about to be defined, one finds: : <math>P \times X ~=~ \{ (x, y, z) \in X^3 ~:~ (x, y) \in P ~\mathrm{and}~ z \in X \},\!</math> : <math>X \times Q ~=~ \{ (x, y, z) \in X^3 ~:~ x \in X ~\mathrm{and}~ (y, z) \in Q \}.\!</math> These are just the appropriate special cases of the tacit extensions already defined. : <math>P \times X ~=~ \mathrm{te}_{12}^3 (P),~\!</math> : <math>X \times Q ~=~ \mathrm{te}_{23}^1 (Q).~\!</math> In summary, then, the expression: : <math>\mathrm{proj}_{13} (P \times X ~\cap~ X \times Q)\!</math> is equivalent to the expression: : <math>\mathrm{proj}_{13} (\mathrm{te}_{12}^3 (P) ~\cap~ \mathrm{te}_{23}^1 (Q))\!</math> and this form is generalized &mdash; although, relative to one's school of thought, perhaps inessentially so &mdash; by the form that was given above as follows: : '''Definition.''' <math>P \circ Q ~=~ \mathrm{proj}_{XZ} (\mathrm{te}_{XY}^Z (P) ~\cap~ \mathrm{te}_{YZ}^X (Q)).\!</math> Figure&nbsp;3 presents a geometric picture of what is involved in formulating a definition of the triadic relation <math>F \subseteq X \times Y \times Z\!</math> by way of a conjunction between the dyadic relation <math>G \subseteq X \times Y\!</math> and the dyadic relation <math>H \subseteq Y \times Z,\!</math> as done for example by means of an expression of the following form: :* <math>F(x, y, z) ~=~ G(x, y) \land H(y, z).\!</math> {| align="center" border="0" cellpadding="10" | <pre> o-------------------------------------------------o | | | o | | /|\ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | o o o | | |\ / \ /| | | | \ / F \ / | | | | \ / * \ / | | | | \ /*\ / | | | | / \//*\\/ \ | | | | / /\/ \/\ \ | | | |/ ///\ /\\\ \| | | o X /// Y \\\ Z o | | |\ \/// | \\\/ /| | | | \ /// | \\\ / | | | | \ ///\ | /\\\ / | | | | \ /// \ | / \\\ / | | | | \/// \ | / \\\/ | | | | /\/ \ | / \/\ | | | | *//\ \|/ /\\* | | | X */ Y o Y \* Z | | \ * | | * / | | \ G | | H / | | \ | | / | | \ | | / | | \ | | / | | \ | | / | | \| |/ | | o o | | | o-------------------------------------------------o Figure 3. Projections of F onto G and H </pre> |} To interpret the Figure, visualize the triadic relation <math>F \subseteq X \times Y \times Z\!</math> as a body in <math>XYZ\!</math>-space, while <math>G\!</math> is a figure in <math>XY\!</math>-space and <math>H\!</math> is a figure in <math>YZ\!</math>-space. The dyadic '''projections''' that accompany a triadic relation over <math>X, Y, Z\!</math> are defined as follows: :* <math>\mathrm{proj}_{XY} (L) ~=~ \{ (x, y) \in X \times Y : (x, y, z) \in L ~\text{for some}~ z \in Z) \},\!</math> :* <math>\mathrm{proj}_{XZ} (L) ~=~ \{ (x, z) \in X \times Z : (x, y, z) \in L ~\text{for some}~ y \in Y) \},\!</math> :* <math>\mathrm{proj}_{YZ} (L) ~=~ \{ (y, z) \in Y \times Z : (x, y, z) \in L ~\text{for some}~ x \in X) \}.\!</math> For many purposes it suffices to indicate the dyadic projections of a triadic relation <math>L\!</math> by means of the briefer equivalents listed next: :* <math>L_{XY} ~=~ \mathrm{proj}_{XY}(L),\!</math> :* <math>L_{XZ} ~=~ \mathrm{proj}_{XZ}(L),\!</math> :* <math>L_{YZ} ~=~ \mathrm{proj}_{YZ}(L).\!</math> In light of these definitions, <math>\mathrm{proj}_{XY}\!</math> is a mapping from the set <math>\mathcal{L}_{XYZ}\!</math> of triadic relations over the domains <math>X, Y, Z\!</math> to the set <math>\mathcal{L}_{XY}\!</math> of dyadic relations over the domains <math>X, Y,\!</math> with similar relationships holding for the other projections. To formalize these relationships in a concise but explicit manner, it serves to add a few more definitions. The set <math>\mathcal{L}_{XYZ},~\!</math> whose members are just the triadic relations over <math>X, Y, Z,\!</math> can be recognized as the set of all subsets of the cartesian product <math>X \times Y \times Z,\!</math> also known as the ''power set'' of <math>X \times Y \times Z,\!</math> and notated here as <math>\mathrm{Pow} (X \times Y \times Z).\!</math> :* <math>\mathcal{L}_{XYZ} ~=~ \{ L : L \subseteq X \times Y \times Z \} ~=~ \mathrm{Pow} (X \times Y \times Z).\!</math> Likewise, the power sets of the pairwise cartesian products encompass all the dyadic relations on pairs of distinct domains that can be chosen from <math>\{ X, Y, Z \}.\!</math> :* <math>\mathcal{L}_{XY} ~=~ \{L : L \subseteq X \times Y \} ~=~ \mathrm{Pow} (X \times Y),~\!</math> :* <math>\mathcal{L}_{XZ} ~=~ \{L : L \subseteq X \times Z \} ~=~ \mathrm{Pow} (X \times Z),~\!</math> :* <math>\mathcal{L}_{YZ} ~=~ \{L : L \subseteq Y \times Z \} ~=~ \mathrm{Pow} (Y \times Z).~\!</math> In mathematics, the inverse relation corresponding to a projection map is usually called an ''extension''. To avoid confusion with other senses of the word, however, it is probably best for the sake of this discussion to stick with the more specific term ''tacit extension''. Given three sets, <math>X, Y, Z,\!</math> and three dyadic relations, :* <math>U \subseteq X \times Y,~\!</math> :* <math>V \subseteq X \times Z,~\!</math> :* <math>W \subseteq Y \times Z,~\!</math> the ''tacit extensions'', <math>\mathrm{te}_{XY}^Z, \mathrm{te}_{XZ}^Y, \mathrm{te}_{YZ}^X,~\!</math> of <math>U, V, W,\!</math> respectively, are defined as follows: :* <math>\mathrm{te}_{XY}^Z (U) ~=~ \{ (x, y, z) : (x, y) \in U \},\!</math> :* <math>\mathrm{te}_{XZ}^Y (V) ~=~ \{ (x, y, z) : (x, z) \in V \},\!</math> :* <math>\mathrm{te}_{YZ}^X (W) ~=~ \{ (x, y, z) : (y, z) \in W \}.\!</math> So long as the intended indices attaching to the tacit extensions can be gathered from context, it is usually clear enough to use the abbreviated forms, <math>\mathrm{te}(U), \mathrm{te}(V), \mathrm{te}(W).\!</math> The definition and illustration of relational composition presently under way makes use of the tacit extension of <math>G \subseteq X \times Y\!</math> to <math>\mathrm{te}(G) \subseteq X \times Y \times Z\!</math> and the tacit extension of <math>H \subseteq Y \times Z\!</math> to <math>\mathrm{te}(H) \subseteq X \times Y \times Z,\!</math> only. Geometric illustrations of <math>\mathrm{te}(G)\!</math> and <math>\mathrm{te}(H)\!</math> are afforded by Figures&nbsp;4 and 5, respectively. {| align="center" border="0" cellpadding="10" | <pre> o-------------------------------------------------o | | | o | | /|\ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | / | * \ | | o o ** o | | |\ / \*** /| | | | \ / *** / | | | | \ / ***\ / | | | | \ *** / | | | | / \*** / \ | | | | / *** / \ | | | |/ ***\ / \| | | o X /** Y Z o | | |\ \//* | / /| | | | \ /// | / / | | | | \ ///\ | / / | | | | \ /// \ | / / | | | | \/// \ | / / | | | | /\/ \ | / / | | | | *//\ \|/ / * | | | X */ Y o Y * Z | | \ * | | * / | | \ G | | H / | | \ | | / | | \ | | / | | \ | | / | | \ | | / | | \| |/ | | o o | | | o-------------------------------------------------o Figure 4. Tacit Extension of G to X x Y x Z </pre> |} {| align="center" border="0" cellpadding="10" | <pre> o-------------------------------------------------o | | | o | | /|\ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | / * | \ | | o ** o o | | |\ ***/ \ /| | | | \ *** \ / | | | | \ /*** \ / | | | | \ *** / | | | | / \ ***/ \ | | | | / \ *** \ | | | |/ \ /*** \| | | o X Y **\ Z o | | |\ \ | *\\/ /| | | | \ \ | \\\ / | | | | \ \ | /\\\ / | | | | \ \ | / \\\ / | | | | \ \ | / \\\/ | | | | \ \ | / \/\ | | | | * \ \|/ /\\* | | | X * Y o Y \* Z | | \ * | | * / | | \ G | | H / | | \ | | / | | \ | | / | | \ | | / | | \ | | / | | \| |/ | | o o | | | o-------------------------------------------------o Figure 5. Tacit Extension of H to X x Y x Z </pre> |} A geometric interpretation can now be given that fleshes out in graphic form the meaning of a formula like the following: :* <math>F(x, y, z) ~=~ G(x, y) \land H(y, z).\!</math> The conjunction that is indicated by &ldquo;<math>\land\!</math>&rdquo; corresponds as usual to an intersection of two sets, however, in this case it is the intersection of the tacit extensions <math>\mathrm{te}(G)\!</math> and <math>\mathrm{te}(H).\!</math> {| align="center" border="0" cellpadding="10" | <pre> o-------------------------------------------------o | | | o | | /|\ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | o o o | | |\ / \ /| | | | \ / F \ / | | | | \ / * \ / | | | | \ /*\ / | | | | / \//*\\/ \ | | | | / /\/ \/\ \ | | | |/ ///\ /\\\ \| | | o X /// Y \\\ Z o | | |\ \/// | \\\/ /| | | | \ /// | \\\ / | | | | \ ///\ | /\\\ / | | | | \ /// \ | / \\\ / | | | | \/// \ | / \\\/ | | | | /\/ \ | / \/\ | | | | *//\ \|/ /\\* | | | X */ Y o Y \* Z | | \ * | | * / | | \ G | | H / | | \ | | / | | \ | | / | | \ | | / | | \ | | / | | \| |/ | | o o | | | o-------------------------------------------------o Figure 6. F as the Intersection of te(G) and te(H) </pre> |} ==Algebraic construction== The transition from a geometric picture of relation composition to an algebraic formulation is accomplished through the introduction of coordinates, in other words, identifiable names for the objects that are related through the various forms of relations, dyadic and triadic in the present case. Adding coordinates to the running Example produces the following Figure: {| align="center" border="0" cellpadding="10" | <pre> o-------------------------------------------------o | | | o | | /|\ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | o o o | | |\ / \ /| | | | \ / F \ / | | | | \ / * \ / | | | | \ /*\ / | | | | / \//*\\/ \ | | | | / /\/ \/\ \ | | | |/ ///\ /\\\ \| | | o X /// Y \\\ Z o | | |\ 7\/// | \\\/7 /| | | | \ 6// | \\6 / | | | | \ //5\ | /5\\ / | | | | \ /// 4\ | /4 \\\ / | | | | \/// 3\ | /3 \\\/ | | | | G/\/ 2\ | /2 \/\H | | | | *//\ 1\|/1 /\\* | | | X *\ Y o Y /* Z | | 7\ *\\ |7 7| //* /7 | | 6\ |\\\|6 6|///| /6 | | 5\| \\@5 5@// |/5 | | 4@ \@4 4@/ @4 | | 3\ @3 3@ /3 | | 2\ |2 2| /2 | | 1\|1 1|/1 | | o o | | | o-------------------------------------------------o Figure 7. F as the Intersection of te(G) and te(H) </pre> |} Thinking of relations in operational terms is facilitated by using variant notations for ordered tuples and sets of ordered tuples, namely, the ordered pair <math>(x, y)\!</math> is written <math>x\!:\!y,\!</math> the ordered triple <math>(x, y, z)\!</math> is written <math>x\!:\!y\!:\!z,\!</math> and so on, and a set of tuples is conceived as a logical-algebraic sum, which can be written out in the smaller finite cases in forms like <math>a\!:\!b ~+~ b\!:\!c ~+~ c\!:\!d\!</math> and so on. For example, translating the relations <math>F \subseteq X \times Y \times Z, ~ G \subseteq X \times Y, ~ H \subseteq Y \times Z\!</math> into this notation produces the following summary of the data: {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix} F & = & 4:3:4 & + & 4:4:4 & + & 4:5:4 \\ G & = & 4:3 & + & 4:4 & + & 4:5 \\ H & = & 3:4 & + & 4:4 & + & 5:4 \end{matrix}</math> |} As often happens with abstract notations for functions and relations, the ''type information'', in this case, the fact that <math>G\!</math> and <math>H\!</math> live in different spaces, is left implicit in the context of use. Let us now verify that all of the proposed definitions, formulas, and other relationships check out against the concrete data of the current composition example. The ultimate goal is to develop a clearer picture of what is going on in the formula that expresses the relational composition of a couple of dyadic relations in terms of the medial projection of the intersection of their tacit extensions: {| align="center" cellpadding="8" width="90%" | <math>G \circ H ~=~ \mathrm{proj}_{XZ} (\mathrm{te}_{XY}^Z (G) ~\cap~ \mathrm{te}_{YZ}^X (H)).\!</math> |} Here is the big picture, with all the pieces in place: {| align="center" border="0" cellpadding="10" | <pre> o-------------------------------------------------o | | | o | | / \ | | / \ | | / \ | | / \ | | / \ | | / \ | | / G o H \ | | X * Z | | 7\ /|\ /7 | | 6\ / | \ /6 | | 5\ / | \ /5 | | 4@ | @4 | | 3\ | /3 | | 2\ | /2 | | 1\|/1 | | | | | | | | | | | /|\ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | o | o | | |\ /|\ /| | | | \ / F \ / | | | | \ / * \ / | | | | \ /*\ / | | | | / \//*\\/ \ | | | | / /\/ \/\ \ | | | |/ ///\ /\\\ \| | | o X /// Y \\\ Z o | | |\ \/// | \\\/ /| | | | \ /// | \\\ / | | | | \ ///\ | /\\\ / | | | | \ /// \ | / \\\ / | | | | \/// \ | / \\\/ | | | | G/\/ \ | / \/\H | | | | *//\ \|/ /\\* | | | X *\ Y o Y /* Z | | 7\ *\\ |7 7| //* /7 | | 6\ |\\\|6 6|///| /6 | | 5\| \\@5 5@// |/5 | | 4@ \@4 4@/ @4 | | 3\ @3 3@ /3 | | 2\ |2 2| /2 | | 1\|1 1|/1 | | o o | | | o-------------------------------------------------o Figure 8. G o H = proj_XZ (te(G) |^| te(H)) </pre> |} All that remains is to check the following collection of data and derivations against the situation represented in Figure&nbsp;8. {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix} F & = & 4:3:4 & + & 4:4:4 & + & 4:5:4 \\ G & = & 4:3 & + & 4:4 & + & 4:5 \\ H & = & 3:4 & + & 4:4 & + & 5:4 \end{matrix}</math> |} {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix} G \circ H & = & (4\!:\!3 ~+~ 4\!:\!4 ~+~ 4\!:\!5)(3\!:\!4 ~+~ 4\!:\!4 ~+~ 5\!:\!4) \\[6pt] & = & 4:4 \end{matrix}</math> |} {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix} \mathrm{te}(G) & = & \mathrm{te}_{XY}^Z (G) \\[4pt] & = & \displaystyle\sum_{z=1}^7 (4\!:\!3\!:\!z ~+~ 4\!:\!4\!:\!z ~+~ 4\!:\!5\!:\!z) \end{matrix}</math> |} {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix} \mathrm{te}(G) & = & 4:3:1 & + & 4:4:1 & + & 4:5:1 & + \\ & & 4:3:2 & + & 4:4:2 & + & 4:5:2 & + \\ & & 4:3:3 & + & 4:4:3 & + & 4:5:3 & + \\ & & 4:3:4 & + & 4:4:4 & + & 4:5:4 & + \\ & & 4:3:5 & + & 4:4:5 & + & 4:5:5 & + \\ & & 4:3:6 & + & 4:4:6 & + & 4:5:6 & + \\ & & 4:3:7 & + & 4:4:7 & + & 4:5:7 \end{matrix}</math> |} {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix} \mathrm{te}(H) & = & \mathrm{te}_{YZ}^X (H) \\[4pt] & = & \displaystyle\sum_{x=1}^7 (x\!:\!3\!:\!4 ~+~ x\!:\!4\!:\!4 ~+~ x\!:\!5\!:\!4) \end{matrix}</math> |} {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix} \mathrm{te}(H) & = & 1:3:4 & + & 1:4:4 & + & 1:5:4 & + \\ & & 2:3:4 & + & 2:4:4 & + & 2:5:4 & + \\ & & 3:3:4 & + & 3:4:4 & + & 3:5:4 & + \\ & & 4:3:4 & + & 4:4:4 & + & 4:5:4 & + \\ & & 5:3:4 & + & 5:4:4 & + & 5:5:4 & + \\ & & 6:3:4 & + & 6:4:4 & + & 6:5:4 & + \\ & & 7:3:4 & + & 7:4:4 & + & 7:5:4 \end{matrix}</math> |} {| align="center" cellpadding="8" width="90%" | <math>\begin{array}{ccl} \mathrm{te}(G) \cap \mathrm{te}(H) & = & 4\!:\!3:\!4 ~+~ 4\!:\!4\!:\!4 ~+~ 4\!:\!5\!:\!4 \\[4pt] G \circ H & = & \mathrm{proj}_{XZ} (\mathrm{te}(G) \cap \mathrm{te}(H)) \\[4pt] & = & \mathrm{proj}_{XZ} (4\!:\!3:\!4 ~+~ 4\!:\!4\!:\!4 ~+~ 4\!:\!5\!:\!4) \\[4pt] & = & 4:4 \end{array}</math> |} ==Matrix representation== We have it within our reach to pick up another way of representing dyadic relations, namely, the representation as ''[[logical matrix|logical matrices]]'', and also to grasp the analogy between relational composition and ordinary [[matrix multiplication]] as it appears in linear algebra. First of all, while we still have the data of a very simple concrete case in mind, let us reflect on what we did in our last Example in order to find the composition <math>G \circ H\!</math> of the dyadic relations <math>G\!</math> and <math>H.\!</math> Here is the setup that we had before: {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix}X & = & \{ 1, 2, 3, 4, 5, 6, 7 \}\end{matrix}</math> |} {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix} G & = & 4:3 & + & 4:4 & + & 4:5 & \subseteq & X \times X \\ H & = & 3:4 & + & 4:4 & + & 5:4 & \subseteq & X \times X \end{matrix}</math> |} Let us recall the rule for finding the relational composition of a pair of dyadic relations. Given the dyadic relations <math>P \subseteq X \times Y\!</math> and <math>Q \subseteq Y \times Z,\!</math> the composition of <math>P ~\text{on}~ Q\!</math> is written as <math>P \circ Q,\!</math> or more simply as <math>PQ,\!</math> and obtained as follows: To compute <math>PQ,\!</math> in general, where <math>P\!</math> and <math>Q\!</math> are dyadic relations, simply multiply out the two sums in the ordinary distributive algebraic way, but subject to the following rule for finding the product of two elementary relations of shapes <math>a:b\!</math> and <math>c:d.\!</math> {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix} (a:b)(c:d) & = & (a:d) & \text{if}~ b = c \\ (a:b)(c:d) & = & 0 & \text{otherwise} \end{matrix}</math> |} To find the relational composition <math>G \circ H,\!</math> one may begin by writing it as a quasi-algebraic product: {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix} G \circ H & = & (4\!:\!3 ~+~ 4\!:\!4 ~+~ 4\!:\!5)(3\!:\!4 ~+~ 4\!:\!4 ~+~ 5\!:\!4) \end{matrix}</math> |} Multiplying this out in accord with the applicable form of distributive law one obtains the following expansion: {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix} G \circ H & = & (4:3)(3:4) & + & (4:3)(4:4) & + & (4:3)(5:4) & + \\ & & (4:4)(3:4) & + & (4:4)(4:4) & + & (4:4)(5:4) & + \\ & & (4:5)(3:4) & + & (4:5)(4:4) & + & (4:5)(5:4) \end{matrix}</math> |} Applying the rule that determines the product of elementary relations produces the following array: {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix} G \circ H & = & 4:4 & + & 0 & + & 0 & + \\ & & 0 & + & 4:4 & + & 0 & + \\ & & 0 & + & 0 & + & 4:4 \end{matrix}</math> |} Since the plus sign in this context represents an operation of logical disjunction or set-theoretic aggregation, all of the positive multiplicites count as one, and this gives the ultimate result: {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix}G \circ H & = & 4:4\end{matrix}</math> |} With an eye toward extracting a general formula for relation composition, viewed here on analogy to algebraic multiplication, let us examine what we did in multiplying the dyadic relations <math>G\!</math> and <math>H\!</math> together to obtain their relational composite <math>G \circ H.\!</math> Given the space <math>X = \{ 1, 2, 3, 4, 5, 6, 7 \},\!</math> whose cardinality <math>|X|\!</math> is <math>7,\!</math> there are <math>|X \times X| = |X| \cdot |X|\!</math> <math>=\!</math> <math>7 \cdot 7 = 49\!</math> elementary relations of the form <math>i:j,\!</math> where <math>i\!</math> and <math>j\!</math> range over the space <math>X.\!</math> Although they might be organized in many different ways, it is convenient to regard the collection of elementary relations as arranged in a lexicographic block of the following form: {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix} 1\!:\!1 & 1\!:\!2 & 1\!:\!3 & 1\!:\!4 & 1\!:\!5 & 1\!:\!6 & 1\!:\!7 \\ 2\!:\!1 & 2\!:\!2 & 2\!:\!3 & 2\!:\!4 & 2\!:\!5 & 2\!:\!6 & 2\!:\!7 \\ 3\!:\!1 & 3\!:\!2 & 3\!:\!3 & 3\!:\!4 & 3\!:\!5 & 3\!:\!6 & 3\!:\!7 \\ 4\!:\!1 & 4\!:\!2 & 4\!:\!3 & 4\!:\!4 & 4\!:\!5 & 4\!:\!6 & 4\!:\!7 \\ 5\!:\!1 & 5\!:\!2 & 5\!:\!3 & 5\!:\!4 & 5\!:\!5 & 5\!:\!6 & 5\!:\!7 \\ 6\!:\!1 & 6\!:\!2 & 6\!:\!3 & 6\!:\!4 & 6\!:\!5 & 6\!:\!6 & 6\!:\!7 \\ 7\!:\!1 & 7\!:\!2 & 7\!:\!3 & 7\!:\!4 & 7\!:\!5 & 7\!:\!6 & 7\!:\!7 \end{matrix}</math> |} The relations <math>G\!</math> and <math>H\!</math> may then be regarded as logical sums of the following forms: {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix} G & = & \displaystyle\sum_{ij} G_{ij} (i\!:\!j) \\[20pt] H & = & \displaystyle\sum_{ij} H_{ij} (i\!:\!j) \end{matrix}\!</math> |} The notation <math>\textstyle\sum_{ij}\!</math> indicates a logical sum over the collection of elementary relations <math>i\!:\!j\!</math> while the factors <math>G_{ij}\!</math> and <math>H_{ij}\!</math> are values in the ''[[boolean domain]]'' <math>\mathbb{B} = \{ 0, 1 \}~\!</math> that are called the ''coefficients'' of the relations <math>G\!</math> and <math>H,\!</math> respectively, with regard to the corresponding elementary relations <math>i\!:\!j.\!</math> In general, for a dyadic relation <math>L,\!</math> the coefficient <math>L_{ij}\!</math> of the elementary relation <math>i\!:\!j\!</math> in the relation <math>L\!</math> will be <math>0\!</math> or <math>1,\!</math> respectively, as <math>i\!:\!j\!</math> is excluded from or included in <math>L.\!</math> With these conventions in place, the expansions of <math>G\!</math> and <math>H\!</math> may be written out as follows: {| align="center" cellpadding="4" width="90%" | <math>\begin{matrix}G & = & 4:3 & + & 4:4 & + & 4:5 & =\end{matrix}</math> |- | <math>\begin{smallmatrix} 0 \cdot (1:1) & + & 0 \cdot (1:2) & + & 0 \cdot (1:3) & + & 0 \cdot (1:4) & + & 0 \cdot (1:5) & + & 0 \cdot (1:6) & + & 0 \cdot (1:7) & + \\ 0 \cdot (2:1) & + & 0 \cdot (2:2) & + & 0 \cdot (2:3) & + & 0 \cdot (2:4) & + & 0 \cdot (2:5) & + & 0 \cdot (2:6) & + & 0 \cdot (2:7) & + \\ 0 \cdot (3:1) & + & 0 \cdot (3:2) & + & 0 \cdot (3:3) & + & 0 \cdot (3:4) & + & 0 \cdot (3:5) & + & 0 \cdot (3:6) & + & 0 \cdot (3:7) & + \\ 0 \cdot (4:1) & + & 0 \cdot (4:2) & + & \mathbf{1} \cdot (4:3) & + & \mathbf{1} \cdot (4:4) & + & \mathbf{1} \cdot (4:5) & + & 0 \cdot (4:6) & + & 0 \cdot (4:7) & + \\ 0 \cdot (5:1) & + & 0 \cdot (5:2) & + & 0 \cdot (5:3) & + & 0 \cdot (5:4) & + & 0 \cdot (5:5) & + & 0 \cdot (5:6) & + & 0 \cdot (5:7) & + \\ 0 \cdot (6:1) & + & 0 \cdot (6:2) & + & 0 \cdot (6:3) & + & 0 \cdot (6:4) & + & 0 \cdot (6:5) & + & 0 \cdot (6:6) & + & 0 \cdot (6:7) & + \\ 0 \cdot (7:1) & + & 0 \cdot (7:2) & + & 0 \cdot (7:3) & + & 0 \cdot (7:4) & + & 0 \cdot (7:5) & + & 0 \cdot (7:6) & + & 0 \cdot (7:7) \end{smallmatrix}</math> |} {| align="center" cellpadding="4" width="90%" | <math>\begin{matrix}H & = & 3:4 & + & 4:4 & + & 5:4 & =\end{matrix}</math> |- | <math>\begin{smallmatrix} 0 \cdot (1:1) & + & 0 \cdot (1:2) & + & 0 \cdot (1:3) & + & 0 \cdot (1:4) & + & 0 \cdot (1:5) & + & 0 \cdot (1:6) & + & 0 \cdot (1:7) & + \\ 0 \cdot (2:1) & + & 0 \cdot (2:2) & + & 0 \cdot (2:3) & + & 0 \cdot (2:4) & + & 0 \cdot (2:5) & + & 0 \cdot (2:6) & + & 0 \cdot (2:7) & + \\ 0 \cdot (3:1) & + & 0 \cdot (3:2) & + & 0 \cdot (3:3) & + & \mathbf{1} \cdot (3:4) & + & 0 \cdot (3:5) & + & 0 \cdot (3:6) & + & 0 \cdot (3:7) & + \\ 0 \cdot (4:1) & + & 0 \cdot (4:2) & + & 0 \cdot (4:3) & + & \mathbf{1} \cdot (4:4) & + & 0 \cdot (4:5) & + & 0 \cdot (4:6) & + & 0 \cdot (4:7) & + \\ 0 \cdot (5:1) & + & 0 \cdot (5:2) & + & 0 \cdot (5:3) & + & \mathbf{1} \cdot (5:4) & + & 0 \cdot (5:5) & + & 0 \cdot (5:6) & + & 0 \cdot (5:7) & + \\ 0 \cdot (6:1) & + & 0 \cdot (6:2) & + & 0 \cdot (6:3) & + & 0 \cdot (6:4) & + & 0 \cdot (6:5) & + & 0 \cdot (6:6) & + & 0 \cdot (6:7) & + \\ 0 \cdot (7:1) & + & 0 \cdot (7:2) & + & 0 \cdot (7:3) & + & 0 \cdot (7:4) & + & 0 \cdot (7:5) & + & 0 \cdot (7:6) & + & 0 \cdot (7:7) \end{smallmatrix}</math> |} Stripping down to the bare essentials, one obtains the following matrices of coefficients for the relations <math>G\!</math> and <math>H.\!</math> {| align="center" cellpadding="8" width="90%" | <math>G ~=~ \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}</math> |} {| align="center" cellpadding="8" width="90%" | <math>H ~=~ \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}</math> |} These are the logical matrix representations of the dyadic relations <math>G\!</math> and <math>H.\!</math> If the dyadic relations <math>G\!</math> and <math>H\!</math> are viewed as logical sums then their relational composition <math>G \circ H\!</math> can be regarded as a product of sums, a fact that can be indicated as follows: {| align="center" cellpadding="8" width="90%" | <math>G \circ H ~=~ (\sum_{ij} G_{ij} (i\!:\!j))(\sum_{ij} H_{ij} (i\!:\!j)).\!</math> |} The composite relation <math>G \circ H\!</math> is itself a dyadic relation over the same space <math>X,\!</math> in other words, <math>G \circ H \subseteq X \times X,\!</math> and this means that <math>G \circ H\!</math> must be amenable to being written as a logical sum of the following form: {| align="center" cellpadding="8" width="90%" | <math>G \circ H ~=~ \sum_{ij} (G \circ H)_{ij} (i\!:\!j).\!</math> |} In this formula, <math>(G \circ H)_{ij}\!</math> is the coefficient of <math>G \circ H\!</math> with respect to the elementary relation <math>i\!:\!j.\!</math> One of the best ways to reason out what <math>G \circ H\!</math> should be is to ask oneself what its coefficient <math>(G \circ H)_{ij}\!</math> should be for each of the elementary relations <math>i\!:\!j\!</math> in turn. So let us pose the question: {| align="center" cellpadding="8" width="90%" | <math>(G \circ H)_{ij} ~=~ ?\!</math> |} In order to answer this question, it helps to realize that the indicated product given above can be written in the following equivalent form: {| align="center" cellpadding="8" width="90%" | <math>G \circ H ~=~ (\sum_{ik} G_{ik} (i\!:\!k))(\sum_{kj} H_{kj} (k\!:\!j)).\!</math> |} A moment's thought will tell us that <math>(G \circ H)_{ij} = 1\!</math> if and only if there is an element <math>k\!</math> in <math>X\!</math> such that <math>G_{ik} = 1\!</math> and <math>H_{kj} = 1.\!</math> Consequently, we have the result: {| align="center" cellpadding="8" width="90%" | <math>(G \circ H)_{ij} ~=~ \sum_{k} G_{ik} H_{kj}.\!</math> |} This follows from the properties of boolean arithmetic, specifically, from the fact that the product <math>G_{ik} H_{kj}\!</math> is <math>1\!</math> if and only if both <math>G_{ik}\!</math> and <math>H_{kj}\!</math> are <math>1\!</math> and from the fact that <math>\textstyle\sum_{k} F_{k}\!</math> is equal to <math>1\!</math> just in case some <math>F_{k}\!</math> is <math>1.\!</math> All that remains in order to obtain a computational formula for the relational composite <math>G \circ H\!</math> of the dyadic relations <math>G\!</math> and <math>H\!</math> is to collect the coefficients <math>(G \circ H)_{ij}\!</math> as <math>i\!</math> and <math>j\!</math> range over <math>X.\!</math> {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix}G \circ H & = & \displaystyle \sum_{ij} (G \circ H)_{ij} (i\!:\!j) & = & \displaystyle \sum_{ij} (\sum_{k} G_{ik} H_{kj}) (i\!:\!j). \end{matrix}</math> |} This is the logical analogue of matrix multiplication in linear algebra, the difference in the logical setting being that all of the operations performed on coefficients take place in a system of boolean arithmetic where summation corresponds to logical disjunction and multiplication corresponds to logical conjunction. By way of disentangling this formula, one may notice that the form <math>\textstyle \sum_{k} G_{ik} H_{kj}\!</math> is what is usually called a ''scalar product''. In this case it is the scalar product of the <math>i^\text{th}\!</math> row of <math>G\!</math> with the <math>j^\text{th}\!</math> column of <math>H.\!</math> To make this statement more concrete, let us go back to the examples of <math>G\!</math> and <math>H\!</math> we came in with: {| align="center" cellpadding="8" width="90%" | <math>G ~=~ \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}</math> |} {| align="center" cellpadding="8" width="90%" | <math>H ~=~ \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}</math> |} The formula for computing <math>G \circ H\!</math> says the following: {| align="center" cellpadding="8" width="90%" | <math>\begin{array}{ccl} (G \circ H)_{ij} & = & \text{the}~ {ij}^\text{th} ~\text{entry in the matrix representation for}~ G \circ H \\[2pt] & = & \text{the entry in the}~ {i}^\text{th} ~\text{row and the}~ {j}^\text{th} ~\text{column of}~ G \circ H \\[2pt] & = & \text{the scalar product of the}~ {i}^\text{th} ~\text{row of}~ G ~\text{with the}~ {j}^\text{th} ~\text{column of}~ H \\[2pt] & = & \sum_{k} G_{ik} H_{kj} \end{array}</math> |} As it happens, it is possible to make exceedingly light work of this example, since there is only one row of <math>G\!</math> and one column of <math>H\!</math> that are not all zeroes. Taking the scalar product, in a logical way, of the fourth row of <math>G\!</math> with the fourth column of <math>H\!</math> produces the sole non-zero entry for the matrix of <math>G \circ H.\!</math> {| align="center" cellpadding="8" width="90%" | <math>G \circ H ~=~ \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}</math> |} ==Graph-theoretic picture== There is another form of representation for dyadic relations that is useful to keep in mind, especially for its ability to render the logic of many complex formulas almost instantly understandable to the mind's eye. This is the representation in terms of ''bipartite graphs'', or ''bigraphs'' for short. Here is what <math>G\!</math> and <math>H\!</math> look like in the bigraph picture: {| align="center" border="0" cellpadding="10" | <pre> o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | /|\ | | / | \ G | | / | \ | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 9. G = 4:3 + 4:4 + 4:5 </pre> |- | <pre> o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | \ | / | | \ | / H | | \|/ | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 10. H = 3:4 + 4:4 + 5:4 </pre> |} These graphs may be read to say: {| align="center" cellpadding="8" width="90%" | <math>\begin{matrix} G ~\text{puts}~ 4 ~\text{in relation to}~ 3, 4, 5. \\[2pt] H ~\text{puts}~ 3, 4, 5 ~\text{in relation to}~ 4. \end{matrix}</math> |} To form the composite relation <math>G \circ H,\!</math> one simply follows the bigraph for <math>G\!</math> by the bigraph for <math>H,\!</math> here arranging the bigraphs in order down the page, and then treats any non-empty set of paths of length two between two nodes as being equivalent to a single directed edge between those nodes in the composite bigraph for <math>G \circ H.\!</math> Here's how it looks in pictures: {| align="center" border="0" cellpadding="10" | <pre> o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | /|\ | | / | \ G | | / | \ | | o o o o o o o X | | \ | / | | \ | / H | | \|/ | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 11. G Followed By H </pre> |- | <pre> o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | | | | | G o H | | | | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 12. G Composed With H </pre> |} Once again we find that <math>G \circ H = 4:4.\!</math> We have now seen three different representations of dyadic relations. If one has a strong preference for letters, or numbers, or pictures, then one may be tempted to take one or another of these as being canonical, but each of them will be found to have its peculiar advantages and disadvantages in any given application, and the maximum advantage is therefore approached by keeping all three of them in mind. To see the promised utility of the bigraph picture of dyadic relations, let us devise a slightly more complex example of a composition problem, and use it to illustrate the logic of the matrix multiplication formula. Keeping to the same space <math>X = \{ 1, 2, 3, 4, 5, 6, 7 \},\!</math> define the dyadic relations <math>M, N \subseteq X \times X\!</math> as follows: {| align="center" cellpadding="8" width="90%" | <math>\begin{array}{*{19}{c}} M & = & 2\!:\!1 & + & 2\!:\!2 & + & 2\!:\!3 & + & 4\!:\!3 & + & 4\!:\!4 & + & 4\!:\!5 & + & 6\!:\!5 & + & 6\!:\!6 & + & 6\!:\!7 \\[2pt] N & = & 1\!:\!1 & + & 2\!:\!1 & + & 3\!:\!3 & + & 4\!:\!3 & ~ & + & ~ & 4\!:\!5 & + & 5\!:\!5 & + & 6\!:\!7 & + & 7\!:\!7 \end{array}</math> |} Here are the bigraph pictures: {| align="center" border="0" cellpadding="10" | <pre> o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | /|\ /|\ /|\ | | / | \ / | \ / | \ M | | / | \ / | \ / | \ | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 13. Dyadic Relation M </pre> |- | <pre> o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | | / | / \ | \ | | | | / | / \ | \ | N | | |/ |/ \| \| | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 14. Dyadic Relation N </pre> |} To form the composite relation <math>M \circ N,\!</math> one simply follows the bigraph for <math>M\!</math> by the bigraph for <math>N,\!</math> arranging the bigraphs in order down the page, and then counts any non-empty set of paths of length two between two nodes as being equivalent to a single directed edge between those two nodes in the composite bigraph for <math>M \circ N.\!</math> Here's how it looks in pictures: {| align="center" border="0" cellpadding="10" | <pre> o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | /|\ /|\ /|\ | | / | \ / | \ / | \ M | | / | \ / | \ / | \ | | o o o o o o o X | | | / | / \ | \ | | | | / | / \ | \ | N | | |/ |/ \| \| | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 15. M Followed By N </pre> |- | <pre> o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | / \ / \ / \ | | / \ / \ / \ M o N | | / \ / \ / \ | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 16. M Composed With N </pre> |} Let us hark back to that mysterious matrix multiplication formula, and see how it appears in the light of the bigraph representation. The coefficient of the composition <math>M \circ N\!</math> between <math>i\!</math> and <math>j\!</math> in <math>X\!</math> is given as follows: {| align="center" cellpadding="8" width="90%" | <math>(M \circ N)_{ij} ~=~ \sum_{k} M_{ik} N_{kj}\!</math> |} Graphically interpreted, this is a ''sum over paths''. Starting at the node <math>i,\!</math> <math>M_{ik}\!</math> being <math>1\!</math> indicates that there is an edge in the bigraph of <math>M\!</math> from node <math>i\!</math> to node <math>k\!</math> and <math>N_{kj}\!</math> being <math>1\!</math> indicates that there is an edge in the bigraph of <math>N\!</math> from node <math>k\!</math> to node <math>j.\!</math> So the <math>\textstyle\sum_{k}\!</math> ranges over all possible intermediaries <math>k,\!</math> ascending from <math>0\!</math> to <math>1\!</math> just as soon as there happens to be a path of length two between nodes <math>i\!</math> and <math>j.\!</math> It is instructive at this point to compute the other possible composition that can be formed from <math>M\!</math> and <math>N,\!</math> namely, the composition <math>N \circ M,\!</math> that takes <math>M\!</math> and <math>N\!</math> in the opposite order. Here is the graphic computation: {| align="center" border="0" cellpadding="10" | <pre> o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | | / | / \ | \ | | | | / | / \ | \ | N | | |/ |/ \| \| | | o o o o o o o X | | /|\ /|\ /|\ | | / | \ / | \ / | \ M | | / | \ / | \ / | \ | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 17. N Followed By M </pre> |- | <pre> o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | | | N o M | | | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 18. N Composed With M </pre> |} In sum, <math>N \circ M = 0.\!</math> This example affords sufficient evidence that relational composition, just like its kindred, matrix multiplication, is a ''non-commutative'' algebraic operation. ==References== * Ulam, S.M., and Bednarek, A.R., &ldquo;On the Theory of Relational Structures and Schemata for Parallel Computation&rdquo; (1977), pp. 477&ndash;508 in A.R. Bednarek and Françoise Ulam (eds.), ''Analogies Between Analogies : The Mathematical Reports of S.M. Ulam and His Los&nbsp;Alamos Collaborators'', University of California Press, Berkeley, CA, 1990. ==Bibliography== * Mathematical Society of Japan, ''Encyclopedic Dictionary of Mathematics'', 2nd edition, 2&nbsp;volumes., Kiyosi Itô (ed.), MIT Press, Cambridge, MA, 1993. * Mili, A., Desharnais, J., Mili, F., with Frappier, M., ''Computer Program Construction'', Oxford University Press, New York, NY, 1994. * Ulam, S.M., ''Analogies Between Analogies : The Mathematical Reports of S.M. Ulam and His Los&nbsp;Alamos Collaborators'', A.R. Bednarek and Françoise Ulam (eds.), University of California Press, Berkeley, CA, 1990. ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Relation_composition Relation Composition @ InterSciWiki] * [http://mywikibiz.com/Relation_composition Relation Composition @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Relation_composition Relation Composition @ Subject Wikis] * [http://en.wikiversity.org/wiki/Relation_composition Relation Composition @ Wikiversity] * [http://beta.wikiversity.org/wiki/Relation_composition Relation Composition @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Relation_composition Relation Composition], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Relation_composition Relation Composition], [http://mywikibiz.com/ MyWikiBiz] * [http://semanticweb.org/wiki/Relation_composition Relation Composition], [http://semanticweb.org/ Semantic Web] * [http://ref.subwiki.org/wiki/Relation_composition Relation Composition], [http://ref.subwiki.org/ Subject Wikis] * [http://wikinfo.org/w/index.php/Relation_composition Relation Composition], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Relation_composition Relation Composition], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Relation_composition Relation Composition], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Relation_composition&oldid=43467878 Relation Composition], [http://en.wikipedia.org/ Wikipedia] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Database Theory]] [[Category:Discrete Mathematics]] [[Category:Formal Sciences]] [[Category:Inquiry]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Relation Theory]] [[Category:Set Theory]] d7aee4132bf7fa33a7b529f659e23ef1b9165f10 0 336 753 558 2015-11-14T19:24:03Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. In logic and mathematics, '''relation construction''' and '''relational constructibility''' have to do with the ways that one [[relation (mathematics)|relation]] is determined by an indexed family or a sequence of other relations, called the ''relation dataset''.&nbsp; The relation in the focus of consideration is called the ''faciendum''.&nbsp; The relation dataset typically consists of a specified relation over sets of relations, called the ''constructor'', the ''factor'', or the ''method of construction'', plus a specified set of other relations, called the ''faciens'', the ''ingredients'', or the ''makings''. [[Relation composition]] and [[relation reduction]] are special cases of relation constructions. ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Relation_construction Relation Construction @ InterSciWiki] * [http://mywikibiz.com/Relation_construction Relation Construction @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Relation_construction Relation Construction @ Subject Wikis] * [http://en.wikiversity.org/wiki/Relation_construction Relation Construction @ Wikiversity] * [http://beta.wikiversity.org/wiki/Relation_construction Relation Construction @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Relation_construction Relation Construction], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Relation_construction Relation Construction], [http://mywikibiz.com/ MyWikiBiz] * [http://planetmath.org/RelationConstruction Relation Construction], [http://planetmath.org/ PlanetMath] * [http://semanticweb.org/wiki/Relation_construction Relation Construction], [http://semanticweb.org/ Semantic Web] * [http://wikinfo.org/w/index.php/Relation_construction Relation Construction], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Relation_construction Relation Construction], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Relation_construction Relation Construction], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Relation_construction&oldid=39070184 Relation Construction], [http://en.wikipedia.org/ Wikipedia] [[Category:Charles Sanders Peirce]] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Database Theory]] [[Category:Discrete Mathematics]] [[Category:Formal Sciences]] [[Category:Inquiry]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Relation Theory]] [[Category:Set Theory]] 08607ac7db24b3c674ede515b4a7566a344983c6 0 337 754 653 2015-11-15T03:36:00Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. In logic and mathematics, '''relation reduction''' and '''relational reducibility''' have to do with the extent to which a given [[relation (mathematics)|relation]] is determined by a set of other relations, called the ''relation dataset''. The relation under examination is called the ''reductandum''. The relation dataset typically consists of a specified relation over sets of relations, called the ''reducer'', the ''method of reduction'', or the ''relational step'', plus a set of other relations, called the ''reduciens'' or the ''relational base'', each of which is properly simpler in a specified way than the relation under examination. A question of relation reduction or relational reducibility is sometimes posed as a question of '''relation reconstruction''' or '''relational reconstructibility''', since a useful way of stating the question is to ask whether the reductandum can be reconstructed from the reduciens. A relation that is not uniquely determined by a particular relation dataset is said to be ''irreducible'' in just that respect. A relation that is not uniquely determined by any relation dataset in a particular class of relation datasets is said to be ''irreducible'' in respect of that class. ==Discussion== The main thing that keeps the general problem of relational reducibility from being fully well-defined is that one would have to survey all of the conceivable ways of &ldquo;getting new relations from old&rdquo; in order to say precisely what is meant by the claim that the relation <math>L\!</math> is reducible to the set of relations <math>\{ L_j : j \in J \}.\!</math> This amounts to claiming one can be given a set of ''properly simpler'' relations <math>L_j\!</math> for values <math>j\!</math> in a given index set <math>J\!</math> and that this collection of data would suffice to fix the original relation <math>L\!</math> that one is seeking to analyze, determine, specify, or synthesize. In practice, however, apposite discussion of a particular application typically settles on either one of two different notions of reducibility as capturing the pertinent issues, namely: # Reduction under composition. # Reduction under projections. As it happens, there is an interesting relationship between these two notions of reducibility, the implications of which may be taken up partly in parallel with the discussion of the basic concepts. ==Projective reducibility of relations== It is convenient to begin with the ''projective reduction'' of relations, partly because this type of reduction is simpler and more intuitive (in the visual sense), but also because a number of conceptual tools that are needed in any case arise quite naturally in the projective setting. The work of intuiting how projections operate on multidimensional relations is often facilitated by keeping in mind the following sort of geometric image: * Picture a <math>k\!</math>-adic relation <math>L\!</math> as a body that resides in a <math>k\!</math>-dimensional space <math>X.\!</math> If the domains of the relation <math>L\!</math> are <math>X_1, \ldots, X_k,\!</math> then the ''extension'' of the relation <math>L\!</math> is a subset of the cartesian product <math>X = X_1 \times \ldots \times X_k.\!</math> In this setting the interval <math>K = [1, k] = \{ 1, \ldots, k \}\!</math> is called the ''index set'' of the ''indexed family'' of sets <math>X_1, \ldots, X_k.\!</math> For any subset <math>F\!</math> of the index set <math>K,\!</math> there is the corresponding subfamily of sets, <math>\{ X_j : j \in F \},\!</math> and there is the corresponding cartesian product over this subfamily, notated and defined as <math>\textstyle X_F = \prod_{j \in F} X_j.\!</math> For any point <math>x\!</math> in <math>X,\!</math> the ''projection'' of <math>x\!</math> on the subspace <math>X_F\!</math> is notated as <math>\mathrm{proj}_F (x).\!</math> More generally, for any relation <math>L \subseteq X,\!</math> the projection of <math>L\!</math> on the subspace <math>X_F\!</math> is written as <math>\mathrm{proj}_F (L)\!</math> or still more simply as <math>\mathrm{proj}_F L.\!</math> The question of ''projective reduction'' for <math>k\!</math>-adic relations can be stated with moderate generality in the following way: * Given a set of <math>k\!</math>-place relations in the same space <math>X\!</math> and a set of projections from <math>X\!</math> to the associated subspaces, do the projections afford sufficient data to tell the different relations apart? ==Projective reducibility of triadic relations== : ''Main article : [[Triadic relation]]'' By way of illustrating the different sorts of things that can occur in considering the projective reducibility of relations, it is convenient to reuse the four examples of 3-adic relations that are discussed in the main article on that subject. ===Examples of projectively irreducible relations=== The 3-adic relations <math>L_0\!</math> and <math>L_1\!</math> are shown in the next two Tables: <br> {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%" |+ style="height:30px" | <math>L_0 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 0 \}\!</math> |- style="height:40px; background:ghostwhite" | <math>X\!</math> || <math>Y\!</math> || <math>Z\!</math> |- | <math>0\!</math> || <math>0\!</math> || <math>0\!</math> |- | <math>0\!</math> || <math>1\!</math> || <math>1\!</math> |- | <math>1\!</math> || <math>0\!</math> || <math>1\!</math> |- | <math>1\!</math> || <math>1\!</math> || <math>0\!</math> |} <br> {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%" |+ style="height:30px" | <math>L_1 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 1 \}\!</math> |- style="height:40px; background:ghostwhite" | <math>X\!</math> || <math>Y\!</math> || <math>Z\!</math> |- | <math>0\!</math> || <math>0\!</math> || <math>1\!</math> |- | <math>0\!</math> || <math>1\!</math> || <math>0\!</math> |- | <math>1\!</math> || <math>0\!</math> || <math>0\!</math> |- | <math>1\!</math> || <math>1\!</math> || <math>1\!</math> |} <br> A ''2-adic projection'' of a 3-adic relation <math>L\!</math> is the 2-adic relation that results from deleting one column of the table for <math>L\!</math> and then deleting all but one row of any resulting rows that happen to be identical in content. In other words, the multiplicity of any repeated row is ignored. In the case of the above two relations, <math>{L_0, L_1 ~\subseteq~ X \times Y \times Z ~\cong~ \mathbb{B}^3},\!</math> the 2-adic projections are indexed by the columns or domains that remain, as shown in the following Tables. <br> {| align="center" style="width:90%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\mathrm{proj}_{XY} L_0\!</math> |- style="height:40px; background:ghostwhite" | <math>X\!</math> || <math>Y\!</math> |- | <math>0\!</math> || <math>0\!</math> |- | <math>0\!</math> || <math>1\!</math> |- | <math>1\!</math> || <math>0\!</math> |- | <math>1\!</math> || <math>1\!</math> |} | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\mathrm{proj}_{XZ} L_0\!</math> |- style="height:40px; background:ghostwhite" | <math>X\!</math> || <math>Z\!</math> |- | <math>0\!</math> || <math>0\!</math> |- | <math>0\!</math> || <math>1\!</math> |- | <math>1\!</math> || <math>1\!</math> |- | <math>1\!</math> || <math>0\!</math> |} | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\mathrm{proj}_{YZ} L_0\!</math> |- style="height:40px; background:ghostwhite" | <math>Y\!</math> || <math>Z\!</math> |- | <math>0\!</math> || <math>0\!</math> |- | <math>1\!</math> || <math>1\!</math> |- | <math>0\!</math> || <math>1\!</math> |- | <math>1\!</math> || <math>0\!</math> |} |} <br> {| align="center" style="width:90%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\mathrm{proj}_{XY} L_1\!</math> |- style="height:40px; background:ghostwhite" | <math>X\!</math> || <math>Y\!</math> |- | <math>0\!</math> || <math>0\!</math> |- | <math>0\!</math> || <math>1\!</math> |- | <math>1\!</math> || <math>0\!</math> |- | <math>1\!</math> || <math>1\!</math> |} | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\mathrm{proj}_{XZ} L_1\!</math> |- style="height:40px; background:ghostwhite" | <math>X\!</math> || <math>Z\!</math> |- | <math>0\!</math> || <math>1\!</math> |- | <math>0\!</math> || <math>0\!</math> |- | <math>1\!</math> || <math>0\!</math> |- | <math>1\!</math> || <math>1\!</math> |} | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\mathrm{proj}_{YZ} L_1\!</math> |- style="height:40px; background:ghostwhite" | <math>Y\!</math> || <math>Z\!</math> |- | <math>0\!</math> || <math>1\!</math> |- | <math>1\!</math> || <math>0\!</math> |- | <math>0\!</math> || <math>0\!</math> |- | <math>1\!</math> || <math>1\!</math> |} |} <br> It is clear on inspection that the following three equations hold: {| align="center" cellpadding="8" style="text-align:center; width:90%" | <math>\mathrm{proj}_{XY}(L_0) ~=~ \mathrm{proj}_{XY}(L_1)~\!</math> | <math>\mathrm{proj}_{XZ}(L_0) ~=~ \mathrm{proj}_{XZ}(L_1)~\!</math> | <math>\mathrm{proj}_{YZ}(L_0) ~=~ \mathrm{proj}_{YZ}(L_1)~\!</math> |} These equations say that <math>L_0\!</math> and <math>L_1\!</math> cannot be distinguished from each other solely on the basis of their 2-adic projection data. In such a case, each relation is said to be ''irreducible with respect to 2-adic projections''. Since reducibility with respect to 2-adic projections is the only interesting case where it concerns the reduction of 3-adic relations, it is customary to say more simply of such a relation that it is ''projectively irreducible'', the 2-adic basis being understood. It is immediate from the definition that projectively irreducible relations always arise in non-trivial multiplets of mutually indiscernible relations. ===Examples of projectively reducible relations=== The 3-adic relations <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> are shown in the next two Tables: <br> {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%" |+ style="height:30px" | <math>L_\mathrm{A} ~=~ \text{Sign Relation of Interpreter A}\!</math> |- style="height:40px; background:ghostwhite" | style="width:33%" | <math>\text{Object}\!</math> | style="width:33%" | <math>\text{Sign}\!</math> | style="width:33%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} <br> {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%" |+ style="height:30px" | <math>L_\mathrm{B} ~=~ \text{Sign Relation of Interpreter B}\!</math> |- style="height:40px; background:ghostwhite" | style="width:33%" | <math>\text{Object}\!</math> | style="width:33%" | <math>\text{Sign}\!</math> | style="width:33%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- |<math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} <br> In the case of the two sign relations, <math>L_\mathrm{A}, L_\mathrm{B} ~\subseteq~ X \times Y \times Z ~\cong~ O \times S \times I,\!</math> the 2-adic projections are indexed by the columns or domains that remain, as shown in the following Tables. <br> {| align="center" style="width:90%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\mathrm{proj}_{XY}(L_\mathrm{A})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Object}\!</math> | width="50%" | <math>\text{Sign}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\mathrm{proj}_{XZ}(L_\mathrm{A})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Object}\!</math> | width="50%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\mathrm{proj}_{YZ}(L_\mathrm{A})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Sign}\!</math> | width="50%" | <math>\text{Interpretant}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} |} <br> {| align="center" style="width:90%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\mathrm{proj}_{XY}(L_\mathrm{B})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Object}\!</math> | width="50%" | <math>\text{Sign}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\mathrm{proj}_{XZ}(L_\mathrm{B})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Object}\!</math> | width="50%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\mathrm{proj}_{YZ}(L_\mathrm{B})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Sign}\!</math> | width="50%" | <math>\text{Interpretant}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} |} <br> It is clear on inspection that the following three inequalities hold: {| align="center" cellpadding="8" style="text-align:left; width:90%" | <math>\mathrm{proj}_{XY}(L_\mathrm{A}) ~\ne~ \mathrm{proj}_{XY}(L_\mathrm{B})\!</math> | <math>\mathrm{proj}_{XZ}(L_\mathrm{A}) ~\ne~ \mathrm{proj}_{XZ}(L_\mathrm{B})\!</math> | <math>\mathrm{proj}_{YZ}(L_\mathrm{A}) ~\ne~ \mathrm{proj}_{YZ}(L_\mathrm{B})\!</math> |} These inequalities say that <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> can be distinguished from each other solely on the basis of their 2-adic projection data. But this is not enough to say that either one of them is projectively reducible to their 2-adic projection data. To say that a 3-adic relation is projectively reducible in that respect, one has to show that it can be distinguished from ''every'' other 3-adic relation on the basis of the 2-adic projection data alone. In other words, to show that a 3-adic relation <math>L\!</math> on <math>O \times S \times I\!</math> is ''reducible'' or ''reconstructible'' in the 2-adic projective sense, it is necessary to show that no distinct <math>L'\!</math> on <math>O \times S \times I\!</math> exists such that <math>L\!</math> and <math>L'\!</math> have the same set of projections. Proving this takes a much more comprehensive or exhaustive investigation of the space of possible relations on <math>O \times S \times I\!</math> than looking merely at one or two relations at a time. '''Fact.''' As it happens, each of the relations <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> is uniquely determined by its 2-adic projections. This can be seen by following the proof that is given below. Before tackling the proof, however, it will speed things along to recall a few ideas and notations from other articles. * If <math>L\!</math> is a relation over a set of domains that includes the domains <math>U\!</math> and <math>V,\!</math> then the abbreviated notation <math>L_{UV}\!</math> can be used for the projection <math>\mathrm{proj}_{UV}(L).\!</math> * The operation of reversing a projection asks what elements of a bigger space project onto given elements of a smaller space. The set of elements that project onto <math>x\!</math> under a given projection <math>f\!</math> is called the ''fiber'' of <math>x\!</math> under <math>f\!</math> and is written <math>f^{-1}(x)\!</math> or <math>f^{-1}x.\!</math> * If <math>X\!</math> is a finite set, the ''cardinality'' of <math>X,\!</math> written <math>\mathrm{card}(X)\!</math> or <math>|X|,\!</math> means the number of elements in <math>X.\!</math> '''Proof.''' Let <math>L\!</math> be either one of the relations <math>L_\mathrm{A}\!</math> or <math>L_\mathrm{B}.\!</math> Consider any coordinate position <math>(s, i)\!</math> in the <math>SI\!</math>-plane <math>S \times I.\!</math> If <math>(s, i)\!</math> is not in <math>L_{SI}\!</math> then there can be no element <math>(o, s, i)\!</math> in <math>L,\!</math> therefore we may restrict our attention to positions <math>(s, i)\!</math> in <math>L_{SI},\!</math> knowing that there exist at least <math>|L_{SI}| = 8\!</math> elements in <math>L,\!</math> and seeking only to determine what objects <math>o\!</math> exist such that <math>(o, s, i)\!</math> is an element in the ''fiber'' of <math>(s, i).\!</math> In other words, for what <math>o\!</math> in <math>O\!</math> is <math>(o, s, i)\!</math> in the fiber <math>\mathrm{proj}_{SI}^{-1}(s, i)?\!</math> Now, the circumstance that <math>L_{OS}\!</math> has exactly one element <math>(o, s)\!</math> for each coordinate <math>s\!</math> in <math>S\!</math> and that <math>L_{OI}\!</math> has exactly one element <math>(o, i)\!</math> for each coordinate <math>i\!</math> in <math>I,\!</math> plus the &ldquo;coincidence&rdquo; of it being the same <math>o\!</math> at any one choice for <math>(s, i),\!</math> tells us that <math>L\!</math> has just the one element <math>(o, s, i)\!</math> over each point of <math>S \times I.\!</math> All together, this proves that both <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> are reducible in an informative sense to 3-tuples of 2-adic relations, that is, they are ''projectively 2-adically reducible''. ===Summary=== The ''projective analysis'' of 3-adic relations, illustrated by means of concrete examples, has been pursued just far enough at this point to state this clearly demonstrated result: * Some 3-adic relations are, and other 3-adic relations are not, reducible to, or reconstructible from, their 2-adic projection data. In short, some 3-adic relations are projectively reducible and some 3-adic relations are projectively irreducible. ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Relation_reduction Relation Reduction @ InterSciWiki] * [http://mywikibiz.com/Relation_reduction Relation Reduction @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Relation_reduction Relation Reduction @ Subject Wikis] * [http://en.wikiversity.org/wiki/Relation_reduction Relation Reduction @ Wikiversity] * [http://beta.wikiversity.org/wiki/Relation_reduction Relation Reduction @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Relation_reduction Relation Reduction], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Relation_reduction Relation Reduction], [http://mywikibiz.com/ MyWikiBiz] * [http://semanticweb.org/wiki/Relation_reduction Relation Reduction], [http://semanticweb.org/ Semantic Web] * [http://ref.subwiki.org/wiki/Relation_reduction Relation Reduction], [http://ref.subwiki.org/ Subject Wikis] * [http://wikinfo.org/w/index.php/Relation_reduction Relation Reduction], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Relation_reduction Relation Reduction], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Relation_reduction Relation Reduction], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Relation_reduction&oldid=39828834 Relation Reduction], [http://en.wikipedia.org/ Wikipedia] [[Category:Charles Sanders Peirce]] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Database Theory]] [[Category:Discrete Mathematics]] [[Category:Formal Sciences]] [[Category:Inquiry]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Relation Theory]] [[Category:Semiotics]] [[Category:Set Theory]] 3c9ebce54f545b248472898da98cf76835a95e87 0 325 755 546 2015-11-15T15:27:02Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''relative term''', also called a '''rhema''' or a '''rheme''', is a logical term that requires reference to any number of other objects, called the ''correlates'' of the term, in order to denote a definite object, called the ''relate'' (pronounced with the accent on the first syllable) of the relative term in question.&nbsp; A relative term is typically expressed in ordinary language by means of a phrase with explicit or implicit blanks, for example, ''lover&nbsp;of&nbsp;__'', or ''giver&nbsp;of&nbsp;__&nbsp;to&nbsp;__''. ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Relative_term Relative Term @ InterSciWiki] * [http://mywikibiz.com/Relative_term Relative Term @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Relative_term Relative Term @ Subject Wikis] * [http://en.wikiversity.org/wiki/Relative_term Relative Term @ Wikiversity] * [http://beta.wikiversity.org/wiki/Relative_term Relative Term @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Relative_term Relative Term], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Relative_term Relative Term], [http://mywikibiz.com/ MyWikiBiz] * [http://semanticweb.org/wiki/Relative_term Relative Term], [http://semanticweb.org/ Semantic Web] * [http://en.wikiversity.org/wiki/Relative_term Relative Term], [http://em.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Relative_term Relative Term], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Relative_term&oldid=35330741 Relative Term], [http://en.wikipedia.org/ Wikipedia] [[Category:Charles Sanders Peirce]] [[Category:Formal Languages]] [[Category:Formal Sciences]] [[Category:Formal Systems]] [[Category:Inquiry]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Philosophy]] [[Category:Pragmatism]] [[Category:Relation Theory]] [[Category:Science]] [[Category:Semiotics]] 7b292afb619a2bc1d8411db4a5bb8ba924ccf872 761 755 2015-11-16T19:58:03Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''relative term''' is a logical term that requires reference to any number of other objects, called the ''correlates'' of the term, in order to denote a definite object, called the ''relate''<sup>1</sup> of the relative term in question.&nbsp; A relative term is typically expressed in ordinary language by means of a phrase with explicit or implicit blanks, for example, &ldquo;lover&nbsp;of&nbsp;__&rdquo;, or &ldquo;giver&nbsp;of&nbsp;__&nbsp;to&nbsp;__&rdquo;. 1. Pronounced with the accent on the first syllable. ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Relative_term Relative Term @ InterSciWiki] * [http://mywikibiz.com/Relative_term Relative Term @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Relative_term Relative Term @ Subject Wikis] * [http://en.wikiversity.org/wiki/Relative_term Relative Term @ Wikiversity] * [http://beta.wikiversity.org/wiki/Relative_term Relative Term @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Relative_term Relative Term], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Relative_term Relative Term], [http://mywikibiz.com/ MyWikiBiz] * [http://semanticweb.org/wiki/Relative_term Relative Term], [http://semanticweb.org/ Semantic Web] * [http://en.wikiversity.org/wiki/Relative_term Relative Term], [http://em.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Relative_term Relative Term], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Relative_term&oldid=35330741 Relative Term], [http://en.wikipedia.org/ Wikipedia] [[Category:Charles Sanders Peirce]] [[Category:Formal Languages]] [[Category:Formal Sciences]] [[Category:Formal Systems]] [[Category:Inquiry]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Philosophy]] [[Category:Pragmatism]] [[Category:Relation Theory]] [[Category:Science]] [[Category:Semiotics]] 9fb9b8301898c64c62a2ec5cf7a493d7a2993b44 762 761 2015-11-16T20:22:04Z Jon Awbrey 3 + link wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''relative term''' is a logical term that requires reference to any number of other objects, called the ''correlates'' of the term, in order to denote a definite object, called the ''relate''<sup>1</sup> of the relative term in question.&nbsp; A relative term is typically expressed in ordinary language by means of a phrase with explicit or implicit blanks, for example, &ldquo;lover&nbsp;of&nbsp;__&rdquo;, or &ldquo;giver&nbsp;of&nbsp;__&nbsp;to&nbsp;__&rdquo;. 1. Pronounced with the accent on the first syllable. ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Relative_term Relative Term @ InterSciWiki] * [http://mywikibiz.com/Relative_term Relative Term @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Relative_term Relative Term @ Subject Wikis] * [http://en.wikiversity.org/wiki/Relative_term Relative Term @ Wikiversity] * [http://beta.wikiversity.org/wiki/Relative_term Relative Term @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Relative_term Relative Term], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Relative_term Relative Term], [http://mywikibiz.com/ MyWikiBiz] * [http://semanticweb.org/wiki/Relative_term Relative Term], [http://semanticweb.org/ Semantic Web] * [http://wikinfo.org/w/index.php/Relative_term Relative Term], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Relative_term Relative Term], [http://em.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Relative_term Relative Term], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Relative_term&oldid=35330741 Relative Term], [http://en.wikipedia.org/ Wikipedia] [[Category:Charles Sanders Peirce]] [[Category:Formal Languages]] [[Category:Formal Sciences]] [[Category:Formal Systems]] [[Category:Inquiry]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Philosophy]] [[Category:Pragmatism]] [[Category:Relation Theory]] [[Category:Science]] [[Category:Semiotics]] 566f2ddf48a2abfe90dfe9f1fb46c8c54b2ab9a0 0 314 756 665 2015-11-15T21:35:04Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''sign relation''' is the basic construct in the theory of signs, also known as [[semeiotic]] or [[semiotics]], as developed by Charles Sanders Peirce. ==Anthesis== {| align="center" cellpadding="6" width="90%" | <p>Thus, if a sunflower, in turning towards the sun, becomes by that very act fully capable, without further condition, of reproducing a sunflower which turns in precisely corresponding ways toward the sun, and of doing so with the same reproductive power, the sunflower would become a Representamen of the sun. (C.S. Peirce, &ldquo;Syllabus&rdquo; (''c''.&nbsp;1902), ''Collected Papers'', CP&nbsp;2.274).</p> |} In his picturesque illustration of a sign relation, along with his tracing of a corresponding sign process, or ''[[semiosis]]'', Peirce uses the technical term ''representamen'' for his concept of a sign, but the shorter word is precise enough, so long as one recognizes that its meaning in a particular theory of signs is given by a specific definition of what it means to be a sign. ==Definition== One of Peirce's clearest and most complete definitions of a sign is one that he gives in the context of providing a definition for ''logic'', and so it is informative to view it in that setting. {| align="center" cellpadding="6" width="90%" | <p>Logic will here be defined as ''formal semiotic''. A definition of a sign will be given which no more refers to human thought than does the definition of a line as the place which a particle occupies, part by part, during a lapse of time. Namely, a sign is something, ''A'', which brings something, ''B'', its ''interpretant'' sign determined or created by it, into the same sort of correspondence with something, ''C'', its ''object'', as that in which itself stands to ''C''. It is from this definition, together with a definition of &ldquo;formal&rdquo;, that I deduce mathematically the principles of logic. I also make a historical review of all the definitions and conceptions of logic, and show, not merely that my definition is no novelty, but that my non-psychological conception of logic has ''virtually'' been quite generally held, though not generally recognized. (C.S. Peirce, NEM&nbsp;4, 20&ndash;21).</p> |} In the general discussion of diverse theories of signs, the question frequently arises whether signhood is an absolute, essential, indelible, or ''ontological'' property of a thing, or whether it is a relational, interpretive, and mutable role that a thing can be said to have only within a particular context of relationships. Peirce's definition of a ''sign'' defines it in relation to its ''object'' and its ''interpretant sign'', and thus it defines signhood in ''[[logic of relatives|relative terms]]'', by means of a predicate with three places. In this definition, signhood is a role in a [[triadic relation]], a role that a thing bears or plays in a given context of relationships &mdash; it is not as an ''absolute'', ''non-relative'' property of a thing-in-itself, one that it possesses independently of all relationships to other things. Some of the terms that Peirce uses in his definition of a sign may need to be elaborated for the contemporary reader. :* '''Correspondence.''' From the way that Peirce uses this term throughout his work, it is clear that he means what he elsewhere calls a &ldquo;triple correspondence&rdquo;, and thus this is just another way of referring to the whole triadic sign relation itself. In particular, his use of this term should not be taken to imply a dyadic correspondence, like the kinds of &ldquo;mirror image&rdquo; correspondence between realities and representations that are bandied about in contemporary controversies about &ldquo;correspondence theories of truth&rdquo;. :* '''Determination.''' Peirce's concept of determination is broader in several directions than the sense of the word that refers to strictly deterministic causal-temporal processes. First, and especially in this context, he is invoking a more general concept of determination, what is called a ''formal'' or ''informational'' determination, as in saying &ldquo;two points determine a line&rdquo;, rather than the more special cases of causal and temporal determinisms. Second, he characteristically allows for what is called ''determination in measure'', that is, an order of determinism that admits a full spectrum of more and less determined relationships. :* '''Non-psychological.''' Peirce's &ldquo;non-psychological conception of logic&rdquo; must be distinguished from any variety of ''anti-psychologism''. He was quite interested in matters of psychology and had much of import to say about them. But logic and psychology operate on different planes of study even when they have occasion to view the same data, as logic is a ''[[normative science]]'' where psychology is a ''[[descriptive science]]'', and so they have very different aims, methods, and rationales. ==Signs and inquiry== : ''Main article : [[Inquiry]]'' There is a close relationship between the pragmatic theory of signs and the pragmatic theory of [[inquiry]]. In fact, the correspondence between the two studies exhibits so many congruences and parallels that it is often best to treat them as integral parts of one and the same subject. In a very real sense, inquiry is the process by which sign relations come to be established and continue to evolve. In other words, inquiry, &ldquo;thinking&rdquo; in its best sense, &ldquo;is a term denoting the various ways in which things acquire significance&rdquo; (John Dewey). Thus, there is an active and intricate form of cooperation that needs to be appreciated and maintained between these converging modes of investigation. Its proper character is best understood by realizing that the theory of inquiry is adapted to study the developmental aspects of sign relations, a subject which the theory of signs is specialized to treat from structural and comparative points of view. ==Examples of sign relations== Because the examples to follow have been artificially constructed to be as simple as possible, their detailed elaboration can run the risk of trivializing the whole theory of sign relations. Despite their simplicity, however, these examples have subtleties of their own, and their careful treatment will serve to illustrate many important issues in the general theory of signs. Imagine a discussion between two people, Ann and Bob, and attend only to that aspect of their interpretive practice that involves the use of the following nouns and pronouns: &ldquo;Ann&rdquo;, &ldquo;Bob&rdquo;, &ldquo;I&rdquo;, &ldquo;you&rdquo;. :* The ''object domain'' of this discussion fragment is the set of two people <math>\{ \text{Ann}, \text{Bob} \}.\!</math> :* The ''syntactic domain'' or the ''sign system'' of their discussion is limited to the set of four signs <math>\{ {}^{\backprime\backprime} \text{Ann} {}^{\prime\prime}, {}^{\backprime\backprime} \text{Bob} {}^{\prime\prime}, {}^{\backprime\backprime} \text{I} {}^{\prime\prime}, {}^{\backprime\backprime} \text{you} {}^{\prime\prime} \}.\!</math> In their discussion, Ann and Bob are not only the passive objects of nominative and accusative references but also the active interpreters of the language that they use. The ''system of interpretation'' (SOI) associated with each language user can be represented in the form of an individual [[three-place relation]] called the ''sign relation'' of that interpreter. Understood in terms of its ''set-theoretic extension'', a sign relation <math>L\!</math> is a ''subset'' of a ''cartesian product'' <math>O \times S \times I.\!</math> Here, <math>O, S, I\!</math> are three sets that are known as the ''object domain'', the ''sign domain'', and the ''interpretant domain'', respectively, of the sign relation <math>L \subseteq O \times S \times I.\!</math> Broadly speaking, the three domains of a sign relation can be any sets whatsoever, but the kinds of sign relations typically contemplated in formal settings are usually constrained to having <math>I \subseteq S\!</math>. In this case interpretants are just a special variety of signs and this makes it convenient to lump signs and interpretants together into a single class called the ''syntactic domain''. In the forthcoming examples <math>S\!</math> and <math>I\!</math> are identical as sets, so the same elements manifest themselves in two different roles of the sign relations in question. When it is necessary to refer to the whole set of objects and signs in the union of the domains <math>O, S, I\!</math> for a given sign relation <math>L,\!</math> one may refer to this set as the ''World'' of <math>L\!</math> and write <math>W = W_L = O \cup S \cup I.\!</math> To facilitate an interest in the abstract structures of sign relations, and to keep the notations as brief as possible as the examples become more complicated, it serves to introduce the following general notations: {| align="center" cellspacing="6" width="90%" | <math>\begin{array}{ccl} O & = & \text{Object Domain} \\[6pt] S & = & \text{Sign Domain} \\[6pt] I & = & \text{Interpretant Domain} \end{array}</math> |} Introducing a few abbreviations for use in considering the present Example, we have the following data: {| align="center" cellspacing="6" width="90%" | <math>\begin{array}{cclcl} O & = & \{ \text{Ann}, \text{Bob} \} & = & \{ \mathrm{A}, \mathrm{B} \} \\[6pt] S & = & \{ {}^{\backprime\backprime} \text{Ann} {}^{\prime\prime}, {}^{\backprime\backprime} \text{Bob} {}^{\prime\prime}, {}^{\backprime\backprime} \text{I} {}^{\prime\prime}, {}^{\backprime\backprime} \text{you} {}^{\prime\prime} \} & = & \{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \} \\[6pt] I & = & \{ {}^{\backprime\backprime} \text{Ann} {}^{\prime\prime}, {}^{\backprime\backprime} \text{Bob} {}^{\prime\prime}, {}^{\backprime\backprime} \text{I} {}^{\prime\prime}, {}^{\backprime\backprime} \text{you} {}^{\prime\prime} \} & = & \{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \} \end{array}</math> |} In the present example, <math>S = I = \text{Syntactic Domain}\!</math>. The next two Tables give the sign relations associated with the interpreters <math>\mathrm{A}\!</math> and <math>\mathrm{B},\!</math> respectively, putting them in the form of ''relational databases''. Thus, the rows of each Table list the ordered triples of the form <math>(o, s, i)\!</math> that make up the corresponding sign relations, <math>L_\mathrm{A}, L_\mathrm{B} \subseteq O \times S \times I.\!</math> <br> {| align="center" style="width:92%" | width="50%" | {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 1a.} ~~ {L_\mathrm{A}} = \text{Sign Relation of Interpreter A}\!</math> |- style="height:40px; background:ghostwhite" | style="width:33%" | <math>\text{Object}\!</math> | style="width:33%" | <math>\text{Sign}\!</math> | style="width:33%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} | width="50%" | {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 1b.} ~~ {L_\mathrm{B}} = \text{Sign Relation of Interpreter B}\!</math> |- style="height:40px; background:ghostwhite" | style="width:33%" | <math>\text{Object}\!</math> | style="width:33%" | <math>\text{Sign}\!</math> | style="width:33%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- |<math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} |} <br> These Tables codify a rudimentary level of interpretive practice for the agents <math>\mathrm{A}\!</math> and <math>\mathrm{B}\!</math> and provide a basis for formalizing the initial semantics that is appropriate to their common syntactic domain. Each row of a Table names an object and two co-referent signs, making up an ordered triple of the form <math>(o, s, i)\!</math> called an ''elementary relation'', that is, one element of the relation's set-theoretic extension. Already in this elementary context, there are several different meanings that might attach to the project of a ''formal semiotics'', or a formal theory of meaning for signs. In the process of discussing these alternatives, it is useful to introduce a few terms that are occasionally used in the philosophy of language to point out the needed distinctions. ==Dyadic aspects of sign relations== For an arbitrary triadic relation <math>L \subseteq O \times S \times I,\!</math> whether it is a sign relation or not, there are six dyadic relations that can be obtained by ''projecting'' <math>L\!</math> on one of the planes of the <math>OSI\!</math>-space <math>O \times S \times I.\!</math> The six dyadic projections of a triadic relation <math>L\!</math> are defined and notated as follows: <br> {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%" |+ style="height:30px" | <math>\text{Table 2.} ~~ \text{Dyadic Projections of Triadic Relations}\!</math> | <math>\begin{matrix} L_{OS} & = & \mathrm{proj}_{OS}(L) & = & \{ (o, s) \in O \times S ~:~ (o, s, i) \in L ~\text{for some}~ i \in I \} \\[6pt] L_{SO} & = & \mathrm{proj}_{SO}(L) & = & \{ (s, o) \in S \times O ~:~ (o, s, i) \in L ~\text{for some}~ i \in I \} \\[6pt] L_{IS} & = & \mathrm{proj}_{IS}(L) & = & \{ (i, s) \in I \times S ~:~ (o, s, i) \in L ~\text{for some}~ o \in O \} \\[6pt] L_{SI} & = & \mathrm{proj}_{SI}(L) & = & \{ (s, i) \in S \times I ~:~ (o, s, i) \in L ~\text{for some}~ o \in O \} \\[6pt] L_{OI} & = & \mathrm{proj}_{OI}(L) & = & \{ (o, i) \in O \times I ~:~ (o, s, i) \in L ~\text{for some}~ s \in S \} \\[6pt] L_{IO} & = & \mathrm{proj}_{IO}(L) & = & \{ (i, o) \in I \times O ~:~ (o, s, i) \in L ~\text{for some}~ s \in S \} \end{matrix}</math> |} <br> By way of unpacking the set-theoretic notation, here is what the first definition says in ordinary language. {| align="center" cellpadding="6" width="90%" | <p>The dyadic relation that results from the projection of <math>L\!</math> on the <math>OS\!</math>-plane <math>O \times S\!</math> is written briefly as <math>L_{OS}\!</math> or written more fully as <math>\mathrm{proj}_{OS}(L),\!</math> and it is defined as the set of all ordered pairs <math>(o, s)\!</math> in the cartesian product <math>O \times S\!</math> for which there exists an ordered triple <math>(o, s, i)\!</math> in <math>L\!</math> for some interpretant <math>i\!</math> in the interpretant domain <math>I.\!</math></p> |} In the case where <math>L\!</math> is a sign relation, which it becomes by satisfying one of the definitions of a sign relation, some of the dyadic aspects of <math>L\!</math> can be recognized as formalizing aspects of sign meaning that have received their share of attention from students of signs over the centuries, and thus they can be associated with traditional concepts and terminology. Of course, traditions may vary as to the precise formation and usage of such concepts and terms. Other aspects of meaning have not received their fair share of attention, and thus remain anonymous on the contemporary scene of sign studies. ===Denotation=== One aspect of a sign's complete meaning is concerned with the reference that a sign has to its objects, which objects are collectively known as the ''denotation'' of the sign. In the pragmatic theory of sign relations, denotative references fall within the projection of the sign relation on the plane that is spanned by its object domain and its sign domain. The dyadic relation that makes up the ''denotative'', ''referential'', or ''semantic'' aspect or component of a sign relation <math>L\!</math> is notated as <math>\mathrm{Den}(L).\!</math> Information about the denotative aspect of meaning is obtained from <math>L\!</math> by taking its projection on the object-sign plane, in other words, on the 2-dimensional space that is generated by the object domain <math>O\!</math> and the sign domain <math>S.\!</math> This component of a sign relation <math>L\!</math> can be written in any of the forms, <math>\mathrm{proj}_{OS} L,\!</math> &nbsp; <math>L_{OS},\!</math> &nbsp; <math>\mathrm{proj}_{12} L,\!</math> &nbsp; <math>L_{12},\!</math> and it is defined as follows: {| align="center" cellpadding="6" width="90%" | <math>\begin{matrix} \mathrm{Den}(L) & = & \mathrm{proj}_{OS} L & = & \{ (o, s) \in O \times S ~:~ (o, s, i) \in L ~\text{for some}~ i \in I \}. \end{matrix}</math> |} Looking to the denotative aspects of <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B},\!</math> various rows of the Tables specify, for example, that <math>\mathrm{A}\!</math> uses <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> to denote <math>\mathrm{A}\!</math> and <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> to denote <math>\mathrm{B},\!</math> whereas <math>\mathrm{B}\!</math> uses <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> to denote <math>\mathrm{B}\!</math> and <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> to denote <math>\mathrm{A}.\!</math> All of these denotative references are summed up in the projections on the <math>OS\!</math>-plane, as shown in the following Tables: <br> {| align="center" style="width:90%" | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 3a.} ~~ \mathrm{Den}(L_\mathrm{A}) = \mathrm{proj}_{OS}(L_\mathrm{A})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Object}\!</math> | width="50%" | <math>\text{Sign}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 3b.} ~~ \mathrm{Den}(L_\mathrm{B}) = \mathrm{proj}_{OS}(L_\mathrm{B})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Object}\!</math> | width="50%" | <math>\text{Sign}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} |} <br> ===Connotation=== Another aspect of meaning concerns the connection that a sign has to its interpretants within a given sign relation. As before, this type of connection can be vacuous, singular, or plural in its collection of terminal points, and it can be formalized as the dyadic relation that is obtained as a planar projection of the triadic sign relation in question. The connection that a sign makes to an interpretant is here referred to as its ''connotation''. In the full theory of sign relations, this aspect of meaning includes the links that a sign has to affects, concepts, ideas, impressions, intentions, and the whole realm of an agent's mental states and allied activities, broadly encompassing intellectual associations, emotional impressions, motivational impulses, and real conduct. Taken at the full, in the natural setting of semiotic phenomena, this complex system of references is unlikely ever to find itself mapped in much detail, much less completely formalized, but the tangible warp of its accumulated mass is commonly alluded to as the connotative import of language. Formally speaking, however, the connotative aspect of meaning presents no additional difficulty. For a given sign relation <math>L,\!</math> the dyadic relation that constitutes the ''connotative aspect'' or ''connotative component'' of <math>L\!</math> is notated as <math>\mathrm{Con}(L).\!</math> The connotative aspect of a sign relation <math>L\!</math> is given by its projection on the plane of signs and interpretants, and is therefore defined as follows: {| align="center" cellpadding="6" width="90%" | <math>\begin{matrix} \mathrm{Con}(L) & = & \mathrm{proj}_{SI} L & = & \{ (s, i) \in S \times I ~:~ (o, s, i) \in L ~\text{for some}~ o \in O \}. \end{matrix}</math> |} All of these connotative references are summed up in the projections on the <math>SI\!</math>-plane, as shown in the following Tables: <br> {| align="center" style="width:90%" | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 4a.} ~~ \mathrm{Con}(L_\mathrm{A}) = \mathrm{proj}_{SI}(L_\mathrm{A})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Sign}\!</math> | width="50%" | <math>\text{Interpretant}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 4b.} ~~ \mathrm{Con}(L_\mathrm{B}) = \mathrm{proj}_{SI}(L_\mathrm{B})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Sign}\!</math> | width="50%" | <math>\text{Interpretant}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} |} <br> ===Ennotation=== The aspect of a sign's meaning that arises from the dyadic relation of its objects to its interpretants has no standard name. If an interpretant is considered to be a sign in its own right, then its independent reference to an object can be taken as belonging to another moment of denotation, but this neglects the mediational character of the whole transaction in which this occurs. Denotation and connotation have to do with dyadic relations in which the sign plays an active role, but here we have to consider a dyadic relation between objects and interpretants that is mediated by the sign from an off-stage position, as it were. As a relation between objects and interpretants that is mediated by a sign, this aspect of meaning may be referred to as the ''ennotation'' of a sign, and the dyadic relation that constitutes the ''ennotative aspect'' of a sign relation <math>L\!</math> may be notated as <math>\mathrm{Enn}(L).\!</math> The ennotational component of meaning for a sign relation <math>L\!</math> is captured by its projection on the plane of the object and interpretant domains, and it is thus defined as follows: {| align="center" cellpadding="6" width="90%" | <math>\begin{matrix} \mathrm{Enn}(L) & = & \mathrm{proj}_{OI} L & = & \{ (o, i) \in O \times I ~:~ (o, s, i) \in L ~\text{for some}~ s \in S \}. \end{matrix}</math> |} As it happens, the sign relations <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> are fully symmetric with respect to exchanging signs and interpretants, so all the data of <math>\mathrm{proj}_{OS} L_\mathrm{A}\!</math> is echoed unchanged in <math>\mathrm{proj}_{OI} L_\mathrm{A}\!</math> and all the data of <math>\mathrm{proj}_{OS} L_\mathrm{B}\!</math> is echoed unchanged in <math>\mathrm{proj}_{OI} L_\mathrm{B}.\!</math> <br> {| align="center" style="width:90%" | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 5a.} ~~ \mathrm{Enn}(L_\mathrm{A}) = \mathrm{proj}_{OI}(L_\mathrm{A})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Object}\!</math> | width="50%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 5b.} ~~ \mathrm{Enn}(L_\mathrm{B}) = \mathrm{proj}_{OI}(L_\mathrm{B})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>\text{Object}\!</math> | width="50%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} |} <br> ==Semiotic equivalence relations== A ''semiotic equivalence relation'' (SER) is a special type of equivalence relation that arises in the analysis of sign relations. As a general rule, any equivalence relation is closely associated with a family of equivalence classes that partition the underlying set of elements, frequently called the ''domain'' or ''space'' of the relation. In the case of a SER, the equivalence classes are called ''semiotic equivalence classes'' (SECs) and the partition is called a ''semiotic partition'' (SEP). The sign relations <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> have many interesting properties that are not possessed by sign relations in general. Some of these properties have to do with the relation between signs and their interpretant signs, as reflected in the projections of <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> on the <math>SI\!</math>-plane, notated as <math>\mathrm{proj}_{SI} L_\mathrm{A}\!</math> and <math>\mathrm{proj}_{SI} L_\mathrm{B},\!</math> respectively. The 2-adic relations on <math>S \times I\!</math> induced by these projections are also referred to as the ''connotative components'' of the corresponding sign relations, notated as <math>\mathrm{Con}(L_\mathrm{A})\!</math> and <math>\mathrm{Con}(L_\mathrm{B}),\!</math> respectively. Tables&nbsp;6a and 6b show the corresponding connotative components. <br> {| align="center" style="width:90%" | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 6a.} ~~ \mathrm{Con}(L_\mathrm{A}) = \mathrm{proj}_{SI}(L_\mathrm{A})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>S\!</math> | width="50%" | <math>I\!</math> |- | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \end{matrix}</math> | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \end{matrix}</math> |- | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \end{matrix}</math> | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \end{matrix}</math> |} | width="50%" | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 6b.} ~~ \mathrm{Con}(L_\mathrm{B}) = \mathrm{proj}_{SI}(L_\mathrm{B})\!</math> |- style="height:40px; background:ghostwhite" | width="50%" | <math>S\!</math> | width="50%" | <math>I\!</math> |- | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \end{matrix}</math> | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \end{matrix}</math> |- | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \end{matrix}</math> | valign="bottom" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \end{matrix}</math> |} |} <br> One nice property possessed by the sign relations <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> is that their connotative components <math>\mathrm{Con}(L_\mathrm{A})\!</math> and <math>\mathrm{Con}(L_\mathrm{B})\!</math> form a pair of equivalence relations on their common syntactic domain <math>S = I.\!</math> It is convenient to refer to such a structure as a ''semiotic equivalence relation'' (SER) since it equates signs that mean the same thing to some interpreter. Each of the SERs, <math>\mathrm{Con}(L_\mathrm{A}), \mathrm{Con}(L_\mathrm{B}) \subseteq S \times I \cong S \times S\!</math> partitions the whole collection of signs into ''semiotic equivalence classes'' (SECs). This makes for a strong form of representation in that the structure of the interpreters' common object domain <math>\{ \mathrm{A}, \mathrm{B} \}\!</math> is reflected or reconstructed, part for part, in the structure of each of their ''semiotic partitions'' (SEPs) of the syntactic domain <math>\{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}.\!</math> But it needs to be observed that the semiotic partitions for interpreters <math>\mathrm{A}\!</math> and <math>\mathrm{B}\!</math> are not the same, indeed, they are orthogonal to each other. This makes it difficult to interpret either one of the partitions or equivalence relations on the syntactic domain as corresponding to any sort of objective structure or invariant reality, independent of the individual interpreter's point of view. Information about the contrasting patterns of semiotic equivalence induced by the interpreters <math>\mathrm{A}\!</math> and <math>\mathrm{B}\!</math> is summarized in Tables&nbsp;7a and 7b. The form of these Tables should suffice to explain what is meant by saying that the SEPs for <math>\mathrm{A}\!</math> and <math>\mathrm{B}\!</math> are orthogonal to each other. <br> {| align="center" style="width:92%" | width="50%" | {| align="center" cellpadding="20" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 7a.} ~~ \text{Semiotic Partition for Interpreter A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | style="border-top:1px solid black" | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | style="border-top:1px solid black" | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |} | width="50%" | {| align="center" cellpadding="20" cellspacing="1" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%" |+ style="height:30px" | <math>\text{Table 7b.} ~~ \text{Semiotic Partition for Interpreter B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | style="border-left:1px solid black" | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | style="border-left:1px solid black" | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |} |} <br> A few items of notation are useful in discussing equivalence relations in general and semiotic equivalence relations in particular. As a general consideration, if <math>E\!</math> is an equivalence relation on a set <math>X,\!</math> then every element <math>x\!</math> of <math>X\!</math> belongs to a unique equivalence class under <math>E\!</math> called ''the equivalence class of <math>x\!</math> under <math>E\!</math>''. Convention provides the ''square bracket notation'' for denoting this equivalence class, either in the subscripted form <math>[x]_E\!</math> or in the simpler form <math>[x]\!</math> when the subscript <math>E\!</math> is understood. A statement that the elements <math>x\!</math> and <math>y\!</math> are equivalent under <math>E\!</math> is called an ''equation'' or an ''equivalence'' and may be expressed in any of the following ways: {| align="center" style="text-align:center; width:100%" | <math>\begin{array}{clc} (x, y) & \in & E \\[4pt] x & \in & [y]_E \\[4pt] y & \in & [x]_E \\[4pt] [x]_E & = & [y]_E \\[4pt] x & =_E & y \end{array}</math> |} Thus we have the following definitions: {| align="center" style="text-align:center; width:100%" | <math>\begin{array}{ccc} [x]_E & = & \{ y \in X : (x, y) \in E \} \\[6pt] x =_E y & \Leftrightarrow & (x, y) \in E \end{array}</math> |} In the application to sign relations it is useful to extend the square bracket notation in the following ways. If <math>L\!</math> is a sign relation whose connotative component or syntactic projection <math>L_{SI}\!</math> is an equivalence relation on <math>S = I,\!</math> let <math>[s]_L\!</math> be the equivalence class of <math>s\!</math> under <math>L_{SI}.\!</math> That is to say, <math>[s]_L = [s]_{L_{SI}}.\!</math> A statement that the signs <math>x\!</math> and <math>y\!</math> are equivalent under a semiotic equivalence relation <math>L_{SI}\!</math> is called a ''semiotic equation'' (SEQ) and may be written in either of the following equivalent forms: {| align="center" style="text-align:center; width:100%" | <math>\begin{array}{clc} [x]_L & = & [y]_L \\[6pt] x & =_L & y \end{array}</math> |} In many situations there is one further adaptation of the square bracket notation for semiotic equivalence classes that can be useful. Namely, when there is known to exist a particular triple <math>(o, s, i)\!</math> in a sign relation <math>L,\!</math> it is permissible to let <math>[o]_L\!</math> be defined as <math>[s]_L.\!</math> These modifications are designed to make the notation for semiotic equivalence classes harmonize as well as possible with the frequent use of similar devices for the denotations of signs and expressions. The semiotic equivalence relation for interpreter <math>\mathrm{A}\!</math> yields the following semiotic equations: {| align="center" style="text-align:center; width:100%" | <math>\begin{matrix} [ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} ]_{L_\mathrm{A}} & = & [ {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} ]_{L_\mathrm{A}} \\[6pt] [ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} ]_{L_\mathrm{A}} & = & [ {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} ]_{L_\mathrm{A}} \end{matrix}</math> |} or {| align="center" style="text-align:center; width:100%" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} & =_{L_\mathrm{A}} & {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} & =_{L_\mathrm{A}} & {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \end{matrix}</math> |} Thus it induces the semiotic partition: {| align="center" cellpadding="12" style="text-align:center; width:100%" | <math>\{ \{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \}, \{ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \} \}.\!</math> |} The semiotic equivalence relation for interpreter <math>\mathrm{B}\!</math> yields the following semiotic equations: {| align="center" style="text-align:center; width:100%" | <math>\begin{matrix} [ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} ]_{L_\mathrm{B}} & = & [ {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} ]_{L_\mathrm{B}} \\[6pt] [ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} ]_{L_\mathrm{B}} & = & [ {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} ]_{L_\mathrm{B}} \end{matrix}</math> |} or {| align="center" style="text-align:center; width:100%" | <math>\begin{matrix} {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} & =_{L_\mathrm{B}} & {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} & =_{L_\mathrm{B}} & {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \end{matrix}</math> |} Thus it induces the semiotic partition: {| align="center" cellpadding="12" style="text-align:center; width:100%" | <math>\{ \{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}, \{ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \} \}.\!</math> |} ==Graphical representations== The dyadic components of sign relations can be given graph-theoretic representations, as ''digraphs'' (or ''directed graphs''), that provide concise pictures of their structural and potential dynamic properties. By way of terminology, a directed edge <math>(x, y)\!</math> is called an ''arc'' from point <math>x\!</math> to point <math>y,\!</math> and a self-loop <math>(x, x)\!</math> is called a ''sling'' at <math>x.\!</math> The denotative components <math>\mathrm{Den}(L_\mathrm{A})\!</math> and <math>\mathrm{Den}(L_\mathrm{B})\!</math> can be represented as digraphs on the six points of their common world set <math>W = O \cup S \cup I =\!</math> <math>\{ \mathrm{A}, \mathrm{B}, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}.\!</math> The arcs are given as follows: {| align="center" cellspacing="6" width="90%" | <p><math>\mathrm{Den}(L_\mathrm{A})\!</math> has an arc from each point of <math>\{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \}\!</math> to <math>\mathrm{A}\!</math> and an arc from each point of <math>\{ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}\!</math> to <math>\mathrm{B}.\!</math></p> |- | <p><math>\mathrm{Den}(L_\mathrm{B})\!</math> has an arc from each point of <math>\{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}\!</math> to <math>\mathrm{A}\!</math> and an arc from each point of <math>\{ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \}\!</math> to <math>\mathrm{B}.\!</math></p> |} <math>\mathrm{Den}(L_\mathrm{A})\!</math> and <math>\mathrm{Den}(L_\mathrm{B})\!</math> can be interpreted as ''transition digraphs'' that chart the succession of steps or the connection of states in a computational process. If the graphs are read this way, the denotational arcs summarize the ''upshots'' of the computations that are involved when the interpreters <math>\mathrm{A}\!</math> and <math>\mathrm{B}\!</math> evaluate the signs in <math>S\!</math> according to their own frames of reference. The connotative components <math>\mathrm{Con}(L_\mathrm{A})\!</math> and <math>\mathrm{Con}(L_\mathrm{B})\!</math> can be represented as digraphs on the four points of their common syntactic domain <math>S = I =\!</math> <math>\{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}.\!</math> Since <math>\mathrm{Con}(L_\mathrm{A})\!</math> and <math>\mathrm{Con}(L_\mathrm{B})\!</math> are semiotic equivalence relations, their digraphs conform to the pattern that is manifested by all digraphs of equivalence relations. In general, a digraph of an equivalence relation falls into connected components that correspond to the parts of the associated partition, with a complete digraph on the points of each part, and no other arcs. In the present case, the arcs are given as follows: {| align="center" cellspacing="6" width="90%" | <p><math>\mathrm{Con}(L_\mathrm{A})\!</math> has the structure of a semiotic equivalence relation on <math>S,\!</math> with a sling at each point of <math>S,\!</math> arcs in both directions between the points of <math>\{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \},\!</math> and arcs in both directions between the points of <math>\{ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}.\!</math></p> |- | <p><math>\mathrm{Con}(L_\mathrm{B})\!</math> has the structure of a semiotic equivalence relation on <math>S,\!</math> with a sling at each point of <math>S,\!</math> arcs in both directions between the points of <math>\{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \},\!</math> and arcs in both directions between the points of <math>\{ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \}.\!</math></p> |} Taken as transition digraphs, <math>\mathrm{Con}(L_\mathrm{A})\!</math> and <math>\mathrm{Con}(L_\mathrm{B})\!</math> highlight the associations that are permitted between equivalent signs, as this equivalence is judged by the interpreters <math>\mathrm{A}\!</math> and <math>\mathrm{B},\!</math> respectively. ==Six ways of looking at a sign relation== In the context of 3-adic relations in general, Peirce provides the following illustration of the six ''converses'' of a 3-adic relation, that is, the six differently ordered ways of stating what is logically the same 3-adic relation: : So in a triadic fact, say, for example <br> {| align="center" cellspacing="8" style="width:72%" | align="center" | ''A'' gives ''B'' to ''C'' |} : we make no distinction in the ordinary logic of relations between the ''subject nominative'', the ''direct object'', and the ''indirect object''. We say that the proposition has three ''logical subjects''. We regard it as a mere affair of English grammar that there are six ways of expressing this: <br> {| align="center" cellspacing="8" style="width:72%" | style="width:36%" | ''A'' gives ''B'' to ''C'' | style="width:36%" | ''A'' benefits ''C'' with ''B'' |- | ''B'' enriches ''C'' at expense of ''A'' | ''C'' receives ''B'' from ''A'' |- | ''C'' thanks ''A'' for ''B'' | ''B'' leaves ''A'' for ''C'' |} : These six sentences express one and the same indivisible phenomenon. (C.S. Peirce, &ldquo;The Categories Defended&rdquo;, MS&nbsp;308 (1903), EP&nbsp;2, 170&ndash;171). ===IOS=== (Text in Progress) ===ISO=== (Text in Progress) ===OIS=== {| align="center" cellpadding="6" width="90%" | <p>Words spoken are symbols or signs (σύμβολα) of affections or impressions (παθήματα) of the soul (ψυχή); written words are the signs of words spoken. As writing, so also is speech not the same for all races of men. But the mental affections themselves, of which these words are primarily signs (σημεια), are the same for the whole of mankind, as are also the objects (πράγματα) of which those affections are representations or likenesses, images, copies (ομοιώματα). (Aristotle, ''De Interpretatione'', 1.16<sup>a</sup>4).</p> |} ===OSI=== (Text in Progress) ===SIO=== {| align="center" cellpadding="6" width="90%" | <p>Logic will here be defined as ''formal semiotic''. A definition of a sign will be given which no more refers to human thought than does the definition of a line as the place which a particle occupies, part by part, during a lapse of time. Namely, a sign is something, ''A'', which brings something, ''B'', its ''interpretant'' sign determined or created by it, into the same sort of correspondence with something, ''C'', its ''object'', as that in which itself stands to ''C''. It is from this definition, together with a definition of &ldquo;formal&rdquo;, that I deduce mathematically the principles of logic. I also make a historical review of all the definitions and conceptions of logic, and show, not merely that my definition is no novelty, but that my non-psychological conception of logic has ''virtually'' been quite generally held, though not generally recognized. (C.S. Peirce, &ldquo;Application to the Carnegie Institution&rdquo;, L75 (1902), NEM 4, 20&ndash;21).</p> |} ===SOI=== {| align="center" cellpadding="6" width="90%" | <p>A ''Sign'' is anything which is related to a Second thing, its ''Object'', in respect to a Quality, in such a way as to bring a Third thing, its ''Interpretant'', into relation to the same Object, and that in such a way as to bring a Fourth into relation to that Object in the same form, ''ad infinitum''. (CP&nbsp;2.92, quoted in Fisch 1986, p.&nbsp;274)</p> |} ==References== ==Bibliography== ===Primary sources=== * [[Charles Sanders Peirce (Bibliography)]] ===Secondary sources=== * Awbrey, J.L., and Awbrey, S.M. (1995), &ldquo;Interpretation as Action : The Risk of Inquiry&rdquo;, ''Inquiry : Critical Thinking Across the Disciplines'' 15(1), pp. 40&ndash;52. [http://web.archive.org/web/19970626071826/http://chss.montclair.edu/inquiry/fall95/awbrey.html Archive]. [http://independent.academia.edu/JonAwbrey/Papers/1302117/Interpretation_as_Action_The_Risk_of_Inquiry Online]. * Deledalle, Gérard (2000), ''C.S. Peirce's Philosophy of Signs'', Indiana University Press, Bloomington, IN. * Eisele, Carolyn (1979), in ''Studies in the Scientific and Mathematical Philosophy of C.S. Peirce'', Richard Milton Martin (ed.), Mouton, The Hague. * Esposito, Joseph (1980), ''Evolutionary Metaphysics : The Development of Peirce's Theory of Categories'', Ohio University Press (?). * Fisch, Max (1986), ''Peirce, Semeiotic, and Pragmatism'', Indiana University Press, Bloomington, IN. * Houser, N., Roberts, D.D., and Van Evra, J. (eds.)(1997), ''Studies in the Logic of C.S. Peirce'', Indiana University Press, Bloomington, IN. * Liszka, J.J. (1996), ''A General Introduction to the Semeiotic of C.S. Peirce'', Indiana University Press, Bloomington, IN. * Misak, C. (ed.)(2004), ''Cambridge Companion to C.S. Peirce'', Cambridge University Press. * Moore, E., and Robin, R. (1964), ''Studies in the Philosophy of C.S. Peirce, Second Series'', University of Massachusetts Press, Amherst, MA. * Murphey, M. (1961), ''The Development of Peirce's Thought''. Reprinted, Hackett, Indianapolis, IN, 1993. * Percy, Walker (2000), pp. 271&ndash;291 in ''Signposts in a Strange Land'', P. Samway (ed.), Saint Martin's Press. ==Resources== * [http://www.helsinki.fi/science/commens/dictionary.html The Commens Dictionary of Peirce's Terms] ** [http://www.helsinki.fi/science/commens/terms/semeiotic.html Semeiotic, Semiotic, Semeotic, Semeiotics] * [http://forum.wolframscience.com/archive/ A New Kind Of Science &bull; Forum Archive] ** [http://forum.wolframscience.com/archive/topic/647.html Excerpts from Peirce on Sign Relations] ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Sign_relation Sign Relation @ InterSciWiki] * [http://mywikibiz.com/Sign_relation Sign Relation @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Sign_relation Sign Relation @ Subject Wikis] * [http://en.wikiversity.org/wiki/Sign_relation Sign Relation @ Wikiversity] * [http://beta.wikiversity.org/wiki/Sign_relation Sign Relation @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Sign_relation Sign Relation], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Sign_relation Sign Relation], [http://mywikibiz.com/ MyWikiBiz] * [http://planetmath.org/SignRelation Sign Relation], [http://planetmath.org/ PlanetMath] * [http://wikinfo.org/w/index.php/Sign_relation Sign Relation], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Sign_relation Sign Relation], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Sign_relation Sign Relation], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Sign_relation&oldid=161631069 Sign Relation], [http://en.wikipedia.org/ Wikipedia] [[Category:Artificial Intelligence]] [[Category:Charles Sanders Peirce]] [[Category:Cognitive Sciences]] [[Category:Computer Science]] [[Category:Formal Sciences]] [[Category:Hermeneutics]] [[Category:Information Systems]] [[Category:Information Theory]] [[Category:Inquiry]] [[Category:Intelligence Amplification]] [[Category:Knowledge Representation]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Philosophy]] [[Category:Pragmatics]] [[Category:Pragmatism]] [[Category:Relation Theory]] [[Category:Semantics]] [[Category:Semiotics]] [[Category:Syntax]] 8e892da1cab8797c9958849c4fa411ce6029de57 0 313 757 652 2015-11-15T22:04:05Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. In logic, mathematics, and semiotics, a '''triadic relation''' is an important special case of a [[relation (mathematics)|polyadic or finitary relation]], one in which the number of places in the relation is three. In other language that is often used, a triadic relation is called a '''ternary relation'''. One may also see the adjectives ''3-adic'', ''3-ary'', ''3-dimensional'', or ''3-place'' being used to describe these relations. Mathematics is positively rife with examples of 3-adic relations, and a [[sign relation]], the arch-idea of the whole field of semiotics, is a special case of a 3-adic relation. Therefore it will be useful to consider a few concrete examples from each of these two realms. ==Examples from mathematics== For the sake of topics to be taken up later, it is useful to examine a pair of 3-adic relations in tandem, <math>L_0\!</math> and <math>L_1,\!</math> that can be described in the following manner. The first order of business is to define the space in which the relations <math>L_0\!</math> and <math>L_1\!</math> take up residence. This space is constructed as a 3-fold [[cartesian power]] in the following way. The ''[[boolean domain]]'' is the set <math>\mathbb{B} = \{ 0, 1 \}.\!</math> The ''plus sign'' <math>{}^{\backprime\backprime} + {}^{\prime\prime},\!</math> used in the context of the boolean domain <math>\mathbb{B},\!</math> denotes addition modulo 2. Interpreted for logic, the plus sign can be used to indicate either the boolean operation of ''[[exclusive disjunction]]'', <math>\mathrm{XOR} : \mathbb{B} \times \mathbb{B} \to \mathbb{B},\!</math> or the boolean relation of ''logical inequality'', <math>\mathrm{NEQ} \subseteq \mathbb{B} \times \mathbb{B}.\!</math> The third cartesian power of <math>\mathbb{B}\!</math> is the set <math>\mathbb{B}^3 = \mathbb{B} \times \mathbb{B} \times \mathbb{B} = \{ (x_1, x_2, x_3) : x_j \in \mathbb{B} ~\text{for}~ j = 1, 2, 3 \}.\!</math> In what follows, the space <math>X \times Y \times Z\!</math> is isomorphic to <math>\mathbb{B} \times \mathbb{B} \times \mathbb{B} ~=~ \mathbb{B}^3.\!</math> The relation <math>L_0\!</math> is defined as follows: {| align="center" cellpadding="6" width="90%" | <math>L_0 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 0 \}.~\!</math> |} The relation <math>L_0\!</math> is the set of four triples enumerated here: {| align="center" cellpadding="6" width="90%" | <math>L_0 ~=~ \{ (0, 0, 0), (0, 1, 1), (1, 0, 1), (1, 1, 0) \}.\!</math> |} The relation <math>L_1\!</math> is defined as follows: {| align="center" cellpadding="6" width="90%" | <math>L_1 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 1 \}.~\!</math> |} The relation <math>L_1\!</math> is the set of four triples enumerated here: {| align="center" cellpadding="6" width="90%" | <math>L_1 ~=~ \{ (0, 0, 1), (0, 1, 0), (1, 0, 0), (1, 1, 1) \}.\!</math> |} The triples that make up the relations <math>L_0\!</math> and <math>L_1\!</math> are conveniently arranged in the form of ''relational data tables'', as follows: <br> {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%" |+ style="height:30px" | <math>L_0 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 0 \}\!</math> |- style="height:40px; background:ghostwhite" | <math>X\!</math> || <math>Y\!</math> || <math>Z\!</math> |- | <math>0\!</math> || <math>0\!</math> || <math>0\!</math> |- | <math>0\!</math> || <math>1\!</math> || <math>1\!</math> |- | <math>1\!</math> || <math>0\!</math> || <math>1\!</math> |- | <math>1\!</math> || <math>1\!</math> || <math>0\!</math> |} <br> {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%" |+ style="height:30px" | <math>L_1 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 1 \}\!</math> |- style="height:40px; background:ghostwhite" | <math>X\!</math> || <math>Y\!</math> || <math>Z\!</math> |- | <math>0\!</math> || <math>0\!</math> || <math>1\!</math> |- | <math>0\!</math> || <math>1\!</math> || <math>0\!</math> |- | <math>1\!</math> || <math>0\!</math> || <math>0\!</math> |- | <math>1\!</math> || <math>1\!</math> || <math>1\!</math> |} <br> ==Examples from semiotics== The study of signs &mdash; the full variety of significant forms of expression &mdash; in relation to the things that signs are significant ''of'', and in relation to the beings that signs are significant ''to'', is known as ''[[semiotics]]'' or the ''theory of signs''. As just described, semiotics treats of a 3-place relation among ''signs'', their ''objects'', and their ''interpreters''. The term ''[[semiosis]]'' refers to any activity or process that involves signs. Studies of semiosis that deal with its more abstract form are not concerned with every concrete detail of the entities that act as signs, as objects, or as agents of semiosis, but only with the most salient patterns of relationship among these three roles. In particular, the formal theory of signs does not consider all of the properties of the interpretive agent but only the more striking features of the impressions that signs make on a representative interpreter. In its formal aspects, that impact or influence may be treated as just another sign, called the ''interpretant sign'', or the ''interpretant'' for short. Such a 3-adic relation, among objects, signs, and interpretants, is called a ''[[sign relation]]''. For example, consider the aspects of sign use that concern two people &mdash; let us say <math>\mathrm{Ann}\!</math> and <math>\mathrm{Bob}\!</math> &mdash; in using their own proper names, <math>{}^{\backprime\backprime} \mathrm{Ann} {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} \mathrm{Bob} {}^{\prime\prime},\!</math> together with the pronouns, <math>{}^{\backprime\backprime} \mathrm{I} {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} \mathrm{you} {}^{\prime\prime}.\!</math> For brevity, these four signs may be abbreviated to the set <math>\{ \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \, \}.\!</math> The abstract consideration of how <math>\mathrm{A}\!</math> and <math>\mathrm{B}\!</math> use this set of signs to refer to themselves and each other leads to the contemplation of a pair of 3-adic relations, the sign relations <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B},\!</math> that reflect the differential use of these signs by <math>\mathrm{A}\!</math> and <math>\mathrm{B},\!</math> respectively. Each of the sign relations, <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B},\!</math> consists of eight triples of the form <math>(x, y, z),\!</math> where the ''object'' <math>x\!</math> is an element of the ''object domain'' <math>O = \{ \mathrm{A}, \mathrm{B} \},\!</math> where the ''sign'' <math>y\!</math> is an element of the ''sign domain'' <math>S\!,</math> where the ''interpretant sign'' <math>z\!</math> is an element of the interpretant domain <math>I,\!</math> and where it happens in this case that <math>S = I = \{ \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \, \}.\!</math> In general, it is convenient to refer to the union <math>S \cup I\!</math> as the ''syntactic domain'', but in this case <math>S ~=~ I ~=~ S \cup I.\!</math> The set-up so far is summarized as follows: {| align="center" cellpadding="8" width="90%" | <math>\begin{array}{ccc} L_\mathrm{A}, L_\mathrm{B} & \subseteq & O \times S \times I \\ \\ O & = & \{ \mathrm{A}, \mathrm{B} \} \\ \\ S & = & \{ \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \, \} \\ \\ I & = & \{ \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \, \} \\ \\ \end{array}</math> |} The relation <math>L_\mathrm{A}\!</math> is the set of eight triples enumerated here: {| align="center" cellpadding="8" width="90%" | <math>\begin{array}{cccccc} \{ & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}), & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}), & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}), & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}), & \\ & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}), & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}), & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}), & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}) & \}. \end{array}</math> |} The triples in <math>L_\mathrm{A}\!</math> represent the way that interpreter <math>\mathrm{A}\!</math> uses signs. For example, the listing of the triple <math>(\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime})\!</math> in <math>L_\mathrm{A}\!</math> represents the fact that <math>\mathrm{A}\!</math> uses <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> to mean the same thing that <math>\mathrm{A}\!</math> uses <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> to mean, namely, <math>\mathrm{B}.\!</math> The relation <math>L_\mathrm{B}\!</math> is the set of eight triples enumerated here: {| align="center" cellpadding="8" width="90%" | <math>\begin{array}{cccccc} \{ & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}), & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}), & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}), & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}), & \\ & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}), & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}), & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}), & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}) & \}. \end{array}</math> |} The triples in <math>L_\mathrm{B}\!</math> represent the way that interpreter <math>\mathrm{B}\!</math> uses signs. For example, the listing of the triple <math>(\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime})\!</math> in <math>L_\mathrm{B}\!</math> represents the fact that <math>\mathrm{B}\!</math> uses <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> to mean the same thing that <math>\mathrm{B}\!</math> uses <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> to mean, namely, <math>\mathrm{B}.\!</math> The triples that make up the relations <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> are conveniently arranged in the form of ''relational data tables'', as follows: <br> {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%" |+ style="height:30px" | <math>L_\mathrm{A} ~=~ \text{Sign Relation of Interpreter A}\!</math> |- style="height:40px; background:ghostwhite" | style="width:33%" | <math>\text{Object}\!</math> | style="width:33%" | <math>\text{Sign}\!</math> | style="width:33%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} <br> {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%" |+ style="height:30px" | <math>L_\mathrm{B} ~=~ \text{Sign Relation of Interpreter B}\!</math> |- style="height:40px; background:ghostwhite" | style="width:33%" | <math>\text{Object}\!</math> | style="width:33%" | <math>\text{Sign}\!</math> | style="width:33%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- |<math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} <br> ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Triadic_relation Triadic Relation @ InterSciWiki] * [http://mywikibiz.com/Triadic_relation Triadic Relation @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Triadic_relation Triadic Relation @ Subject Wikis] * [http://en.wikiversity.org/wiki/Triadic_relation Triadic Relation @ Wikiversity] * [http://beta.wikiversity.org/wiki/Triadic_relation Triadic Relation @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Triadic_relation Triadic Relation], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Triadic_relation Triadic Relation], [http://mywikibiz.com/ MyWikiBiz] * [http://planetmath.org/TriadicRelation Triadic Relation], [http://planetmath.org/ PlanetMath] * [http://en.wikiversity.org/wiki/Triadic_relation Triadic Relation], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Triadic_relation Triadic Relation], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Triadic_relation&oldid=108548758 Triadic Relation], [http://en.wikipedia.org/ Wikipedia] [[Category:Artificial Intelligence]] [[Category:Boolean Functions]] [[Category:Charles Sanders Peirce]] [[Category:Cognitive Sciences]] [[Category:Computer Science]] [[Category:Formal Sciences]] [[Category:Hermeneutics]] [[Category:Information Systems]] [[Category:Information Theory]] [[Category:Inquiry]] [[Category:Intelligence Amplification]] [[Category:Knowledge Representation]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Philosophy]] [[Category:Pragmatics]] [[Category:Pragmatism]] [[Category:Relation Theory]] [[Category:Semantics]] [[Category:Semiotics]] [[Category:Syntax]] e1af846d820ee5e16ee9feba336726ecf39ec6cf 764 757 2015-11-16T21:25:38Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. In logic, mathematics, and semiotics, a '''triadic relation''' is an important special case of a [[relation (mathematics)|polyadic or finitary relation]], one in which the number of places in the relation is three. In other language that is often used, a triadic relation is called a '''ternary relation'''. One may also see the adjectives ''3-adic'', ''3-ary'', ''3-dimensional'', or ''3-place'' being used to describe these relations. Mathematics is positively rife with examples of 3-adic relations, and a [[sign relation]], the arch-idea of the whole field of semiotics, is a special case of a 3-adic relation. Therefore it will be useful to consider a few concrete examples from each of these two realms. ==Examples from mathematics== For the sake of topics to be taken up later, it is useful to examine a pair of 3-adic relations in tandem, <math>L_0\!</math> and <math>L_1,\!</math> that can be described in the following manner. The first order of business is to define the space in which the relations <math>L_0\!</math> and <math>L_1\!</math> take up residence. This space is constructed as a 3-fold [[cartesian power]] in the following way. The ''[[boolean domain]]'' is the set <math>\mathbb{B} = \{ 0, 1 \}.\!</math> The ''plus sign'' <math>{}^{\backprime\backprime} + {}^{\prime\prime},\!</math> used in the context of the boolean domain <math>\mathbb{B},\!</math> denotes addition modulo 2. Interpreted for logic, the plus sign can be used to indicate either the boolean operation of ''[[exclusive disjunction]]'', <math>\mathrm{XOR} : \mathbb{B} \times \mathbb{B} \to \mathbb{B},\!</math> or the boolean relation of ''logical inequality'', <math>\mathrm{NEQ} \subseteq \mathbb{B} \times \mathbb{B}.\!</math> The third cartesian power of <math>\mathbb{B}\!</math> is the set <math>\mathbb{B}^3 = \mathbb{B} \times \mathbb{B} \times \mathbb{B} = \{ (x_1, x_2, x_3) : x_j \in \mathbb{B} ~\text{for}~ j = 1, 2, 3 \}.\!</math> In what follows, the space <math>X \times Y \times Z\!</math> is isomorphic to <math>\mathbb{B} \times \mathbb{B} \times \mathbb{B} ~=~ \mathbb{B}^3.\!</math> The relation <math>L_0\!</math> is defined as follows: {| align="center" cellpadding="6" width="90%" | <math>L_0 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 0 \}.~\!</math> |} The relation <math>L_0\!</math> is the set of four triples enumerated here: {| align="center" cellpadding="6" width="90%" | <math>L_0 ~=~ \{ (0, 0, 0), (0, 1, 1), (1, 0, 1), (1, 1, 0) \}.\!</math> |} The relation <math>L_1\!</math> is defined as follows: {| align="center" cellpadding="6" width="90%" | <math>L_1 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 1 \}.~\!</math> |} The relation <math>L_1\!</math> is the set of four triples enumerated here: {| align="center" cellpadding="6" width="90%" | <math>L_1 ~=~ \{ (0, 0, 1), (0, 1, 0), (1, 0, 0), (1, 1, 1) \}.\!</math> |} The triples that make up the relations <math>L_0\!</math> and <math>L_1\!</math> are conveniently arranged in the form of ''relational data tables'', as follows: <br> {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%" |+ style="height:30px" | <math>L_0 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 0 \}\!</math> |- style="height:40px; background:ghostwhite" | <math>X\!</math> || <math>Y\!</math> || <math>Z\!</math> |- | <math>0\!</math> || <math>0\!</math> || <math>0\!</math> |- | <math>0\!</math> || <math>1\!</math> || <math>1\!</math> |- | <math>1\!</math> || <math>0\!</math> || <math>1\!</math> |- | <math>1\!</math> || <math>1\!</math> || <math>0\!</math> |} <br> {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%" |+ style="height:30px" | <math>L_1 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 1 \}\!</math> |- style="height:40px; background:ghostwhite" | <math>X\!</math> || <math>Y\!</math> || <math>Z\!</math> |- | <math>0\!</math> || <math>0\!</math> || <math>1\!</math> |- | <math>0\!</math> || <math>1\!</math> || <math>0\!</math> |- | <math>1\!</math> || <math>0\!</math> || <math>0\!</math> |- | <math>1\!</math> || <math>1\!</math> || <math>1\!</math> |} <br> ==Examples from semiotics== The study of signs &mdash; the full variety of significant forms of expression &mdash; in relation to the things that signs are significant ''of'', and in relation to the beings that signs are significant ''to'', is known as ''[[semiotics]]'' or the ''theory of signs''. As just described, semiotics treats of a 3-place relation among ''signs'', their ''objects'', and their ''interpreters''. The term ''[[semiosis]]'' refers to any activity or process that involves signs. Studies of semiosis that deal with its more abstract form are not concerned with every concrete detail of the entities that act as signs, as objects, or as agents of semiosis, but only with the most salient patterns of relationship among these three roles. In particular, the formal theory of signs does not consider all of the properties of the interpretive agent but only the more striking features of the impressions that signs make on a representative interpreter. In its formal aspects, that impact or influence may be treated as just another sign, called the ''interpretant sign'', or the ''interpretant'' for short. Such a 3-adic relation, among objects, signs, and interpretants, is called a ''[[sign relation]]''. For example, consider the aspects of sign use that concern two people &mdash; let us say <math>\mathrm{Ann}\!</math> and <math>\mathrm{Bob}\!</math> &mdash; in using their own proper names, <math>{}^{\backprime\backprime} \mathrm{Ann} {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} \mathrm{Bob} {}^{\prime\prime},\!</math> together with the pronouns, <math>{}^{\backprime\backprime} \mathrm{I} {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} \mathrm{you} {}^{\prime\prime}.\!</math> For brevity, these four signs may be abbreviated to the set <math>\{ \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \, \}.\!</math> The abstract consideration of how <math>\mathrm{A}\!</math> and <math>\mathrm{B}\!</math> use this set of signs to refer to themselves and each other leads to the contemplation of a pair of 3-adic relations, the sign relations <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B},\!</math> that reflect the differential use of these signs by <math>\mathrm{A}\!</math> and <math>\mathrm{B},\!</math> respectively. Each of the sign relations, <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B},\!</math> consists of eight triples of the form <math>(x, y, z),\!</math> where the ''object'' <math>x\!</math> is an element of the ''object domain'' <math>O = \{ \mathrm{A}, \mathrm{B} \},\!</math> where the ''sign'' <math>y\!</math> is an element of the ''sign domain'' <math>S\!,</math> where the ''interpretant sign'' <math>z\!</math> is an element of the interpretant domain <math>I,\!</math> and where it happens in this case that <math>S = I = \{ \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \, \}.\!</math> In general, it is convenient to refer to the union <math>S \cup I\!</math> as the ''syntactic domain'', but in this case <math>S ~=~ I ~=~ S \cup I.\!</math> The set-up so far is summarized as follows: {| align="center" cellpadding="8" width="90%" | <math>\begin{array}{ccc} L_\mathrm{A}, L_\mathrm{B} & \subseteq & O \times S \times I \\ \\ O & = & \{ \mathrm{A}, \mathrm{B} \} \\ \\ S & = & \{ \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \, \} \\ \\ I & = & \{ \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \, \} \\ \\ \end{array}</math> |} The relation <math>L_\mathrm{A}\!</math> is the set of eight triples enumerated here: {| align="center" cellpadding="8" width="90%" | <math>\begin{array}{cccccc} \{ & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}), & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}), & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}), & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}), & \\ & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}), & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}), & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}), & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}) & \}. \end{array}</math> |} The triples in <math>L_\mathrm{A}\!</math> represent the way that interpreter <math>\mathrm{A}\!</math> uses signs. For example, the listing of the triple <math>(\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime})\!</math> in <math>L_\mathrm{A}\!</math> represents the fact that <math>\mathrm{A}\!</math> uses <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> to mean the same thing that <math>\mathrm{A}\!</math> uses <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> to mean, namely, <math>\mathrm{B}.\!</math> The relation <math>L_\mathrm{B}\!</math> is the set of eight triples enumerated here: {| align="center" cellpadding="8" width="90%" | <math>\begin{array}{cccccc} \{ & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}), & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}), & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}), & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}), & \\ & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}), & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}), & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}), & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}) & \}. \end{array}</math> |} The triples in <math>L_\mathrm{B}\!</math> represent the way that interpreter <math>\mathrm{B}\!</math> uses signs. For example, the listing of the triple <math>(\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime})\!</math> in <math>L_\mathrm{B}\!</math> represents the fact that <math>\mathrm{B}\!</math> uses <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> to mean the same thing that <math>\mathrm{B}\!</math> uses <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> to mean, namely, <math>\mathrm{B}.\!</math> The triples that make up the relations <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> are conveniently arranged in the form of ''relational data tables'', as follows: <br> {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%" |+ style="height:30px" | <math>L_\mathrm{A} ~=~ \text{Sign Relation of Interpreter A}\!</math> |- style="height:40px; background:ghostwhite" | style="width:33%" | <math>\text{Object}\!</math> | style="width:33%" | <math>\text{Sign}\!</math> | style="width:33%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |} <br> {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%" |+ style="height:30px" | <math>L_\mathrm{B} ~=~ \text{Sign Relation of Interpreter B}\!</math> |- style="height:40px; background:ghostwhite" | style="width:33%" | <math>\text{Object}\!</math> | style="width:33%" | <math>\text{Sign}\!</math> | style="width:33%" | <math>\text{Interpretant}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- |<math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math> |- | <math>\mathrm{A}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> |- | <math>\mathrm{B}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> |} <br> ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Triadic_relation Triadic Relation @ InterSciWiki] * [http://mywikibiz.com/Triadic_relation Triadic Relation @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Triadic_relation Triadic Relation @ Subject Wikis] * [http://en.wikiversity.org/wiki/Triadic_relation Triadic Relation @ Wikiversity] * [http://beta.wikiversity.org/wiki/Triadic_relation Triadic Relation @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Triadic_relation Triadic Relation], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Triadic_relation Triadic Relation], [http://mywikibiz.com/ MyWikiBiz] * [http://planetmath.org/TriadicRelation Triadic Relation], [http://planetmath.org/ PlanetMath] * [http://wikinfo.org/w/index.php/Triadic_relation Triadic Relation], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Triadic_relation Triadic Relation], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Triadic_relation Triadic Relation], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Triadic_relation&oldid=108548758 Triadic Relation], [http://en.wikipedia.org/ Wikipedia] [[Category:Artificial Intelligence]] [[Category:Boolean Functions]] [[Category:Charles Sanders Peirce]] [[Category:Cognitive Sciences]] [[Category:Computer Science]] [[Category:Formal Sciences]] [[Category:Hermeneutics]] [[Category:Information Systems]] [[Category:Information Theory]] [[Category:Inquiry]] [[Category:Intelligence Amplification]] [[Category:Knowledge Representation]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Philosophy]] [[Category:Pragmatics]] [[Category:Pragmatism]] [[Category:Relation Theory]] [[Category:Semantics]] [[Category:Semiotics]] [[Category:Syntax]] 1a8cbf849298ee9c327099af56192a9661740a5e 0 342 758 564 2015-11-15T22:36:11Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. '''Inquiry''' is any proceeding or process that has the aim of augmenting knowledge, resolving doubt, or solving a problem. A theory of inquiry is an account of the various types of inquiry and a treatment of the ways that each type of inquiry achieves its aim. ==Classical sources== ===Deduction=== {| align="center" cellpadding="4" width="90%" <!--QUOTE--> | <p>When three terms are so related to one another that the last is wholly contained in the middle and the middle is wholly contained in or excluded from the first, the extremes must admit of perfect syllogism. By 'middle term' I mean that which both is contained in another and contains another in itself, and which is the middle by its position also; and by 'extremes' (a) that which is contained in another, and (b) that in which another is contained. For if ''A'' is predicated of all ''B'', and ''B'' of all ''C'', ''A'' must necessarily be predicated of all ''C''. &hellip; I call this kind of figure the First. (Aristotle, ''Prior Analytics'', 1.4).</p> |} ===Induction=== {| align="center" cellpadding="4" width="90%" <!--QUOTE--> | <p>Induction, or inductive reasoning, consists in establishing a relation between one extreme term and the middle term by means of the other extreme; for example, if ''B'' is the middle term of ''A'' and ''C'', in proving by means of ''C'' that ''A'' applies to ''B''; for this is how we effect inductions. (Aristotle, ''Prior Analytics'', 2.23).</p> |} ===Abduction=== The ''locus classicus'' for the study of [[abductive reasoning]] is found in [[Aristotle]]'s ''[[Prior Analytics]]'', Book 2, Chapt. 25. It begins this way: {| align="center" cellpadding="4" width="90%" <!--QUOTE--> | <p>We have Reduction (&#945;&#960;&#945;&#947;&#969;&#947;&#951;, [[abductive reasoning|abduction]]):</p> <ol> <li><p>When it is obvious that the first term applies to the middle, but that the middle applies to the last term is not obvious, yet is nevertheless more probable or not less probable than the conclusion;</p></li> <li><p>Or if there are not many intermediate terms between the last and the middle;</p></li> </ol> <p>For in all such cases the effect is to bring us nearer to knowledge.</p> |} By way of explanation, [[Aristotle]] supplies two very instructive examples, one for each of the two varieties of abductive inference steps that he has just described in the abstract: {| align="center" cellpadding="4" width="90%" <!--QUOTE--> | <ol> <li><p>For example, let ''A'' stand for "that which can be taught", ''B'' for "knowledge", and ''C'' for "morality". Then that knowledge can be taught is evident; but whether virtue is knowledge is not clear. Then if ''BC'' is not less probable or is more probable than ''AC'', we have reduction; for we are nearer to knowledge for having introduced an additional term, whereas before we had no knowledge that ''AC'' is true.<p></li> <li><p>Or again we have reduction if there are not many intermediate terms between ''B'' and ''C''; for in this case too we are brought nearer to knowledge. For example, suppose that ''D'' is "to square", ''E'' "rectilinear figure", and ''F'' "circle". Assuming that between ''E'' and ''F'' there is only one intermediate term &mdash; that the circle becomes equal to a rectilinear figure by means of [[lunule]]s &mdash; we should approximate to knowledge.</p></li> </ol> <p>([[Aristotle]], "[[Prior Analytics]]", 2.25, with minor alterations)</p> |} Aristotle's latter variety of abductive reasoning, though it will take some explaining in the sequel, is well worth our contemplation, since it hints already at streams of inquiry that course well beyond the syllogistic source from which they spring, and into regions that Peirce will explore more broadly and deeply. ==Inquiry in the pragmatic paradigm== In the pragmatic philosophies of [[Charles Sanders Peirce]], [[William James]], [[John Dewey]], and others, inquiry is closely associated with the [[normative science]] of [[logic]]. In its inception, the pragmatic model or theory of inquiry was extracted by Peirce from its raw materials in classical logic, with a little bit of help from [[Kant]], and refined in parallel with the early development of symbolic logic by [[Boole]], [[De Morgan]], and Peirce himself to address problems about the nature and conduct of scientific reasoning. Borrowing a brace of concepts from [[Aristotle]], Peirce examined three fundamental modes of reasoning that play a role in inquiry, commonly known as [[abductive reasoning|abductive]], [[deductive reasoning|deductive]], and [[inductive reasoning|inductive]] [[inference]]. In rough terms, ''[[abductive reasoning|abduction]]'' is what we use to generate a likely [[hypothesis]] or an initial [[diagnosis]] in response to a phenomenon of interest or a problem of concern, while ''[[deductive reasoning|deduction]]'' is used to clarify, to derive, and to explicate the relevant consequences of the selected hypothesis, and ''[[inductive reasoning|induction]]'' is used to test the sum of the predictions against the sum of the data. It needs to be observed that the classical and pragmatic treatments of the types of reasoning, dividing the generic territory of inference as they do into three special parts, arrive at a different characterization of the environs of reason than do those accounts that count only two. These three processes typically operate in a cyclic fashion, systematically operating to reduce the uncertainties and the difficulties that initiated the inquiry in question, and in this way, to the extent that inquiry is successful, leading to an increase in knowledge or in skills. In the pragmatic way of thinking everything has a purpose, and the purpose of each thing is the first thing we should try to note about it. The purpose of inquiry is to reduce doubt and lead to a state of belief, which a person in that state will usually call ''[[knowledge]]'' or ''[[certainty]]''. As they contribute to the end of inquiry, we should appreciate that the three kinds of inference describe a cycle that can be understood only as a whole, and none of the three makes complete sense in isolation from the others. For instance, the purpose of abduction is to generate guesses of a kind that deduction can explicate and that induction can evaluate. This places a mild but meaningful constraint on the production of hypotheses, since it is not just any wild guess at explanation that submits itself to reason and bows out when defeated in a match with reality. In a similar fashion, each of the other types of inference realizes its purpose only in accord with its proper role in the whole cycle of inquiry. No matter how much it may be necessary to study these processes in abstraction from each other, the integrity of inquiry places strong limitations on the effective [[modularity]] of its principal components. ===Art and science of inquiry=== For our present purposes, the first feature to note in distinguishing the three principal modes of reasoning from each other is whether each of them is exact or approximate in character. In this light, deduction is the only one of the three types of reasoning that can be made exact, in essence, always deriving true conclusions from true premisses, while abduction and induction are unavoidably approximate in their modes of operation, involving elements of fallible judgment in practice and inescapable error in their application. The reason for this is that deduction, in the ideal limit, can be rendered a purely internal process of the reasoning agent, while the other two modes of reasoning essentially demand a constant interaction with the outside world, a source of phenomena and problems that will no doubt continue to exceed the capacities of any finite resource, human or machine, to master. Situated in this larger reality, approximations can be judged appropriate only in relation to their context of use and can be judged fitting only with regard to a purpose in view. A parallel distinction that is often made in this connection is to call deduction a ''[[demonstrative]]'' form of inference, while abduction and induction are classed as ''[[non-demonstrative]]'' forms of reasoning. Strictly speaking, the latter two modes of reasoning are not properly called inferences at all. They are more like controlled associations of words or ideas that just happen to be successful often enough to be preserved as useful heuristic strategies in the repertoire of the agent. But [[non-demonstrative]] ways of thinking are inherently subject to error, and must be constantly checked out and corrected as needed in practice. In classical terminology, forms of judgment that require attention to the context and the purpose of the judgment are said to involve an element of 'art', in a sense that is judged to distinguish them from 'science', and in their renderings as expressive judgments to implicate arbiters in styles of [[rhetoric]], as contrasted with [[logic]]. In a figurative sense, this means that only deductive logic can be reduced to an exact theoretical science, while the practice of any empirical science will always remain to some degree an art. ===Zeroth order inquiry=== Many aspects of inquiry can be recognized and usefully studied in very basic logical settings, even simpler than the level of [[syllogism]], for example, in the realm of reasoning that is variously known as ''[[boolean algebra]]'', ''[[propositional logic|propositional calculus]]'', ''[[sentential calculus]]'', or ''[[zeroth-order logic]]''. By way of approaching the learning curve on the gentlest availing slope, we may well begin at the level of ''[[zeroth-order inquiry]]'', in effect, taking the syllogistic approach to inquiry only so far as the propositional or sentential aspects of the associated reasoning processes are concerned. One of the bonuses of doing this in the context of Peirce's logical work is that it provides us with doubly instructive exercises in the use of his [[logical graph]]s, taken at the level of his so-called '[[alpha graph]]s'. In the case of propositional calculus or sentential logic, deduction comes down to applications of the [[transitive law]] for conditional implications and the approximate forms of inference hang on the properties that derive from these. In describing the various types of inference I will employ a few old terms of art from classical logic that are still of use in treating these kinds of simple problems in reasoning. : '''Deduction''' takes a Case, the [[minor premiss]] <math>X \Rightarrow Y</math> : and combines it with a Rule,the [[major premiss]] <math>Y \Rightarrow Z</math> : to arrive at a Fact, the demonstrative [[conclusion]] <math>X \Rightarrow Z.</math> : '''Induction''' takes a Case of the form <math>X \Rightarrow Y</math> : and matches it with a Fact of the form <math>X \Rightarrow Z</math> : to infer a Rule of the form <math>Y \Rightarrow Z.</math> : '''Abduction''' takes a Fact of the form <math>X \Rightarrow Z</math> : and matches it with a Rule of the form <math>Y \Rightarrow Z</math> : to infer a Case of the form <math>X \Rightarrow Y.</math> For ease of reference, Figure 1 and the Legend beneath it summarize the classical terminology for the three types of inference and the relationships among them. {| align="center" cellpadding="8" style="text-align:center" | <pre> o-------------------------------------------------o | | | Z | | o | | |\ | | | \ | | | \ | | | \ | | | \ | | | \ R U L E | | | \ | | | \ | | F | \ | | | \ | | A | \ | | | o Y | | C | / | | | / | | T | / | | | / | | | / | | | / C A S E | | | / | | | / | | | / | | | / | | |/ | | o | | X | | | | Deduction takes a Case of the form X => Y, | | matches it with a Rule of the form Y => Z, | | then adverts to a Fact of the form X => Z. | | | | Induction takes a Case of the form X => Y, | | matches it with a Fact of the form X => Z, | | then adverts to a Rule of the form Y => Z. | | | | Abduction takes a Fact of the form X => Z, | | matches it with a Rule of the form Y => Z, | | then adverts to a Case of the form X => Y. | | | | Even more succinctly: | | | | Abduction Deduction Induction | | | | Premiss: Fact Rule Case | | Premiss: Rule Case Fact | | Outcome: Case Fact Rule | | | o-------------------------------------------------o Figure 1. Elementary Structure and Terminology </pre> |} In its original usage a statement of Fact has to do with a deed done or a record made, that is, a type of event that is openly observable and not riddled with speculation as to its very occurrence. In contrast, a statement of Case may refer to a hidden or a hypothetical cause, that is, a type of event that is not immediately observable to all concerned. Obviously, the distinction is a rough one and the question of which mode applies can depend on the points of view that different observers adopt over time. Finally, a statement of a Rule is called that because it states a regularity or a regulation that governs a whole class of situations, and not because of its syntactic form. So far in this discussion, all three types of constraint are expressed in the form of conditional propositions, but this is not a fixed requirement. In practice, these modes of statement are distinguished by the roles that they play within an argument, not by their style of expression. When the time comes to branch out from the syllogistic framework, we will find that propositional constraints can be discovered and represented in arbitrary syntactic forms. ===Kinds of inference=== The three kinds of inference that Peirce would come to refer to as ''abductive'', ''deductive'', and ''inductive'' inference he gives his earliest systematic treatment in two series of lectures on the logic of science: the [[Harvard University]] Lectures of 1865 and the [[Lowell Institute]] Lectures of 1866. There he sums up the characters of the three kinds of reasoning in the following terms: {| align="center" cellpadding="4" width="90%" <!--QUOTE--> | <p>We have then three different kinds of inference:</p> <p>&nbsp;&nbsp;&nbsp;Deduction or inference ''[[a priori and a posteriori (philosophy)|à priori]]'',</p> <p>&nbsp;&nbsp;&nbsp;Induction or inference ''[[à particularis]]'', and</p> <p>&nbsp;&nbsp;&nbsp;Hypothesis or inference ''[[à posteriori]]''.</p> <p>(Peirce, "On the Logic of Science" (1865), CE 1, 267).</p> |} Early in the first series of lectures Peirce gives a very revealing illustration of how he then thinks of the natures, operations, and relationships of this trio of inference types: {| align="center" cellpadding="4" width="90%" <!--QUOTE--> | <p> If I reason that certain conduct is wise <br>because it has a character which belongs <br>''only'' to wise things, I reason ''à priori''.</p> <p> If I think it is wise because it once turned out <br>to be wise, that is, if I infer that it is wise on <br>this occasion because it was wise on that occasion, <br>I reason inductively [''à particularis''].</p> <p> But if I think it is wise because a wise man does it, <br>I then make the pure hypothesis that he does it <br>because he is wise, and I reason ''à posteriori''.</p> <p>(Peirce, "On the Logic of Science" (1865), CE 1, 180).</p> |} We may begin the analysis of Peirce's example by making the following assignments of letters to the qualitative attributes mentioned in it: :* A = 'Wisdom', :* B = 'a certain character', :* C = 'a certain conduct', :* D = 'done by a wise man', :* E = 'a certain occasion'. Recognizing that a little more concreteness will serve as an aid to the understanding, let's augment the Spartan features of Peirce's illustration in the following way: :* B = 'Benevolence', a certain character, :* C = 'Contributes to Charity', a certain conduct, :* E = 'Earlier today', a certain occasion. The converging operation of all three reasonings is shown in Figure 2. {| align="center" cellpadding="8" style="text-align:center" | <pre> o---------------------------------------------------------------------o | | | D ("done by a wise man") | | o | | \* | | \ * | | \ * | | \ * | | \ * | | \ * | | \ * A ("a wise act") | | \ o | | \ /| * | | \ / | * | | \ / | * | | . | o B ("benevolence", a certain character) | | / \ | * | | / \ | * | | / \| * | | / o | | / * C ("contributes to charity", a certain conduct) | | / * | | / * | | / * | | / * | | / * | | /* | | o | | E ("earlier today", a certain occasion) | | | o---------------------------------------------------------------------o Figure 2. A Thrice Wise Act </pre> |} One of the styles of syntax that Aristotle uses for syllogistic propositions suggests the composite symbols that geometers have long used for labeling line intervals in a geometric figure, and it comports quite nicely with the Figure that we have just drawn. Specifically, the proposition that predicates X of the subject Y is represented by the digram 'XY' and associated with the line interval XY that descends from the point X to the point Y in the corresponding lattice diagram. In this wise we make the following observations: The common proposition that concludes each argument is AC. Introducing the symbol "&rArr;" to denote the relation of logical implication, the proposition AC can be written as C &rArr; A, and read as "C implies A". Adopting the parenthetical form of Peirce's alpha graphs, in their ''existential interpretation'', AC can be written as (C (A)), and most easily comprehended as "not C without A". In the context of the present example, all of these forms are equally good ways of expressing the same concrete proposition, namely, "contributing to charity is wise". :* Deduction could have obtained the Fact AC from the Rule AB, "benevolence is wisdom", along with the Case BC, "contributing to charity is benevolent". :* Induction could have gathered the Rule AC, after a manner of saying that "contributing to charity is exemplary of wisdom", from the Fact AE, "the act of earlier today is wise", along with the Case CE, "the act of earlier today was an instance of contributing to charity". :* Abduction could have guessed the Case AC, in a style of expression stating that "contributing to charity is explained by wisdom", from the Fact DC, "contributing to charity is done by this wise man", and the Rule DA, "everything that is wise is done by this wise man". Thus, a wise man, who happens to do all of the wise things that there are to do, may nevertheless contribute to charity for no good reason, and even be known to be charitable to a fault. But all of this notwithstanding, on seeing the wise man contribute to charity we may find it natural to conjecture, in effect, to consider it as a possibility worth examining further, that charity is indeed a mark of his wisdom, and not just the accidental trait or the immaterial peculiarity of his character &mdash; in essence, that wisdom is the ''cause'' of his contribution or the ''reason'' for his charity. As a general rule, and despite many obvious exceptions, an English word that ends in ''-ion'' denotes equivocally either a process or its result. In our present application, this means that each of the words ''abduction'', ''deduction'', ''induction'' can be used to denote either the process of inference or the product of that inference, that is, the proposition to which the inference in question leads. One of the morals of Peirce's illustration can now be drawn. It demonstrates in a very graphic fashion that the three kinds of inference are three kinds of process and not three kinds of proposition, not if one takes the word ''kind'' in its literal sense as denoting a ''genus'' of being, essence, or substance. Said another way, it means that being an abductive Case, a deductive Fact, or an inductive Rule is a category of relation, indeed, one that involves at the very least a triadic relation among propositions, and not a category of essence or substance, that is, not a property that inheres in the proposition alone. This category distinction between the absolute, essential, or monadic predicates and the more properly relative predicates constitutes a very important theme in Peirce's architectonic. There is of course a parallel application of it in the theory of sign relations, or semiotics, where the distinctions among the sign relational roles of Object, Sign, and Interpretant are distinct ways of relating to other things, modes of relation that may vary from moment to moment in the extended trajectory of a sign process, and not distinctions that mark some fixed and eternal essence of the thing in itself. In the normal course of inquiry, the elementary types of inference proceed in the order: Abduction, Deduction, Induction. However, the same building blocks can be assembled in other ways to yield different types of complex inferences. Of particular importance, reasoning by analogy can be analyzed as a combination of induction and deduction, in other words, as the abstraction and the application of a rule. Because a complicated pattern of analogical inference will be used in our example of a complete inquiry, it will help to prepare the ground if we first stop to consider an example of analogy in its simplest form. ====Abduction==== : ''Main article : [[Abductive reasoning]]'' Much of Peirce's work deals with the scientific and logical questions of [[knowledge]] and [[truth]], questions grounded in his experience as a working logician and experimental scientist, one who was a member of the international community of scientists and thinkers of his day. He made important contributions to [[deductive logic]] (see below), but was primarily interested in the logic of science and specifically in what he called [[abduction (logic)|abduction]] or "hypothesis", as opposed to [[deductive reasoning|deduction]] and [[inductive reasoning|induction]]. Abduction is the process whereby a hypothesis is generated, so that surprising facts may be explained. "There is a more familiar name for it than abduction", Peirce wrote, "for it is neither more nor less than guessing". Indeed, Peirce considered abduction to be at the heart not only of scientific research but of native human intelligence as well. In his "Illustrations of the Logic of Science" (CE 3, 325-326), Peirce gives the following example of how abduction nests with deductive and inductive reasoning. Peirce begins by positing the following three statements: :*''Rule'': "All the beans from this bag are white." :*''Case'': "These beans are from this bag." :*''Result'': "These beans are white." Now let any two of these statements be Givens (their order not mattering), and let the remaining statement be the Conclusion. The result is an ''argument'', of which three kinds are possible: {| align="center" cellpadding="4" |- ! &nbsp; !! Deduction !! Induction !! Abduction |- |- style="border-top:1px solid #999;" |- | ''Premiss'' || Rule || Case || Rule |- | ''Premiss'' || Case || Fact || Fact |- | ''Conclusion'' || Fact || Rule || Case |} ====Deduction==== : ''Main article : [[Deductive reasoning]]'' ====Induction==== : ''Main article : [[Inductive reasoning]]'' ====Analogy==== The classic description of analogy in the syllogistic frame comes from Aristotle, who called this form of inference by the name ''paradeigma'', that is, reasoning by way of example or through the parallel comparison of cases. {| align="center" cellpadding="4" width="90%" <!--QUOTE--> | <p>We have an Example [&#960;&#945;&#961;&#945;&#948;&#949;&#953;&#947;&#956;&#945;, analogy] when the major extreme is shown to be applicable to the middle term by means of a term similar to the third. It must be known both that the middle applies to the third term and that the first applies to the term similar to the third. (Aristotle, "Prior Analytics", 2.24).</p> |} Aristotle illustrates this pattern of argument with the following sample of reasoning. The setting is a discussion, taking place in Athens, on the issue of going to war with Thebes. It is apparently accepted that a war between Thebes and Phocis is or was a bad thing, perhaps from the objectivity lent by non-involvement or perhaps as a lesson of history. {| align="center" cellpadding="4" width="90%" <!--QUOTE--> | <p>For example, let ''A'' be 'bad', ''B'' 'to make war on neighbors', ''C'' 'Athens against Thebes', and ''D'' 'Thebes against Phocis'. Then if we require to prove that war against Thebes is bad, we must be satisfied that war against neighbors is bad. Evidence of this can be drawn from similar examples, for example, that war by Thebes against Phocis is bad. Then since war against neighbors is bad, and war against Thebes is war against neighbors, it is evident that war against Thebes is bad. (Aristotle, "Prior Analytics", 2.24, with minor alterations).</p> |} Aristotle's sample of argument from analogy may be analyzed in the following way: First, a Rule is induced from the consideration of a similar Case and a relevant Fact: :* Case: D &rArr; B, Thebes vs Phocis is war against neighbors. :* Fact: D &rArr; A, Thebes vs Phocis is bad. :* Rule: B &rArr; A, War against neighbors is bad. Next, the Fact to be proved is deduced from the application of the previously induced Rule to the present Case: :* Case: C &rArr; B, Athens vs Thebes is war against neighbors. :* Rule: B &rArr; A, War against neighbors is bad. :* Fact: C &rArr; A, Athens vs Thebes is bad. In practice, of course, it would probably take a mass of comparable cases to establish a rule. As far as the logical structure goes, however, this quantitative confirmation only amounts to 'gilding the lily'. Perfectly valid rules can be guessed on the first try, abstracted from a single experience or adopted vicariously with no personal experience. Numerical factors only modify the degree of confidence and the strength of habit that govern the application of previously learned rules. Figure 3 gives a graphical illustration of Aristotle's example of 'Example', that is, the form of reasoning that proceeds by Analogy or according to a Paradigm. {| align="center" cellpadding="8" style="text-align:center" | <pre> o-----------------------------------------------------------o | | | A | | o | | /*\ | | / * \ | | / * \ | | / * \ | | / * \ | | / * \ | | / R u l e \ | | / * \ | | / * \ | | / * \ | | / * \ | | F a c t o F a c t | | / * B * \ | | / * * \ | | / * * \ | | / * * \ | | / C a s e C a s e \ | | / * * \ | | / * * \ | | / * * \ | | / * * \ | | / * * \ | | o o | | C D | | | | A = Atrocious, Adverse to All, A bad thing | | B = Belligerent Battle Between Brethren | | C = Contest of Athens against Thebes | | D = Debacle of Thebes against Phocis | | | | A is a major term | | B is a middle term | | C is a minor term | | D is a minor term, similar to C | | | o-----------------------------------------------------------o Figure 3. Aristotle's "War Against Neighbors" Example </pre> |} In this analysis of reasoning by Analogy, it is a complex or a mixed form of inference that can be seen as taking place in two steps: :* The first step is an Induction that abstracts a Rule from a Case and a Fact. :: Case: D &rArr; B, Thebes vs Phocis is a battle between neighbors. :: Fact: D &rArr; A, Thebes vs Phocis is adverse to all. :: Rule: B &rArr; A, A battle between neighbors is adverse to all. :* The final step is a Deduction that applies this Rule to a Case to arrive at a Fact. :: Case: C &rArr; B, Athens vs Thebes is a battle between neighbors. :: Rule: B &rArr; A, A battle between neighbors is adverse to all. :: Fact: C &rArr; A, Athens vs Thebes is adverse to all. As we see, Aristotle analyzed analogical reasoning into a phase of inductive reasoning followed by a phase of deductive reasoning. Peirce would pick up the story at this juncture and eventually parse analogy in a couple of different ways, both of them involving all three types of inference: abductive, deductive, and inductive. ==Example of inquiry== Examples of inquiry, that illustrate the full cycle of its abductive, deductive, and inductive phases, and yet are both concrete and simple enough to be suitable for a first (or zeroth) exposition, are somewhat rare in Peirce's writings, and so let us draw one from the work of fellow pragmatician John Dewey, analyzing it according to the model of zeroth-order inquiry that we developed above. {| align="center" cellpadding="4" width="90%" <!--QUOTE--> | <p>A man is walking on a warm day. The sky was clear the last time he observed it; but presently he notes, while occupied primarily with other things, that the air is cooler. It occurs to him that it is probably going to rain; looking up, he sees a dark cloud between him and the sun, and he then quickens his steps. What, if anything, in such a situation can be called thought? Neither the act of walking nor the noting of the cold is a thought. Walking is one direction of activity; looking and noting are other modes of activity. The likelihood that it will rain is, however, something ''suggested''. The pedestrian ''feels'' the cold; he ''thinks of'' clouds and a coming shower. (John Dewey, ''How We Think'', pp. 6&ndash;7).</p> |} ===Once over quickly=== Let's first give Dewey's elegant example of inquiry in everyday life the quick once over, hitting just the high points of its analysis into Peirce's three kinds of reasoning. ====Abductive phase==== In Dewey's 'Rainy Day' or 'Sign of Rain' story, we find our peripatetic hero presented with a surprising Fact: :* Fact: C &rArr; A, In the Current situation the Air is cool. Responding to an intellectual reflex of puzzlement about the situation, his resource of common knowledge about the world is impelled to seize on an approximate Rule: :* Rule: B &rArr; A, Just Before it rains, the Air is cool. This Rule can be recognized as having a potential relevance to the situation because it matches the surprising Fact, C &rArr; A, in its consequential feature A. All of this suggests that the present Case may be one in which it is just about to rain: :* Case: C &rArr; B, The Current situation is just Before it rains. The whole mental performance, however automatic and semi-conscious it may be, that leads up from a problematic Fact and a previously settled knowledge base of Rules to the plausible suggestion of a Case description, is what we are calling an [[abductive inference]]. ====Deductive phase==== The next phase of inquiry uses deductive inference to expand the implied consequences of the abductive hypothesis, with the aim of testing its truth. For this purpose, the inquirer needs to think of other things that would follow from the consequence of his precipitate explanation. Thus, he now reflects on the Case just assumed: :* Case: C &rArr; B, The Current situation is just Before it rains. He looks up to scan the sky, perhaps in a random search for further information, but since the sky is a logical place to look for details of an imminent rainstorm, symbolized in our story by the letter B, we may safely suppose that our reasoner has already detached the consequence of the abduced Case, C &rArr; B, and has begun to expand on its further implications. So let us imagine that our up-looker has a more deliberate purpose in mind, and that his search for additional data is driven by the new-found, determinate Rule: :* Rule: B &rArr; D, Just Before it rains, Dark clouds appear. Contemplating the assumed Case in combination with this new Rule leads him by an immediate deduction to predict an additional Fact: :* Fact: C &rArr; D, In the Current situation Dark clouds appear. The reconstructed picture of reasoning assembled in this second phase of inquiry is true to the pattern of [[deductive inference]]. ====Inductive phase==== Whatever the case, our subject observes a Dark cloud, just as he would expect on the basis of the new hypothesis. The explanation of imminent rain removes the discrepancy between observations and expectations and thereby reduces the shock of surprise that made this process of inquiry necessary. ===Looking more closely=== ====Seeding hypotheses==== Figure 4 gives a graphical illustration of Dewey's example of inquiry, isolating for the purposes of the present analysis the first two steps in the more extended proceedings that go to make up the whole inquiry. {| align="center" cellpadding="8" style="text-align:center" | <pre> o-----------------------------------------------------------o | | | A D | | o o | | \ * * / | | \ * * / | | \ * * / | | \ * * / | | \ * * / | | \ R u l e R u l e / | | \ * * / | | \ * * / | | \ * * / | | \ * B * / | | F a c t o F a c t | | \ * / | | \ * / | | \ * / | | \ * / | | \ C a s e / | | \ * / | | \ * / | | \ * / | | \ * / | | \ * / | | \*/ | | o | | C | | | | A = the Air is cool | | B = just Before it rains | | C = the Current situation | | D = a Dark cloud appears | | | | A is a major term | | B is a middle term | | C is a minor term | | D is a major term, associated with A | | | o-----------------------------------------------------------o Figure 4. Dewey's "Rainy Day" Inquiry </pre> |} In this analysis of the first steps of Inquiry, we have a complex or a mixed form of inference that can be seen as taking place in two steps: :* The first step is an Abduction that abstracts a Case from the consideration of a Fact and a Rule. :: Fact: C &rArr; A, In the Current situation the Air is cool. :: Rule: B &rArr; A, Just Before it rains, the Air is cool. :: Case: C &rArr; B, The Current situation is just Before it rains. :* The final step is a Deduction that admits this Case to another Rule and so arrives at a novel Fact. :: Case: C &rArr; B, The Current situation is just Before it rains. :: Rule: B &rArr; D, Just Before it rains, a Dark cloud will appear. :: Fact: C &rArr; D, In the Current situation, a Dark cloud will appear. This is nowhere near a complete analysis of the Rainy Day inquiry, even insofar as it might be carried out within the constraints of the syllogistic framework, and it covers only the first two steps of the relevant inquiry process, but maybe it will do for a start. One other thing needs to be noticed here, the formal [[duality]] between this expansion phase of inquiry and the argument from [[analogy]]. This can be seen most clearly in the propositional [[lattice]] diagrams shown in Figures 3 and 4, where analogy exhibits a rough "A" shape and the first two steps of inquiry exhibit a rough "V" shape, respectively. Since we find ourselves repeatedly referring to this expansion phase of inquiry as a unit, let's give it a name that suggests its duality with [[analogical reasoning|analogy]] — '[[catalogical reasoning|catalogy]]' will do for the moment. This usage is apt enough if one thinks of a catalogue entry for an item as a text that lists its salient features. Notice that [[analogical reasoning|analogy]] has to do with the examples of a given quality, while [[catalogical reasoning|catalogy]] has to do with the qualities of a given example. Peirce noted similar forms of duality in many of his early writings, leading to the consummate treatment in his 1867 paper [http://members.door.net/arisbe/menu/library/bycsp/newlist/nl-frame.htm "On a New List of Categories"] (CP 1.545-559, CE 2, 49-59). ====Weeding hypotheses==== In order to comprehend the bearing of [[inductive reasoning]] on the closing phases of inquiry there are a couple of observations that we need to make: :* First, we need to recognize that smaller inquiries are typically woven into larger inquiries, whether we view the whole pattern of inquiry as carried on by a single agent or by a complex community. :* Further, we need to consider the different ways in which the particular instances of inquiry can be related to ongoing inquiries at larger scales. Three modes of inductive interaction between the micro-inquiries and the macro-inquiries that are salient here can be described under the headings of the 'Learning', the 'Transfer', and the 'Testing' of rules. ====Analogy of experience==== Throughout inquiry the reasoner makes use of rules that have to be transported across intervals of experience, from the masses of experience where they are learned to the moments of experience where they are applied. Inductive reasoning is involved in the learning and the transfer of these rules, both in accumulating a knowledge base and in carrying it through the times between acquisition and application. :* Learning. The principal way that induction contributes to an ongoing inquiry is through the learning of rules, that is, by creating each of the rules that goes into the knowledge base, or ever gets used along the way. :* Transfer. The continuing way that induction contributes to an ongoing inquiry is through the exploit of analogy, a two-step combination of induction and deduction that serves to transfer rules from one context to another. :* Testing. Finally, every inquiry that makes use of a knowledge base constitutes a 'field test' of its accumulated contents. If the knowledge base fails to serve any live inquiry in a satisfactory manner, then there is a prima facie reason to reconsider and possibly to amend some of its rules. Let's now consider how these principles of learning, transfer, and testing apply to John Dewey's 'Sign of Rain' example. =====Learning===== Rules in a knowledge base, as far as their effective content goes, can be obtained by any mode of inference. For example, a rule like: :* Rule: B &rArr; A, Just Before it rains, the Air is cool, is usually induced from a consideration of many past events, in a manner that can be rationally reconstructed as follows: :* Case: C &rArr; B, In Certain events, it is just Before it rains, :* Fact: C &rArr; A, In Certain events, the Air is cool, : ------------------------------------------------------------------------------------------ :* Rule: B &rArr; A, Just Before it rains, the Air is cool. However, the very same proposition could also be abduced as an explanation of a singular occurrence or deduced as a conclusion of a presumptive theory. =====Transfer===== What is it that gives a distinctively inductive character to the acquisition of a knowledge base? It is evidently the 'analogy of experience' that underlies its useful application. Whenever we find ourselves prefacing an argument with the phrase 'If past experience is any guide …' then we can be sure that this principle has come into play. We are invoking an analogy between past experience, considered as a totality, and present experience, considered as a point of application. What we mean in practice is this: 'If past experience is a fair sample of possible experience, then the knowledge gained in it applies to present experience'. This is the mechanism that allows a knowledge base to be carried across gulfs of experience that are indifferent to the effective contents of its rules. Here are the details of how this notion of transfer works out in the case of the 'Sign of Rain' example: Let K(pres) be a portion of the reasoner's knowledge base that is logically equivalent to the conjunction of two rules, as follows: :* K(pres) = (B &rArr; A) and (B &rArr; D). K(pres) is the present knowledge base, expressed in the form of a logical constraint on the present universe of discourse. It is convenient to have the option of expressing all logical statements in terms of their [[logical model]]s, that is, in terms of the primitive circumstances or the elements of experience over which they hold true. :* Let E(past) be the chosen set of experiences, or the circumstances that we have in mind when we refer to 'past experience'. :* Let E(poss) be the collective set of experiences, or the projective total of possible circumstances. :* Let E(pres) be the present experience, or the circumstances that are present to the reasoner at the current moment. If we think of the knowledge base K(pres) as referring to the 'regime of experience' over which it is valid, then all of these sets of models can be compared by the simple relations of [[set inclusion]] or [[logical implication]]. Figure 5 schematizes this way of viewing the 'analogy of experience'. {| align="center" cellpadding="8" style="text-align:center" | <pre> o-----------------------------------------------------------o | | | K(pres) | | o | | /|\ | | / | \ | | / | \ | | / | \ | | / Rule \ | | / | \ | | / | \ | | / | \ | | / E(poss) \ | | Fact / o \ Fact | | / * * \ | | / * * \ | | / * * \ | | / * * \ | | / * * \ | | / * Case Case * \ | | / * * \ | | / * * \ | | /* *\ | | o<<<---------------<<<---------------<<<o | | E(past) Analogy Morphism E(pres) | | More Known Less Known | | | o-----------------------------------------------------------o Figure 5. Analogy of Experience </pre> |} In these terms, the ''analogy of experience'' proceeds by inducing a Rule about the validity of a current knowledge base and then deducing a Fact, its applicability to a current experience, as in the following sequence: Inductive Phase: :* Given Case: E(past) &rArr; E(poss), Chosen events fairly sample Collective events. :* Given Fact: E(past) &rArr; K(pres), Chosen events support the Knowledge regime. : ----------------------------------------------------------------------------------------------------------------------------- :* Induce Rule: E(poss) &rArr; K(pres), Collective events support the Knowledge regime. Deductive Phase: :* Given Case: E(pres) &rArr; E(poss), Current events fairly sample Collective events. :* Given Rule: E(poss) &rArr; K(pres), Collective events support the Knowledge regime. : -------------------------------------------------------------------------------------------------------------------------------- :* Deduce Fact: E(pres) &rArr; K(pres), Current events support the Knowledge regime. =====Testing===== If the observer looks up and does not see dark clouds, or if he runs for shelter but it does not rain, then there is fresh occasion to question the utility or the validity of his knowledge base. But we must leave our foulweather friend for now and defer the logical analysis of this testing phase to another occasion. ==References== * [[Dana Angluin|Angluin, Dana]] (1989), "Learning with Hints", pp. 167&ndash;181 in David Haussler and Leonard Pitt (eds.), ''Proceedings of the 1988 Workshop on Computational Learning Theory'', MIT, 3&ndash;5 August 1988, Morgan Kaufmann, San Mateo, CA, 1989. * [[Aristotle]], "[[Prior Analytics]]", [[Hugh Tredennick]] (trans.), pp. 181&ndash;531 in ''Aristotle, Volume 1'', [[Loeb Classical Library]], [[Heinemann (book publisher)|William Heinemann]], London, UK, 1938. * Awbrey, S.M., and Awbrey, J.L. (May 2001), "Conceptual Barriers to Creating Integrative Universities", ''Organization : The Interdisciplinary Journal of Organization, Theory, and Society'' 8(2), Sage Publications, London, UK, pp. 269&ndash;284. [http://org.sagepub.com/cgi/content/abstract/8/2/269 Abstract]. * Awbrey, S.M., and Awbrey, J.L. (September 18, 1999), "Organizations of Learning or Learning Organizations : The Challenge of Creating Integrative Universities for the Next Century", ''Second International Conference of the Journal ''Organization'' '', ''Re-Organizing Knowledge, Trans-Forming Institutions : Knowing, Knowledge, and the University in the 21st Century'', University of Massachusetts, Amherst, MA. [http://www.cspeirce.com/menu/library/aboutcsp/awbrey/integrat.htm Online]. * Awbrey, J.L., and Awbrey, S.M. (Autumn 1995), "Interpretation as Action : The Risk of Inquiry", ''Inquiry : Critical Thinking Across the Disciplines'' 15(1), pp. 40&ndash;52. [http://www.chss.montclair.edu/inquiry/fall95/awbrey.html Online]. * Awbrey, J.L., and Awbrey, S.M. (June 1992), "Interpretation as Action : The Risk of Inquiry", ''The Eleventh International Human Science Research Conference'', Oakland University, Rochester, Michigan. * Awbrey, S.M., and Awbrey, J.L. (May 1991), "An Architecture for Inquiry : Building Computer Platforms for Discovery", ''Proceedings of the Eighth International Conference on Technology and Education'', Toronto, Canada, pp. 874&ndash;875. [http://home.m04.itscom.net/hhomey/tmp-a.html Online]. * Awbrey, J.L., and Awbrey, S.M. (January 1991), "Exploring Research Data Interactively : Developing a Computer Architecture for Inquiry", Poster presented at the ''Annual Sigma Xi Research Forum'', University of Texas Medical Branch, Galveston, TX. * Awbrey, J.L., and Awbrey, S.M. (August 1990), "Exploring Research Data Interactively. Theme One : A Program of Inquiry", ''Proceedings of the Sixth Annual Conference on Applications of Artificial Intelligence and CD-ROM in Education and Training'', Society for Applied Learning Technology, Washington, DC, pp. 9&ndash;15. * [[Cornelius F. Delaney|Delaney, C.F.]] (1993), ''Science, Knowledge, and Mind: A Study in the Philosophy of C.S. Peirce'', University of Notre Dame Press, Notre Dame, IN. * [[John Dewey|Dewey, John]] (1910), ''How We Think'', [[D.C. Heath]], Lexington, MA, 1910. Reprinted, Prometheus Books, Buffalo, NY, 1991. * Dewey, John (1938), ''Logic: The Theory of Inquiry'', Henry Holt and Company, New York, NY, 1938. Reprinted as pp. 1–527 in ''John Dewey, The Later Works, 1925&ndash;1953, Volume 12 : 1938'', Jo Ann Boydston (ed.), Kathleen Poulos (text. ed.), [[Ernest Nagel]] (intro.), Southern Illinois University Press, Carbondale and Edwardsville, IL, 1986. * [[Susan Haack|Haack, Susan]] (1993), ''Evidence and Inquiry&nbsp;: Towards Reconstruction in Epistemology'', Blackwell Publishers, Oxford, UK. * [[Norwood Russell Hanson|Hanson, Norwood Russell]] (1958), ''Patterns of Discovery, An Inquiry into the Conceptual Foundations of Science'', Cambridge University Press, Cambridge, UK. * [[Vincent F. Hendricks|Hendricks, Vincent F.]] (2005), ''Thought 2 Talk&nbsp;: A Crash Course in Reflection and Expression'', Automatic Press, New York, NY. * [[Cheryl J. Misak|Misak, Cheryl J.]] (1991), ''Truth and the End of Inquiry, A Peircean Account of Truth'', Oxford University Press, Oxford, UK. * [[Charles Peirce (Bibliography)|Peirce, C.S., Bibliography]]. * [[Charles Sanders Peirce|Peirce, C.S.]], (1931&ndash;1935, 1958), ''Collected Papers of Charles Sanders Peirce'', vols. 1&ndash;6, [[Charles Hartshorne]] and [[Paul Weiss (philosopher)|Paul Weiss]] (eds.), vols. 7&ndash;8, [[Arthur W. Burks]] (ed.), Harvard University Press, Cambridge, MA. Cited as CP&nbsp;volume.paragraph. * [[Robert C. Stalnaker|Stalnaker, Robert C.]] (1984), ''Inquiry'', MIT Press, Cambridge, MA. ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry Inquiry @ InterSciWiki] * [http://mywikibiz.com/Inquiry Inquiry @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Inquiry Inquiry @ Subject Wikis] * [http://en.wikiversity.org/wiki/Inquiry Inquiry @ Wikiversity] * [http://beta.wikiversity.org/wiki/Inquiry Inquiry @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Semiotic_Information Semiotic Information] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry Inquiry], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Inquiry Inquiry], [http://mywikibiz.com/ MyWikiBiz] * [http://forum.wolframscience.com/showthread.php?threadid=595 Inquiry], [http://forum.wolframscience.com/ NKS Forum] * [http://semanticweb.org/wiki/Inquiry Inquiry], [http://semanticweb.org/ SemanticWeb] * [http://wikinfo.org/w/index.php/Inquiry Inquiry], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Inquiry Inquiry], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Inquiry Inquiry], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Inquiry&oldid=71880922 Inquiry], [http://en.wikipedia.org/ Wikipedia] [[Category:Artificial Intelligence]] [[Category:Charles Sanders Peirce]] [[Category:Computer Science]] [[Category:Critical Thinking]] [[Category:Cybernetics]] [[Category:Education]] [[Category:Hermeneutics]] [[Category:Information Systems]] [[Category:Information Theory]] [[Category:Inquiry]] [[Category:Inquiry Driven Systems]] [[Category:Intelligence Amplification]] [[Category:Learning Organizations]] [[Category:Knowledge Representation]] [[Category:Logic]] [[Category:Philosophy]] [[Category:Pragmatics]] [[Category:Pragmatism]] [[Category:Relation Theory]] [[Category:Science]] [[Category:Semantics]] [[Category:Semiotics]] [[Category:Systems Science]] bfa493b19b8d9a9860bcdbeb5dddf65d34b00935 0 343 759 565 2015-11-16T00:28:19Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. <blockquote> Every mind which passes from doubt to belief must have ideas which follow after one another in time. Every mind which reasons must have ideas which not only follow after others but are caused by them. Every mind which is capable of logical criticism of its inferences, must be aware of this determination of its ideas by previous ideas. (Peirce, "On Time and Thought", CE&nbsp;3, 68&ndash;69.) </blockquote> All through the 1860s the young Charles Peirce was busy establishing a conceptual base-camp and a technical supply line for the intellectual adventures of a lifetime. Taking the long view of this activity and trying to choose the best titles for the story, it all seems to have something to do with the dynamics of inquiry. This broad subject area has a part that is given by nature and a part that is ruled by nurture. On first approach, it is possible to see a question of articulation and a question of explanation: :* What is needed to articulate the workings of the active form of representation that is known as ''conscious experience''? :* What is needed to account for the workings of the reflective discipline of inquiry that is known as ''science''? The pursuit of answers to these questions finds them to be so entangled with each other that it's ultimately impossible to comprehend them apart from each other, but for the sake of exposition it's convenient to organize our study of Peirce's assault on the ''summa'' by following first the trails of thought that led him to develop a ''[[theory of signs]]'', one that has come to be known as '[[semiotic]]', and tracking next the ways of thinking that led him to develop a ''[[theory of inquiry]]'', one that would be up to the task of saying 'how science works'. Opportune points of departure for exploring the dynamics of representation, such as led to Peirce's theories of [[inference]] and [[information]], inquiry and signs, are those that he took for his own springboards. Perhaps the most significant influences radiate from points on parallel lines of inquiry in [[Aristotle]]'s work, points where the intellectual forerunner focused on many of the same issues and even came to strikingly similar conclusions, at least about the best ways to begin. Staying within the bounds of what will give us a more solid basis for understanding Peirce, it serves to consider the following ''loci'' in Aristotle: :* The basic terminology of [[psychology]], in ''[[On the Soul]]''. :* The founding description of [[sign relations]], in ''[[On Interpretation]]''; :* The differentiation of the genus of reasoning into three species of [[inference]] that are commonly translated into English as ''[[Abductive reasoning|abduction]]'', ''[[Deductive reasoning|deduction]]'', and ''[[Inductive reasoning|induction]]'', in the ''[[Prior Analytics]]''. In addition to the three elements of inference, that Peirce would assay to be [[irreducible]], [[Aristotle]] analyzed several types of [[compound inference]], most importantly the type known as 'reasoning by [[analogy]]' or 'reasoning from [[example]]', employing for the latter description the Greek word 'paradeigma', from which we get our word '[[paradigm]]'. Inquiry is a form of reasoning process, in effect, a particular way of conducting thought, and thus it can be said to institute a specialized manner, style, or turn of thinking. Philosophers of the school that is commonly called 'pragmatic' hold that all thought takes place in signs, where 'sign' is the word they use for the broadest conceivable variety of characters, expressions, formulas, messages, signals, texts, and so on up the line, that might be imagined. Even intellectual concepts and mental ideas are held to be a special class of signs, corresponding to internal states of the thinking agent that both issue in and result from the interpretation of external signs. The subsumption of inquiry within reasoning in general and the inclusion of thinking within the class of sign processes allows us to approach the subject of inquiry from two different perspectives: :* The ''[[syllogistic]]'' approach treats inquiry as a species of logical process, and is limited to those of its aspects that can be related to the most basic laws of inference. :* The ''[[sign-theoretic]]'' approach views inquiry as a genus of ''[[semiosis]]'', an activity taking place within the more general setting of [[sign relation]]s and [[sign process]]es. The distinction between signs denoting and objects denoted is critical to the discussion of Peirce's theory of signs. Wherever needed in the rest of this article, therefore, in order to mark this distinction a little more emphatically than usual, double quotation marks placed around a given sign, for example, a string of zero or more characters, will be used to create a new sign that denotes the given sign as its object. ===Semeiotic : Peirce's theory of signs=== Peirce referred to his general study of signs, based on the concept of a [[triadic relation|triadic]] [[sign relation]], as ''[[semeiotic]]'' or ''[[semiotic]]'', either of which terms are currently used in both singular of plural forms. Peirce began writing on semeiotic in the 1860s, around the time that he devised his system of three categories. He eventually defined ''[[semiosis]]'' as an "action, or influence, which is, or involves, a cooperation of ''three'' subjects, such as a sign, its object, and its interpretant, this tri-relative influence not being in any way resolvable into actions between pairs". (Houser 1998: 411, written 1907). This triadic relation grounds the semeiotic. In order to understand what a ''sign'' is we need to understand what a ''[[sign relation]]'' is, for signhood is a way of being in relation, not a way of being in itself. In order to understand what a sign relation is we need to understand what a ''[[triadic relation]]'' is, for the role of a sign is constituted as one among three, where roles in general are distinct even when the things that fill them are not. In order to understand what a triadic relation is we need to understand what a ''[[relation (mathematics)|relation]]'' is, and here there are traditionally two ways of understanding what a relation is, both of which are necessary if not sufficient to complete understanding, namely, the way of ''[[extension (semantics)|extension]]'' and the way of ''[[intension]]''. To these traditional approximations, Peirce adds a third way, the way of ''[[semiotic information theory|information]]'', that integrates the other two approaches in a unified whole. ====Sign relations==== : ''Main article'' : [[Sign relation]] With that hasty map of relations and relatives sketched above, we may now trek into the terrain of ''sign relations'', the main subject matter of Peirce's ''semeiotic'', or theory of signs. ====Types of signs==== Peirce proposes several typologies and definitions of the signs. More than 76 definitions of what a sign is have been collected throughout Peirce's work. Some canonical typologies can nonetheless be observed, one crucial one being the distinction between "icons", "indices" and "symbols" (CP 2.228, CP 2.229 and CP 5.473). This typology emphasizes the different ways in which the ''representamen'' (or its ''ground'') addresses or refers to its ''object'', through a particular mobilisation of an ''interpretant'' (but Peirce proposes also other typologies based on other criteria). * An '''icon''' is a sign that denotes its objects by virtue of a quality that it shares with them. The sign is perceived as resembling or imitating the object it refers to (e.g. fork on a sign by the road indicating a rest stop). In other words, an icon thus "resembles" to its object. It shares a character or an aspect with it, which allows for it to be interpreted as a sign even if the object does not exist. It signifies essentially on the basis of its "ground". * An '''index''' is a sign that denotes its objects by virtue of an existential connection that it has with them. For an index to signify, the relation to the object is crucial. The ''representamen'' is directly connected in some way (physically or casually) to the object it denotes (e.g. smoke coming from a building is an index of fire). Hence, an index refers to the object because it is really affected or modified by it, and thus may stand as a trace of the existence of the object. * A '''symbol''' is a sign that denotes its objects solely by virtue of the fact that it is interpreted to do so. The ''representamen'' does not resemble the object signified but is fundamentally conventional, so that the signifying relationship must be learned and agreed upon (e.g. the word "cat"). A symbol thus denotes, primarily, by virtue of its ''interpretant''. Its action (''semeiosis'') is ruled by a convention, a more or less systematic set of associations that guarantees its interpretation, independently of any resemblance or any material relation with its object. Note that these definitions are specific to Peirce's theory of signs and are not exactly equivalent to general uses of the notion of "[[icon]]", "[[symbol]]" or "[[index]]". ===Theory of inquiry=== : ''Main article'' : [[Inquiry]] : Upon this first, and in one sense this sole, rule of reason, that in order to learn you must desire to learn, and in so desiring not be satisfied with what you already incline to think, there follows one corollary which itself deserves to be inscribed upon every wall of the city of philosophy: <center> Do not block the way of inquiry.</center> : Although it is better to be methodical in our investigations, and to consider the economics of research, yet there is no positive sin against logic in ''trying'' any theory which may come into our heads, so long as it is adopted in such a sense as to permit the investigation to go on unimpeded and undiscouraged. On the other hand, to set up a philosophy which barricades the road of further advance toward the truth is the one unpardonable offence in reasoning, as it is also the one to which metaphysicians have in all ages shown themselves the most addicted. (Peirce, "F.R.L." (c. 1899), CP 1.135&ndash;136.) Peirce extracted the pragmatic model or [[theory]] of [[inquiry]] from its raw materials in classical logic and refined it in parallel with the early development of symbolic logic to address problems about the nature of scientific reasoning. Borrowing a brace of concepts from [[Aristotle]], Peirce examined three fundamental modes of reasoning that play a role in inquiry, processes that are currently known as ''[[abductive]]'', ''[[deductive]]'', and ''[[inductive]]'' [[inference]]. In the roughest terms, [[abductive reasoning|abduction]] is what we use to generate a likely [[hypothesis]] or an initial [[diagnosis]] in response to a [[phenomenon]] of interest or a [[problem]] of concern, while [[deductive reasoning|deduction]] is used to clarify, to derive, and to explicate the relevant consequences of the selected [[hypothesis]], and [[inductive reasoning|induction]] is used to test the sum of the predictions against the sum of the data. These three processes typically operate in a cyclic fashion, systematically operating to reduce the uncertainties and the difficulties that initiated the inquiry in question, and in this way, to the extent that inquiry is successful, leading to an increase in the [[knowledge]] or [[skills]], in other words, an [[augmentation]] in the [[competence]] or [[performance]], of the agent or community engaged in the inquiry. In the pragmatic way of thinking every thing has a purpose, and the purpose of any thing is the first thing that we should try to note about it. The purpose of inquiry is to reduce doubt and lead to a state of belief, which a person in that state will usually call ''knowledge'' or ''certainty''. It needs to be appreciated that the three kinds of inference, insofar as they contribute to the end of inquiry, describe a cycle that can be understood only as a whole, and none of the three makes complete sense in isolation from the others. For instance, the purpose of [[abductive reasoning|abduction]] is to generate guesses of a kind that [[deductive reasoning|deduction]] can explicate and that [[inductive reasoning|induction]] can evaluate. This places a mild but meaningful [[constraint]] on the production of hypotheses, since it is not just any wild guess at [[explanation]] that submits itself to reason and bows out when defeated in a match with reality. In a similar fashion, each of the other types of [[inference]] realizes its purpose only in accord with its proper role in the whole [[cycle of inquiry]]. No matter how much it may be necessary to study these processes in abstraction from each other, the [[integrity]] of inquiry places strong limitations on the effective modularity of its principal components. If we then think to inquire, "What sort of [[constraint]], exactly, does pragmatic thinking place on our guesses?", we have asked the question that is generally recognized as the problem of "giving a rule to abduction". Peirce's way of answering it is given in terms of the so-called ''[[pragmatic maxim]]'', and this in turn gives us a clue as to the central role of abductive reasoning in Peirce's pragmatic philosophy. ===Logic of information=== : ''Main article'' : [[Logic of information]] <blockquote> Let us now return to the information. The information of a term is the measure of its superfluous comprehension. That is to say that the proper office of the comprehension is to determine the extension of the term. For instance, you and I are men because we possess those attributes &mdash; having two legs, being rational, &tc. &mdash; which make up the comprehension of ''man''. Every addition to the comprehension of a term lessens its extension up to a certain point, after that further additions increase the information instead. (C.S. Peirce, "The Logic of Science, or, Induction and Hypothesis" (1866), CE 1, 467.) </blockquote> ==Source materials== [[C.S. Peirce]], &ldquo;On Time and Thought&rdquo;, MS 215, 8 March 1873. <blockquote> Every mind which passes from doubt to belief must have ideas which follow after one another in time. Every mind which reasons must have ideas which not only follow after others but are caused by them. Every mind which is capable of logical criticism of its inferences, must be aware of this determination of its ideas by previous ideas. But is it pre-supposed in the conception of a logical mind, that the temporal succession in its ideas is continuous, and not by discrete steps? A continuum such as we suppose time and space to be, is defined as something any part of which itself has parts of the same kind. So that the point of time or the point of space is nothing but the ideal limit towards which we approach, but which we can never reach in dividing time or space; and consequently nothing is true of a point which is not true of a space or a time. A discrete quantum, on the other hand, has ultimate parts which differ from any other part of the quantum in their absolute separation from one another. If the succession of images in the mind is by discrete steps, time for that mind will be made up of indivisible instants. Any one idea will be absolutely distinguished from every other idea by its being present only in the passing moment. And the same idea can not exist in two different moments, however similar the ideas felt in the two different moments may, for the sake of argument, be allowed to be. Now an idea exists only so far as the mind thinks it; and only when it is present to the mind. An idea therefore has no characters or qualities but what the mind thinks of it at the time when it is present to the mind. It follows from this that if the succession of time were by separate steps, no idea could resemble another; for these ideas if they are distinct, are present to the mind at different times. Therefore at no time when one is present to the mind, is the other present. Consequently the mind never compares them nor thinks them to be alike; and consequently they are not alike; since they are only what they are thought to be at the time when they are present. It may be objected that though the mind does not directly think them to be alike; yet it may think together reproductions of them, and thus think them to be alike. This would be a valid objection were it not necessary, in the first place, in order that one idea should be the representative of another, that it should resemble that idea, which it could only do by means of some representation of it again, and so on to infinity; the link which is to bind the first two together which are to be pronounced alike, never being found. In short the resemblance of ideas implies that some two ideas are to be thought together which are present to the mind at different times. And this never can be, if instants are separated from one another by absolute steps. This conception is therefore to be abandoned, and it must be acknowledged to be already presupposed in the conception of a logical mind that the flow of time should be continuous. Let us consider then how we are to conceive what is present to the mind. We are accustomed to say that nothing is present but a fleeting instant, a point of time. But this is a wrong view of the matter because a point differs in no respect from a space of time, except that it is the ideal limit which, in the division of time, we never reach. It can not therefore be that it differs from an interval of time in this respect that what is present is only in a fleeting instant, and does not occupy a whole interval of time, unless what is present be an ideal something which can never be reached, and not something real. The true conception is, that ideas which succeed one another during an interval of time, become present to the mind through the successive presence of the ideas which occupy the parts of that time. So that the ideas which are present in each of these parts are more immediately present, or rather less mediately present than those of the whole time. And this division may be carried to any extent. But you never reach an idea which is quite immediately present to the mind, and is not made present by the ideas which occupy the parts of the time that it occupies. Accordingly, it takes time for ideas to be present to the mind. They are present during a time. And they are present by means of the presence of the ideas which are in the parts of that time. Nothing is therefore present to the mind in an instant, but only during a time. The events of a day are less mediately present to the mind than the events of a year; the events of a second less mediately present than the events of a day. (C.S. Peirce, CE 3, pp. 68&ndash;70). </blockquote> Charles Sanders Peirce, MS 215, 1873, [&ldquo;On Time and Thought&rdquo;], pp. 68&ndash;71 in ''Writings of Charles S. Peirce : A Chronological Edition, Volume 3, 1872&ndash;1878'', Peirce Edition Project, Indiana University Press, Bloomington, IN, 1986. ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Dynamics_of_inquiry Dynamics of Inquiry @ InterSciWiki] * [http://mywikibiz.com/Dynamics_of_inquiry Dynamics of Inquiry @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Dynamics_of_inquiry Dynamics of Inquiry @ Subject Wikis] * [http://en.wikiversity.org/wiki/Dynamics_of_inquiry Dynamics of Inquiry @ Wikiversity] * [http://beta.wikiversity.org/wiki/Dynamics_of_inquiry Dynamics of Inquiry @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Semiotic_Information Semiotic Information] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Dynamics_of_inquiry Dynamics of Inquiry], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Dynamics_of_inquiry Dynamics of Inquiry], [http://mywikibiz.com/ MyWikiBiz] * [http://semanticweb.org/wiki/Dynamics_of_inquiry Dynamics of Inquiry], [http://semanticweb.org/ SemanticWeb] * [http://wikinfo.org/w/index.php/Dynamics_of_inquiry Dynamics of Inquiry], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Dynamics_of_inquiry Dynamics of Inquiry], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Dynamics_of_inquiry Dynamics of Inquiry], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Charles_Sanders_Peirce&oldid=111891138#Dynamics_of_inquiry Dynamics of Inquiry], [http://en.wikipedia.org/ Wikipedia] [[Category:Artificial Intelligence]] [[Category:Charles Sanders Peirce]] [[Category:Computer Science]] [[Category:Critical Thinking]] [[Category:Cybernetics]] [[Category:Education]] [[Category:Hermeneutics]] [[Category:Information Systems]] [[Category:Information Theory]] [[Category:Inquiry]] [[Category:Inquiry Driven Systems]] [[Category:Intelligence Amplification]] [[Category:Learning Organizations]] [[Category:Knowledge Representation]] [[Category:Logic]] [[Category:Philosophy]] [[Category:Pragmatics]] [[Category:Pragmatism]] [[Category:Science]] [[Category:Semantics]] [[Category:Semiotics]] [[Category:Systems Science]] d587fe4c3f46e3676a23e723fdeddfd4bfdb606d 0 346 760 568 2015-11-16T03:26:34Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. The '''logic of information''', or the ''logical theory of information'', considers the information content of logical signs &mdash; everything from bits to books and beyond &mdash; along the lines initially developed by [[Charles Sanders Peirce]].&nbsp; In this line of development the concept of information serves to integrate the aspects of logical signs that are separately covered by the concepts of denotation and connotation, or, in roughly equivalent terms, by the concepts of extension and comprehension. Peirce began to develop these ideas in his lectures &ldquo;On the Logic of Science&rdquo; at Harvard University (1865) and the Lowell Institute (1866).&nbsp; Here is one of the starting points: {| align="center" cellpadding="8" width="90%" | <p>Let us now return to the information.&nbsp; The information of a term is the measure of its superfluous comprehension.&nbsp; That is to say that the proper office of the comprehension is to determine the extension of the term.&nbsp; For instance, you and I are men because we possess those attributes &mdash; having two legs, being rational, &tc. &mdash; which make up the comprehension of ''man''.&nbsp; Every addition to the comprehension of a term lessens its extension up to a certain point, after that further additions increase the information instead.</p> <p>Thus, let us commence with the term ''colour'';&nbsp; add to the comprehension of this term, that of ''red''.&nbsp; ''Red colour'' has considerably less extension than ''colour'';&nbsp; add to this the comprehension of ''dark'';&nbsp; ''dark red colour'' has still less [extension].&nbsp; Add to this the comprehension of ''non-blue'' &mdash; ''non-blue dark red colour'' has the same extension as ''dark red colour'', so that the ''non-blue'' here performs a work of supererogation;&nbsp; it tells us that no ''dark red colour'' is blue, but does none of the proper business of connotation, that of diminishing the extension at all.</p> <p>Thus information measures the superfluous comprehension.&nbsp; And, hence, whenever we make a symbol to express any thing or any attribute we cannot make it so empty that it shall have no superfluous comprehension.&nbsp; I am going, next, to show that inference is symbolization and that the puzzle of the validity of scientific inference lies merely in this superfluous comprehension and is therefore entirely removed by a consideration of the laws of ''information''.&nbsp; (C.S. Peirce, &ldquo;The Logic of Science, or, Induction and Hypothesis&rdquo; (1866), CE&nbsp;1, 467).</p> |} ==References== * [[Charles Sanders Peirce (Bibliography)|Peirce, C.S., Bibliography]]. * De Tienne, André (2006), "Peirce's Logic of Information", Seminario del Grupo de Estudios Peirceanos, Universidad de Navarra, 28 Sep 2006. [http://www.unav.es/gep/SeminariodeTienne.html Online]. * Peirce, C.S. (1867), "Upon Logical Comprehension and Extension", [http://www.iupui.edu/~peirce/writings/v2/w2/w2_06/v2_06.htm Online]. ==Resources== * [http://vectors.usc.edu/thoughtmesh/publish/145.php Logic of Information &rarr; ThoughtMesh] ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Logic_of_information Logic of Information @ InterSciWiki] * [http://mywikibiz.com/Logic_of_information Logic of Information @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Logic_of_information Logic of Information @ Subject Wikis] * [http://en.wikiversity.org/wiki/Logic_of_information Logic of Information @ Wikiversity] * [http://beta.wikiversity.org/wiki/Logic_of_information Logic of Information @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Semiotic_Information Semiotic Information] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Logic_of_information Logic of Information], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Logic_of_information Logic of Information], [http://mywikibiz.com/ MyWikiBiz] * [http://semanticweb.org/wiki/Logic_of_information Logic of Information], [http://semanticweb.org/ Semantic Web] * [http://vectors.usc.edu/thoughtmesh/publish/145.php Logic of Information], [http://vectors.usc.edu/thoughtmesh/ ThoughtMesh] * [http://wikinfo.org/w/index.php/Logic_of_information Logic of Information], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Logic_of_information Logic of Information], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Logic_of_information Logic of Information], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Logic_of_information&oldid=67770000 Logic of Information], [http://en.wikipedia.org/ Wikipedia] [[Category:Artificial Intelligence]] [[Category:Charles Sanders Peirce]] [[Category:Cybernetics]] [[Category:Hermeneutics]] [[Category:Information Systems]] [[Category:Information Theory]] [[Category:Inquiry]] [[Category:Intelligence Amplification]] [[Category:Knowledge Representation]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Philosophy]] [[Category:Pragmatics]] [[Category:Pragmatism]] [[Category:Science]] [[Category:Semantics]] [[Category:Semiotics]] [[Category:Systems Science]] 8d3c1c78ad8d98ade27edf56b82ab302fdff1e94 0 312 763 712 2015-11-16T21:05:04Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. This article treats relations from the perspective of combinatorics, in other words, as a subject matter in discrete mathematics, with special attention to finite structures and concrete set-theoretic constructions, many of which arise quite naturally in applications. This approach to relation theory, or the theory of relations, is distinguished from, though closely related to, its study from the perspectives of abstract algebra on the one hand and formal logic on the other. ==Preliminaries== Two definitions of the relation concept are common in the literature. Although it is usually clear in context which definition is being used at a given time, it tends to become less clear as contexts collide, or as discussion moves from one context to another. The same sort of ambiguity arose in the development of the function concept and it may save some effort to follow the pattern of resolution that worked itself out there. When we speak of a function <math>f : X \to Y\!</math> we are thinking of a mathematical object whose articulation requires three pieces of data, specifying the set <math>X,\!</math> the set <math>Y,\!</math> and a particular subset of their cartesian product <math>{X \times Y}.\!</math> So far so good. Let us write <math>f = (\mathrm{obj_1}f, \mathrm{obj_2}f, \mathrm{obj_{12}}f)\!</math> to express what has been said so far. When it comes to parsing the notation <math>{}^{\backprime\backprime} f : X \to Y {}^{\prime\prime},\!</math> everyone takes the part <math>{}^{\backprime\backprime} X \to Y {}^{\prime\prime}\!</math> to specify the ''type'' of the function, that is, the pair <math>(\mathrm{obj_1}f, \mathrm{obj_2}f),\!</math> but <math>{}^{\backprime\backprime} f {}^{\prime\prime}\!</math> is used equivocally to denote both the triple and the subset <math>\mathrm{obj_{12}}f\!</math> that forms one part of it. One way to resolve the ambiguity is to formalize a distinction between a function and its ''graph'', letting <math>\mathrm{graph}(f) := \mathrm{obj_{12}}f.\!</math> Another tactic treats the whole notation <math>{}^{\backprime\backprime} f : X \to Y {}^{\prime\prime}\!</math> as sufficient denotation for the triple, letting <math>{}^{\backprime\backprime} f {}^{\prime\prime}\!</math> denote <math>\mathrm{graph}(f).\!</math> In categorical and computational contexts, at least initially, the type is regarded as an essential attribute or an integral part of the function itself. In other contexts it may be desirable to use a more abstract concept of function, treating a function as a mathematical object that appears in connection with many different types. Following the pattern of the functional case, let the notation <math>{}^{\backprime\backprime} L \subseteq X \times Y {}^{\prime\prime}\!</math> bring to mind a mathematical object that is specified by three pieces of data, the set <math>X,\!</math> the set <math>Y,\!</math> and a particular subset of their cartesian product <math>{X \times Y}.\!</math> As before we have two choices, either let <math>L = (X, Y, \mathrm{graph}(L))\!</math> or let <math>{}^{\backprime\backprime} L {}^{\prime\prime}\!</math> denote <math>\mathrm{graph}(L)\!</math> and choose another name for the triple. ==Definition== It is convenient to begin with the definition of a ''<math>k\!</math>-place relation'', where <math>k\!</math> is a positive integer. '''Definition.''' A ''<math>k\!</math>-place relation'' <math>L \subseteq X_1 \times \ldots \times X_k\!</math> over the nonempty sets <math>X_1, \ldots, X_k\!</math> is a <math>(k+1)\!</math>-tuple <math>(X_1, \ldots, X_k, L)\!</math> where <math>L\!</math> is a subset of the cartesian product <math>X_1 \times \ldots \times X_k.\!</math> ==Remarks== Though usage varies as usage will, there are several bits of optional language that are frequently useful in discussing relations. The sets <math>X_1, \ldots, X_k\!</math> are called the ''domains'' of the relation <math>L \subseteq X_1 \times \ldots \times X_k,\!</math> with <math>{X_j}\!</math> being the <math>j^\text{th}\!</math> domain. If all of the <math>{X_j}\!</math> are the same set <math>X\!</math> then <math>L \subseteq X_1 \times \ldots \times X_k\!</math> is more simply described as a <math>k\!</math>-place relation over <math>X.\!</math> The set <math>L\!</math> is called the ''graph'' of the relation <math>L \subseteq X_1 \times \ldots \times X_k,\!</math> on analogy with the graph of a function. If the sequence of sets <math>X_1, \ldots, X_k\!</math> is constant throughout a given discussion or is otherwise determinate in context, then the relation <math>L \subseteq X_1 \times \ldots \times X_k\!</math> is determined by its graph <math>L,\!</math> making it acceptable to denote the relation by referring to its graph. Other synonyms for the adjective ''<math>k\!</math>-place'' are ''<math>k\!</math>-adic'' and ''<math>k\!</math>-ary'', all of which leads to the integer <math>k\!</math> being called the ''dimension'', ''adicity'', or ''arity'' of the relation <math>L.\!</math> ==Local incidence properties== A ''local incidence property'' (LIP) of a relation <math>L\!</math> is a property that depends in turn on the properties of special subsets of <math>L\!</math> that are known as its ''local flags''. The local flags of a relation are defined in the following way: Let <math>L\!</math> be a <math>k\!</math>-place relation <math>L \subseteq X_1 \times \ldots \times X_k.\!</math> Select a relational domain <math>{X_j}\!</math> and one of its elements <math>x.\!</math> Then <math>L_{x \operatorname{at} j}\!</math> is a subset of <math>L\!</math> that is referred to as the ''flag'' of <math>L\!</math> with <math>x\!</math> at <math>{j},\!</math> or the ''<math>x \operatorname{at} j\!</math>-flag'' of <math>L,\!</math> an object that has the following definition: {| align="center" cellpadding="8" style="text-align:center" | <math>L_{x \operatorname{at} j} ~=~ \{ (x_1, \ldots, x_j, \ldots, x_k) \in L ~:~ x_j = x \}.\!</math> |} Any property <math>C\!</math> of the local flag <math>L_{x \operatorname{at} j} \subseteq L\!</math> is said to be a ''local incidence property'' of <math>L\!</math> with respect to the ''locus'' <math>x \operatorname{at} j.\!</math> A <math>k\!</math>-adic relation <math>L \subseteq X_1 \times \ldots \times X_k\!</math> is said to be ''<math>C\!</math>-regular'' at <math>j\!</math> if and only if every flag of <math>L\!</math> with <math>x\!</math> at <math>j\!</math> has the property <math>C,\!</math> where <math>x\!</math> is taken to vary over the ''theme'' of the fixed domain <math>X_j.\!</math> Expressed in symbols, <math>L\!</math> is ''<math>C\!</math>-regular'' at <math>j\!</math> if and only if <math>C(L_{x \operatorname{at} j})\!</math> is true for all <math>x\!</math> in <math>X_j.\!</math> ==Regional incidence properties== The definition of a local flag can be broadened from a point <math>x\!</math> in <math>{X_j}\!</math> to a subset <math>M\!</math> of <math>X_j,\!</math> arriving at the definition of a ''regional flag'' in the following way: Suppose that <math>L \subseteq X_1 \times \ldots \times X_k,\!</math> and choose a subset <math>M \subseteq X_j.\!</math> Then <math>L_{M \operatorname{at} j}\!</math> is a subset of <math>L\!</math> that is said to be the ''flag'' of <math>L\!</math> with <math>M\!</math> at <math>{j},\!</math> or the ''<math>M \operatorname{at} j\!</math>-flag'' of <math>L,\!</math> an object which has the following definition: {| align="center" cellpadding="8" style="text-align:center" | <math>L_{M \operatorname{at} j} ~=~ \{ (x_1, \ldots, x_j, \ldots, x_k) \in L ~:~ x_j \in M \}.\!</math> |} ==Numerical incidence properties== A ''numerical incidence property'' (NIP) of a relation is a local incidence property that depends on the cardinalities of its local flags. For example, <math>L\!</math> is said to be ''<math>c\!</math>-regular at <math>j\!</math>'' if and only if the cardinality of the local flag <math>L_{x \operatorname{at} j}\!</math> is <math>c\!</math> for all <math>x\!</math> in <math>{X_j},\!</math> or, to write it in symbols, if and only if <math>|L_{x \operatorname{at} j}| = c\!</math> for all <math>{x \in X_j}.\!</math> In a similar fashion, one can define the NIPs, ''<math>(<\!c)\!</math>-regular at <math>{j},\!</math>'' ''<math>(>\!c)\!</math>-regular at <math>{j},\!</math>'' and so on. For ease of reference, a few of these definitions are recorded here: {| align="center" cellpadding="8" style="text-align:center" | <math>\begin{matrix} L & \text{is} & c\textit{-regular} & \text{at}\ j & \text{if and only if} & |L_{x \operatorname{at} j}| & = & c & \text{for all}\ x \in X_j. \\[4pt] L & \text{is} & (<\!c)\textit{-regular} & \text{at}\ j & \text{if and only if} & |L_{x \operatorname{at} j}| & < & c & \text{for all}\ x \in X_j. \\[4pt] L & \text{is} & (>\!c)\textit{-regular} & \text{at}\ j & \text{if and only if} & |L_{x \operatorname{at} j}| & > & c & \text{for all}\ x \in X_j. \end{matrix}</math> |} ==Species of 2-adic relations== Returning to 2-adic relations, it is useful to describe some familiar classes of objects in terms of their local and numerical incidence properties. Let <math>L \subseteq S \times T\!</math> be an arbitrary 2-adic relation. The following properties of <math>L\!</math> can be defined: {| align="center" cellpadding="8" style="text-align:center" | <math>\begin{matrix} L & \text{is} & \textit{total} & \text{at}~ S & \text{if and only if} & L & \text{is} & (\ge 1)\text{-regular} & \text{at}~ S. \\[4pt] L & \text{is} & \textit{total} & \text{at}~ T & \text{if and only if} & L & \text{is} & (\ge 1)\text{-regular} & \text{at}~ T. \\[4pt] L & \text{is} & \textit{tubular} & \text{at}~ S & \text{if and only if} & L & \text{is} & (\le 1)\text{-regular} & \text{at}\ S. \\[4pt] L & \text{is} & \textit{tubular} & \text{at}~ T & \text{if and only if} & L & \text{is} & (\le 1)\text{-regular} & \text{at}~ T. \end{matrix}</math> |} If <math>L \subseteq S \times T\!</math> is tubular at <math>S\!</math> then <math>L\!</math> is called a ''partial function'' or a ''prefunction'' from <math>S\!</math> to <math>T.\!</math> This is sometimes indicated by giving <math>L\!</math> an alternate name, say, <math>{}^{\backprime\backprime} p {}^{\prime\prime},~\!</math> and writing <math>L = p : S \rightharpoonup T.\!</math> Just by way of formalizing the definition: {| align="center" cellpadding="8" style="text-align:center" | <math>\begin{matrix} L & = & p : S \rightharpoonup T & \text{if and only if} & L & \text{is} & \text{tubular} & \text{at}~ S. \end{matrix}</math> |} If <math>L\!</math> is a prefunction <math>p : S \rightharpoonup T\!</math> that happens to be total at <math>S,\!</math> then <math>L\!</math> is called a ''function'' from <math>S\!</math> to <math>T,\!</math> indicated by writing <math>L = f : S \to T.\!</math> To say that a relation <math>L \subseteq S \times T\!</math> is ''totally tubular'' at <math>S\!</math> is to say that it is <math>1\!</math>-regular at <math>S.\!</math> Thus, we may formalize the following definition: {| align="center" cellpadding="8" style="text-align:center" | <math>\begin{matrix} L & = & f : S \to T & \text{if and only if} & L & \text{is} & 1\text{-regular} & \text{at}~ S. \end{matrix}</math> |} In the case of a function <math>f : S \to T,\!</math> one has the following additional definitions: {| align="center" cellpadding="8" style="text-align:center" | <math>\begin{matrix} f & \text{is} & \textit{surjective} & \text{if and only if} & f & \text{is} & \text{total} & \text{at}~ T. \\[4pt] f & \text{is} & \textit{injective} & \text{if and only if} & f & \text{is} & \text{tubular} & \text{at}~ T. \\[4pt] f & \text{is} & \textit{bijective} & \text{if and only if} & f & \text{is} & 1\text{-regular} & \text{at}~ T. \end{matrix}</math> |} ==Variations== Because the concept of a relation has been developed quite literally from the beginnings of logic and mathematics, and because it has incorporated contributions from a diversity of thinkers from many different times and intellectual climes, there is a wide variety of terminology that the reader may run across in connection with the subject. One dimension of variation is reflected in the names that are given to <math>k\!</math>-place relations, for <math>k = 0, 1, 2, 3, \ldots,\!</math> with some writers using the Greek forms, ''medadic'', ''monadic'', ''dyadic'', ''triadic'', ''<math>k\!</math>-adic'', and other writers using the Latin forms, ''nullary'', ''unary'', ''binary'', ''ternary'', ''<math>k\!</math>-ary''. The number of relational domains may be referred to as the ''adicity'', ''arity'', or ''dimension'' of the relation. Accordingly, one finds a relation on a finite number of domains described as a ''polyadic'' relation or a ''finitary'' relation, but others count infinitary relations among the polyadic. If the number of domains is finite, say equal to <math>k,\!</math> then the relation may be described as a ''<math>k\!</math>-adic'' relation, a ''<math>k\!</math>-ary'' relation, or a ''<math>k\!</math>-dimensional'' relation, respectively. A more conceptual than nominal variation depends on whether one uses terms like ''predicate'', ''relation'', and even ''term'' to refer to the formal object proper or else to the allied syntactic items that are used to denote them. Compounded with this variation is still another, frequently associated with philosophical differences over the status in reality accorded formal objects. Among those who speak of numbers, functions, properties, relations, and sets as being real, that is to say, as having objective properties, there are divergences as to whether some things are more real than others, especially whether particulars or properties are equally real or else one is derivative in relationship to the other. Historically speaking, just about every combination of modalities has been used by one school of thought or another, but it suffices here merely to indicate how the options are generated. ==Examples== : ''See the articles on [[relation (mathematics)|relations]], [[relation composition]], [[relation reduction]], [[sign relation]]s, and [[triadic relation]]s for concrete examples of relations.'' Many relations of the greatest interest in mathematics are triadic relations, but this fact is somewhat disguised by the circumstance that many of them are referred to as ''binary operations'', and because the most familiar of these have very specific properties that are dictated by their axioms. This makes it practical to study these operations for quite some time by focusing on their dyadic aspects before being forced to consider their proper characters as triadic relations. ==References== * Peirce, C.S., &ldquo;Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic&rdquo;, ''Memoirs of the American Academy of Arts and Sciences'', 9, 317&ndash;378, 1870. Reprinted, ''Collected Papers'' CP 3.45&ndash;149, ''Chronological Edition'' CE 2, 359&ndash;429. * Ulam, S.M. and Bednarek, A.R., &ldquo;On the Theory of Relational Structures and Schemata for Parallel Computation&rdquo;, pp. 477&ndash;508 in A.R. Bednarek and Françoise Ulam (eds.), ''Analogies Between Analogies : The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators'', University of California Press, Berkeley, CA, 1990. ==Bibliography== * Barr, Michael, and Wells, Charles (1990), ''Category Theory for Computing Science'', Prentice Hall, Hemel Hempstead, UK. * Bourbaki, Nicolas (1994), ''Elements of the History of Mathematics'', John Meldrum (trans.), Springer-Verlag, Berlin, Germany. * Carnap, Rudolf (1958), ''Introduction to Symbolic Logic with Applications'', Dover Publications, New York, NY. * Chang, C.C., and Keisler, H.J. (1973), ''Model Theory'', North-Holland, Amsterdam, Netherlands. * van Dalen, Dirk (1980), ''Logic and Structure'', 2nd edition, Springer-Verlag, Berlin, Germany. * Devlin, Keith J. (1993), ''The Joy of Sets : Fundamentals of Contemporary Set Theory'', 2nd edition, Springer-Verlag, New York, NY. * Halmos, Paul Richard (1960), ''Naive Set Theory'', D. 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Reprinted (CP 3.45&ndash;149), (CE 2, 359&ndash;429). * Peirce, Charles Sanders (1931&ndash;1935, 1958), ''Collected Papers of Charles Sanders Peirce'', vols. 1&ndash;6, Charles Hartshorne and Paul Weiss (eds.), vols. 7&ndash;8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA. Cited as (CP volume.paragraph). * Peirce, Charles Sanders (1981&ndash;), ''Writings of Charles S. Peirce : A Chronological Edition'', Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN. Cited as (CE volume, page). * Poizat, Bruno (2000), ''A Course in Model Theory : An Introduction to Contemporary Mathematical Logic'', Moses Klein (trans.), Springer-Verlag, New York, NY. * Quine, Willard Van Orman (1940/1981), ''Mathematical Logic'', 1940. Revised edition, Harvard University Press, Cambridge, MA, 1951. New preface, 1981. * Royce, Josiah (1961), ''The Principles of Logic'', Philosophical Library, New York, NY. * Runes, Dagobert D. (ed., 1962), ''Dictionary of Philosophy'', Littlefield, Adams, and Company, Totowa, NJ. * Shoenfield, Joseph R. (1967), ''Mathematical Logic'', Addison-Wesley, Reading, MA. * Styazhkin, N.I. (1969), ''History of Mathematical Logic from Leibniz to Peano'', MIT Press, Cambridge, MA. * Suppes, Patrick (1957/1999), ''Introduction to Logic'', 1st published 1957. Reprinted, Dover Publications, New York, NY, 1999. * Suppes, Patrick (1960/1972), ''Axiomatic Set Theory'', 1st published 1960. Reprinted, Dover Publications, New York, NY, 1972. * Tarski, Alfred (1956/1983), ''Logic, Semantics, Metamathematics : Papers from 1923 to 1938'', J.H. Woodger (trans.), Oxford University Press, 1956. 2nd edition, J. Corcoran (ed.), Hackett Publishing, Indianapolis, IN, 1983. * Ulam, Stanislaw Marcin; and Bednarek, A.R. (1977), &ldquo;On the Theory of Relational Structures and Schemata for Parallel Computation&rdquo;, pp. 477&ndash;508 in A.R. Bednarek and Françoise Ulam (eds.), ''Analogies Between Analogies : The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators'', University of California Press, Berkeley, CA, 1990. * Ulam, Stanislaw Marcin (1990), ''Analogies Between Analogies : The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators'', A.R. Bednarek and Françoise Ulam (eds.), University of California Press, Berkeley, CA. * Ullman, Jeffrey D. (1980), ''Principles of Database Systems'', Computer Science Press, Rockville, MD. * Venetus, Paulus (1472/1984), ''Logica Parva : Translation of the 1472 Edition with Introduction and Notes'', Alan R. Perreiah (trans.), Philosophia Verlag, Munich, Germany. ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Relation_theory Relation Theory @ InterSciWiki] * [http://mywikibiz.com/Relation_theory Relation Theory @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Relation_theory Relation Theory @ Subject Wikis] * [http://en.wikiversity.org/wiki/Relation_theory Relation Theory @ Wikiversity] * [http://beta.wikiversity.org/wiki/Relation_theory Relation Theory @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Relation_theory Relation Theory], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Relation_theory Relation Theory], [http://mywikibiz.com/ MyWikiBiz] * [http://planetmath.org/RelationTheory Relation Theory], [http://planetmath.org/ PlanetMath] * [http://semanticweb.org/wiki/Relation_theory Relation Theory], [http://semanticweb.org/ Semantic Web] * [http://wikinfo.org/w/index.php/Relation_theory Relation Theory], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Relation_theory Relation Theory], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Relation_theory Relation Theory], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Theory_of_relations&oldid=45042729 Relation Theory], [http://en.wikipedia.org/ Wikipedia] [[Category:Charles Sanders Peirce]] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Database Theory]] [[Category:Discrete Mathematics]] [[Category:Formal Sciences]] [[Category:Inquiry]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Relation Theory]] [[Category:Set Theory]] 104c4f8867f8179429193f4b5be14c554e26e819 0 350 765 572 2015-11-16T21:56:04Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''descriptive science''', also called a '''special science''', is a form of [[inquiry]], typically involving a community of inquiry and its accumulated body of provisional knowledge, that seeks to discover what is true about a recognized domain of phenomena. ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Descriptive_science Descriptive Science @ InterSciWiki] * [http://mywikibiz.com/Descriptive_science Descriptive Science @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Descriptive_science Descriptive Science @ Subject Wikis] * [http://en.wikiversity.org/wiki/Descriptive_science Descriptive Science @ Wikiversity] * [http://beta.wikiversity.org/wiki/Descriptive_science Descriptive Science @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Semiotic_Information Semiotic Information] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Descriptive_science Descriptive Science], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Descriptive_science Descriptive Science], [http://mywikibiz.com/ MyWikiBiz] * [http://semanticweb.org/wiki/Descriptive_science Descriptive Science], [http://semanticweb.org/ Semantic Web] * [http://wikinfo.org/w/index.php/Descriptive_science Descriptive Science], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Descriptive_science Descriptive Science], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Descriptive_science Descriptive Science], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Descriptive_science&oldid=51990248 Descriptive Science], [http://en.wikipedia.org/ Wikipedia] [[Category:Descriptive Sciences]] [[Category:Inquiry]] [[Category:Philosophy]] [[Category:Philosophy of Science]] [[Category:Science]] 293321bdd963f237d5d5c5050031505b30c015f3 0 349 766 571 2015-11-16T22:44:42Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''normative science''' is a form of [[inquiry]], typically involving a community of inquiry and its accumulated body of provisional knowledge, that seeks to discover good ways of achieving recognized aims, ends, goals, objectives, or purposes. The three '''normative sciences''', according to traditional conceptions in philosophy, are ''aesthetics'', ''ethics'', and ''logic''. ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Normative_science Normative Science @ InterSciWiki] * [http://mywikibiz.com/Normative_science Normative Science @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Normative_science Normative Science @ Subject Wikis] * [http://en.wikiversity.org/wiki/Normative_science Normative Science @ Wikiversity] * [http://beta.wikiversity.org/wiki/Normative_science Normative Science @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Semiotic_Information Semiotic Information] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Normative_science Normative Science], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Normative_science Normative Science], [http://mywikibiz.com/ MyWikiBiz] * [http://semanticweb.org/wiki/Normative_science Normative Science], [http://semanticweb.org/ Semantic Web] * [http://wikinfo.org/w/index.php/Normative_science Normative Science], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Normative_science Normative Science], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Normative_science Normative Science], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Normative_science&oldid=51993011 Normative Science], [http://en.wikipedia.org/ Wikipedia] [[Category:Inquiry]] [[Category:Normative Sciences]] [[Category:Philosophy]] [[Category:Philosophy of Science]] [[Category:Science]] b2f5348867c59a490c041c14450d4c308b258f28 0 347 767 569 2015-11-17T04:35:25Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. The '''pragmatic maxim''', also known as the ''maxim of pragmatism'' or the ''maxim of pragmaticism'', is a maxim of logic formulated by [[Charles Sanders Peirce]]. Serving as a normative recommendation or a regulative principle in the [[normative science]] of logic, its function is to guide the conduct of thought toward the achievement of its purpose, advising the addressee on an optimal way of &ldquo;attaining clearness of apprehension&rdquo;. ==Seven ways of looking at a pragmatic maxim== Peirce stated the pragmatic maxim in many different ways over the years, each of which adds its own bit of clarity or correction to their collective corpus. * The first excerpt appears in the form of a dictionary entry, intended as a definition of ''pragmatism''. <blockquote> Pragmatism. The opinion that metaphysics is to be largely cleared up by the application of the following maxim for attaining clearness of apprehension: &ldquo;Consider what effects, that might conceivably have practical bearings, we conceive the object of our conception to have. Then, our conception of these effects is the whole of our conception of the object.&rdquo; (Peirce, CP&nbsp;5.2, 1878/1902). </blockquote> * The second excerpt presents another version of the pragmatic maxim, a recommendation about a way of clarifying meaning that can be taken to stake out the general philosophy of pragmatism. <blockquote> Pragmaticism was originally enounced in the form of a maxim, as follows: Consider what effects that might ''conceivably'' have practical bearings you ''conceive'' the objects of your ''conception'' to have. Then, your ''conception'' of those effects is the whole of your ''conception'' of the object. (Peirce, CP&nbsp;5.438, 1878/1905). </blockquote> * The third excerpt puts a gloss on the meaning of a ''practical bearing'' and provides an alternative statement of the maxim. <blockquote> <p>Such reasonings and all reasonings turn upon the idea that if one exerts certain kinds of volition, one will undergo in return certain compulsory perceptions. Now this sort of consideration, namely, that certain lines of conduct will entail certain kinds of inevitable experiences is what is called a &ldquo;practical consideration&rdquo;. Hence is justified the maxim, belief in which constitutes pragmatism; namely:</p> <p>''In order to ascertain the meaning of an intellectual conception one should consider what practical consequences might conceivably result by necessity from the truth of that conception; and the sum of these consequences will constitute the entire meaning of the conception.'' (Peirce, CP&nbsp;5.9, 1905).</p> </blockquote> * The fourth excerpt illustrates one of Peirce's many attempts to get the sense of the pragmatic philosophy across by rephrasing the pragmatic maxim in an alternative way. In introducing this version, he addresses an order of prospective critics who do not deem a simple heuristic maxim, much less one that concerns itself with a routine matter of logical procedure, as forming a sufficient basis for a whole philosophy. <blockquote> <p>On their side, one of the faults that I think they might find with me is that I make pragmatism to be a mere maxim of logic instead of a sublime principle of speculative philosophy. In order to be admitted to better philosophical standing I have endeavored to put pragmatism as I understand it into the same form of a philosophical theorem. I have not succeeded any better than this:</p> <p>Pragmatism is the principle that every theoretical judgment expressible in a sentence in the indicative mood is a confused form of thought whose only meaning, if it has any, lies in its tendency to enforce a corresponding practical maxim expressible as a conditional sentence having its apodosis in the imperative mood. (Peirce, CP&nbsp;5.18, 1903).</p> </blockquote> * The fifth excerpt is useful by way of additional clarification, and was aimed to correct a variety of historical misunderstandings that arose over time with regard to the intended meaning of the pragmatic maxim. <blockquote> The doctrine appears to assume that the end of man is action &mdash; a stoical axiom which, to the present writer at the age of sixty, does not recommend itself so forcibly as it did at thirty. If it be admitted, on the contrary, that action wants an end, and that that end must be something of a general description, then the spirit of the maxim itself, which is that we must look to the upshot of our concepts in order rightly to apprehend them, would direct us towards something different from practical facts, namely, to general ideas, as the true interpreters of our thought. (Peirce, CP&nbsp;5.3, 1902). </blockquote> * A sixth excerpt is useful in stating the bearing of the pragmatic maxim on the topic of reflection, namely, that it makes all of pragmatism boil down to nothing more or less than a method of reflection. <blockquote> The study of philosophy consists, therefore, in reflexion, and ''pragmatism'' is that method of reflexion which is guided by constantly holding in view its purpose and the purpose of the ideas it analyzes, whether these ends be of the nature and uses of action or of thought. &hellip; It will be seen that ''pragmatism'' is not a ''Weltanschauung'' but is a method of reflexion having for its purpose to render ideas clear. (Peirce, CP&nbsp;5.13 note&nbsp;1, 1902). </blockquote> * The seventh excerpt is a late reflection on the reception of pragmatism. With a sense of exasperation that is almost palpable, Peirce tries to justify the maxim of pragmatism and to correct its misreadings by pinpointing a number of false impressions that the intervening years have piled on it, and he attempts once more to prescribe against the deleterious effects of these mistakes. Recalling the very conception and birth of pragmatism, he reviews its initial promise and its intended lot in the light of its subsequent vicissitudes and its apparent fate. Adopting the style of a ''post mortem'' analysis, he presents a veritable autopsy of the ways that the main idea of pragmatism, for all its practicality, can be murdered by a host of misdissecting disciplinarians, by what are ostensibly its most devoted followers. <blockquote> This employment five times over of derivates of ''concipere'' must then have had a purpose. In point of fact it had two. One was to show that I was speaking of meaning in no other sense than that of intellectual purport. The other was to avoid all danger of being understood as attempting to explain a concept by percepts, images, schemata, or by anything but concepts. I did not, therefore, mean to say that acts, which are more strictly singular than anything, could constitute the purport, or adequate proper interpretation, of any symbol. I compared action to the finale of the symphony of thought, belief being a demicadence. Nobody conceives that the few bars at the end of a musical movement are the purpose of the movement. They may be called its upshot. (Peirce, CP&nbsp;5.402 note&nbsp;3, 1906). </blockquote> ==References== * Peirce, C.S., ''Collected Papers of Charles Sanders Peirce'', vols. 1&ndash;6, Charles Hartshorne and Paul Weiss (eds.), vols. 7&ndash;8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931&ndash;1935, 1958. (Cited as CP&nbsp;''n''.''m'' for volume&nbsp;''n'', paragraph&nbsp;''m''). ==Resources== * [http://vectors.usc.edu/thoughtmesh/publish/141.php Pragmatic Maxim &rarr; ThoughtMesh] ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Pragmatic_maxim Pragmatic Maxim @ InterSciWiki] * [http://mywikibiz.com/Pragmatic_maxim Pragmatic Maxim @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Pragmatic_maxim Pragmatic Maxim @ Subject Wikis] * [http://en.wikiversity.org/wiki/Pragmatic_maxim Pragmatic Maxim @ Wikiversity] * [http://beta.wikiversity.org/wiki/Pragmatic_maxim Pragmatic Maxim @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Semiotic_Information Semiotic Information] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Pragmatic_maxim Pragmatic Maxim], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Pragmatic_maxim Pragmatic Maxim], [http://mywikibiz.com/ MyWikiBiz] * [http://web.archive.org/web/20050427095007/http://suo.ieee.org/ontology/msg05407.html Pragmatic Maxim], [http://web.archive.org/web/20131214093401/http://suo.ieee.org/ontology/thrd1.html Ontology List] * [http://semanticweb.org/wiki/Pragmatic_maxim Pragmatic Maxim], [http://semanticweb.org/ Semantic Web] * [http://vectors.usc.edu/thoughtmesh/publish/141.php Pragmatic Maxim], [http://vectors.usc.edu/thoughtmesh/ ThoughtMesh] * [http://wikinfo.org/w/index.php/Pragmatic_maxim Pragmatic Maxim], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Pragmatic_maxim Pragmatic Maxim], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Pragmatic_maxim Pragmatic Maxim], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Pragmatic_maxim&oldid=45528828 Pragmatic Maxim], [http://en.wikipedia.org/ Wikipedia] [[Category:Artificial Intelligence]] [[Category:Charles Sanders Peirce]] [[Category:Critical Thinking]] [[Category:Cybernetics]] [[Category:Education]] [[Category:Hermeneutics]] [[Category:Information Systems]] [[Category:Inquiry]] [[Category:Intelligence Amplification]] [[Category:Learning Organizations]] [[Category:Knowledge Representation]] [[Category:Linguistics]] [[Category:Logic]] [[Category:Mathematics]] [[Category:Normative Sciences]] [[Category:Philosophy]] [[Category:Pragmatics]] [[Category:Pragmatism]] [[Category:Semantics]] [[Category:Semiotics]] [[Category:Systems Science]] 3c9e3ea700d0487cdbe8f0eee28a141a52540a2f 0 348 768 570 2015-11-17T14:50:28Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''truth theory''' or a '''theory of truth''' is a conceptual framework that underlies a particular conception of truth, such as those used in art, ethics, logic, mathematics, philosophy, the sciences, or any discussion that either mentions or makes use of a notion of truth. A truth theory can be anything from an informal theory, based on implicit or tacit ideas, to a formal theory, constructed from explicit axioms and definitions and developed by means of definite rules of inference. The scope of a truth theory can be restricted to tightly-controlled and well-bounded universes of discourse or its horizon may extend to the limits of the human imagination. ==Truth in perspective== Notions of truth are notoriously difficult to disentangle from many of our most basic concepts &mdash; meaning, reality, and values in general, to mention just a few. The subjects of meaning and truth are commonly treated together, the idea being that a thing must be meaningful before it can be true or false. This association is found in ancient times, and has become standard in modern times under the heading of ''semantics'', especially ''formal semantics'' and ''model theory''. Another association of longstanding interest is the relation between truth and ''logical validity'', "because the fundamental notion of logic is validity and this is definable in terms of truth and falsehood" (Kneale and Kneale, 16). Though not the main subjects of this article, meaning and validity are truth's neighbors, and incidental inquiries of them can serve to cast light on truth's character. Beyond this minor note of accord, hardly universal, suggesting that meaning is necessary to truth, reflectors on the idea of truth just as quickly disperse into schools of thought that barely comprehend each other's thinking. A few of the more notable points of departure are these: # One of the first partings of the ways occurs at the watershed between literal and symbolic meanings, leading to a corresponding division in truths. People often speak of truth in art, truth in drama, truth in fiction, human truth, moral, religious, and spiritual truth, along with the difference between truth in principle and truth in practice. These topics demand a perspective on meaning, reality, and truth that looks beyond the bounds of literal truth and the branches of philosophy that are limited to it. # Merely resolving that meaning precedes truth, logically speaking, only brings up a host of new questions, since the meaning of the word ''meaning'' is notoriously hard to pin down. There are just to start at least two different dimensions of meaning that are commonly recognized, namely, ''connotative meaning'' and ''denotative meaning''. In one classical formulation, truth is defined as the good of [[logic]], where logic is treated as a [[normative science]], that is, an [[inquiry]] into a ''good'' or a ''value'' that seeks knowledge of it and the means to achieve it. In this scheme of ideas, truth is the positive quality of a sign that indicates the right course of action for reaching a value that we value for its own sake. As such, truth takes its place among justice and beauty, whose normative sciences are ethics and aesthetics, respectively. Viewed in this light, it is pointless to discuss truth in isolation from a frame of reference that encompasses the topics of inquiry, knowledge, logic, meaning, practice, and value, all very broadly conceived. ==Historical overview== In an ancient fragment of text called the ''Dissoi Logoi'', a writer is evidently trying to prove the impossibility of speaking consistently about truth and falsehood. One of the conundrums put forward to confound the reader cites the case of the verbal form, "I am an initiate", which is true when ''A'' says it but false when ''B'' says it. Escape from befuddlement seems easy enough if one observes that it is not the verbal expression, the sentence, to which the predicates of truth and falsity apply but what the sentence expresses, the proposition that it states. (Cf. Kneale and Kneale, 16). This same tension between strings of characters and their meanings remains with us to this day. In his early work &#928;&#949;&#961;&#953; &#917;&#961;&#956;&#951;&#957;&#949;&#953;&#945;s (''Peri Hermeneias'' or ''On Interpretation'') Aristotle strikes a chord that not only sets the key for a number of philosophical movements down through the ages but supplies the initial motif for many themes in the logic of meaning and truth that are still undergoing active development in our time. <blockquote> Words spoken (phoné) are symbols or signs (symbola) of affections or impressions (pathemata) of the soul (psyche); written words (graphomena) are the signs of words spoken. As writing, so also is speech not the same for all races of men. But the mental affections themselves, of which these words are primarily signs (semeia), are the same for the whole of mankind, as are also the objects (pragmata) of which those affections are representations or likenesses, images, copies (homoiomata). (Aristotle, ''On Interpretation'', 1.16^a4). </blockquote> Some of the points to be noted in this passage are these: # Aristotle employs a distinction in Greek that is drawn between natural or physical signs (semeia) and artificial or cultural signs (symbola). # The passage mentions three principal domains of elements, namely, the ''objects'' (pragmata), the ''signs'' (semeia, symbola), and the psychological elements (pathemata). The last domain extends over the full range of a human being's affective and cognitive experiences, for brevity summed up as ''ideas'' and ''impressions'', where these words are taken in their broadest conceivable senses. # This means that the phenomena under investigation have to do with the types of [[three-place relation]]s that conceivably exist among three domains of this sort. As a general rule, three-place relations can be very complex, and a commonly-tried strategy for approaching their complexity is to consider the [[two-place relation]]s that are left when the presence of a selected domain is simply ignored. # There are two types of two-place relation on the face of the overall three-place relation that Aristotle takes the trouble to mention, namely these:<p>Sign <math>\longrightarrow</math> Idea. Words spoken are signs or symbols of pathemata.</p><p>Idea <math>\longrightarrow</math> Object. Pathemata are icons (homoiomata) of pragmata.</p> # More incidentally, but still bearing heavily on many later discussions, Aristotle holds that the relation between writing and speech is analogous to the relation between speech and the realm of experiences, feelings, and thoughts.<p>Writing <math>\longrightarrow</math> Speech. Written words are symbols of spoken words.</p><p> Speech <math>\longrightarrow</math> Ideation. Spoken words are symbols of impressions.</p> ==Elements of theory== It is customary in philosophy to refer to a distinctive treatment of a particular subject matter, frequently summed up in a succinctly stated thesis, as a ''theory'', whether or not it qualifies as a theory by strict empirical or logical standards. When there is any risk of confusion, an informal thesis of this kind may be referred to as an ''account'', a ''perspective'', a ''treatment'', or so on, reserving the term ''theory'' for the type of [[formal system]] that serves in logic and science. Theories of truth can be classified according to the following features: * Primary subjects. What kinds of things are potentially meaningful enough to be asserted or not, believed or not, or considered true or false? * Relevant objects. What kinds of things, in addition to primary subjects, are pertinent to deciding whether to assert them or not, believe them or not, or consider them true or false? * Value predicates. What kinds of things are legitimate to say about primary subjects, either in themselves, or in relation to relevant objects? In some discussions of meaning and truth that consider forms of expression well beyond the limits of literally-interpreted linguistic forms, potentially meaningful elements are called ''[[representation]]s'', or ''[[sign (semiotics)|signs]]'' for short, taking these words in the broadest conceivable senses. Most treatments of truth draw an important distinction at this point, though the language in which they draw it may vary. On the one hand there is a type of incomplete sign that is nevertheless said to be true or false of various objects. For example, in logic there are ''terms'' such as "man" or "woman" that are true of some things and false of others, and there are ''predicates'' such as "__is a man" or "__is a woman" that are true or false in the same way. On the other hand there is a type of complete sign that expresses what grammarians traditionally call a ''complete thought''. Here one speaks of ''sentences'' and ''propositions''. Some considerations of truth admit both types of signs, ''terms'' and ''sentences'', while others admit only the bearers of complete thoughts into the arena of judgment. In a number of recent discussions that focus on linguistic analysis, the vehicles of complete thoughts are described as ''truthbearers'', with no intention of prejudging whether they bear truth or falsehood. The things that can be said about any of these representations, signs, or truthbearers are expressed in what most truth theorists describe as ''truth predicates''. Most inquiries into the character of truth begin with a notion of an informative, meaningful, or significant element, the truth of whose information, meaning, or significance may be put into question and needs to be evaluated. Depending on the context, this element might be called an ''artefact'', ''expression'', ''image'', ''impression'', ''lyric'', ''mark'', ''performance'', ''picture'', ''sentence'', ''sign'', ''string'', ''symbol'', ''text'', ''thought'', ''token'', ''utterance'', ''word'', ''work'', and so on. For the sake of brevity, it is convenient to use the term ''sign'' for any one of these elements. Whatever the case, one has the task of judging whether the bearers of information, meaning, or significance are indeed ''truth-bearers''. This judgment is typically expressed in the form of a specific ''truth predicate'', whose positive application to a sign asserts that the sign is true. Considered within the broadest horizon, there is little reason to imagine that the process of judging a ''work'', that leads to a predication of false or true, is necessarily amenable to formalization, and it may always remain what is commonly called a ''judgment call''. But there are indeed many well-circumscribed domains where it is useful to consider disciplined forms of evaluation, and the observation of these limits allows for the institution of what is called a ''[[method]]'' of judging truth and falsity. One of the first questions that can be asked in this setting is about the relationship between the significant performance and its reflective critique. If one expresses oneself in a particular fashion, and someone says "that's true", is there anything useful at all that can be said in general terms about the relationship between these two acts? For instance, does the critique add value to the expression criticized, does it say something significant in its own right, or is it just an insubstantial echo of the original sign? Theories of truth may be described according to several dimensions of description that affect the character of the predicate "true". The truth predicates that are used in different theories may be classified by the number of things that have to be mentioned in order to assess the truth of a sign, counting the sign itself as the first thing. In formal logic, this number is called the ''[[arity]]'' of the predicate. The kinds of truth predicates may then be subdivided according to any number of more specific characters that various theorists recognize as important. # A ''monadic'' truth predicate is one that applies to its main subject ? typically a concrete representation or its abstract content ? independently of reference to anything else. In this case one can say that a truth bearer is true in and of itself. # A ''dyadic'' truth predicate is one that applies to its main subject only in reference to something else, a second subject. Most commonly, the auxiliary subject is either an ''object'', an ''interpreter'', or a ''language'' to which the representation bears some [[relation (mathematics)|relation]]. # A ''triadic'' truth predicate is one that applies to its main subject only in reference to a second and a third subject. For example, in a pragmatic theory of truth, one has to specify both the object of the sign, and either its interpreter or another sign called the ''interpretant'' before one can say that the sign is true ''of'' its object ''to'' its interpreting agent or sign. Several qualifications must be kept in mind with respect to any such radically simple scheme of classification, as real practice seldom presents any pure types, and there are settings in which it is useful to speak of a theory of truth that is "almost" ''k''-adic, or that "would be" ''k''-adic if certain details can be abstracted away and neglected in a particular context of discussion. That said, given the generic division of truth predicates according to their arity, further species can be differentiated within each genus according to a number of more refined features. The truth predicate of interest in a typical [[correspondence theory of truth]] tells of a relation between representations and objective states of affairs, and is therefore expressed, for the most part, by a dyadic predicate. In general terms, one says that a representation is ''true of'' an objective situation, more briefly, that a sign is true of an object. The nature of the correspondence may vary from theory to theory in this family. The correspondence can be fairly arbitrary or it can take on the character of an ''[[analogy]]'', an ''[[icon]]'', or a ''[[morphism]]'', whereby a representation is rendered true of its object by the existence of corresponding elements and a similar structure. ===Signs=== In some branches of philosophy and fields of science the domain of potentially meaningful entities may include almost any kind of informative or significant element. The generic terms ''sign'' or ''representation'' suffice for these, with the qualification that the terms are used equivocally up and down a full spectrum from the more abstract ''types'' to the more concrete ''tokens'' that are associated with each other. More specifically, the linguistic turn in analytic philosophy begins with a focus on the syntactic character of the ''sentence'', from which is abstracted its meaningful content, referred to as the corresponding ''proposition''. A proposition is the content expressed by a sentence, held in a belief, or affirmed in an assertion or judgment. ''Truthbearer'' is used by a number of writers to refer to any entity that can be judged true or false. The term ''truthbearer'' may be applied to propositions, sentences, statements, ideas, beliefs, and judgments. Some writers exclude one or more of these categories, or argue that some of them are true (or false) only in a derivative sense. Other writers may add additional entities to the list. Truthbearers typically have two possible values, true or false. Fictional forms of expression are usually regarded as false if interpreted literally, but may be said to bear a species of truth if interpreted suitably. Still other truthbearers may be judged true or false to a greater or lesser degree. ===Higher order signs=== As ''predicate terms'', most discussions of truth allow for a number of phrases that are used to say in what ways signs or sentences or their abstract senses are regarded as true, either by themselves or in relation to other things. Theorists who admit the term call these phrases ''[[truth predicate]]s''. A truth predicate that is used to ascribe truth to something, in and of itself, in effect treating truth as an [[intrinsic property (philosophy)|intrinsic property]] of the thing, is called a ''one-place'' or ''monadic'' truth predicate. Other forms of truth predicates may be used to say that something is true in relation to specified numbers and types of other things. These are called ''many-place'' or ''polyadic'' truth predicates. In ordinary parlance, the things that one says about a subject are expressed in predicates. If one says that a sentence is true, then one is predicating truth of that sentence. Is this the same thing as asserting the sentence? This question serves as useful touchstone for sorting out some of the theories of truth. ===Propositional attitudes=== <blockquote> What sort of name shall we give to verbs like 'believe' and 'wish' and so forth? I should be inclined to call them 'propositional verbs'. This is merely a suggested name for convenience, because they are verbs which have the ''form'' of relating an object to a proposition. As I have been explaining, that is not what they really do, but it is convenient to call them propositional verbs. Of course you might call them 'attitudes', but I should not like that because it is a psychological term, and although all the instances in our experience are psychological, there is no reason to suppose that all the verbs I am talking of are psychological. There is never any reason to suppose that sort of thing. (Russell 1918, 227). </blockquote> What a proposition is, is one thing. How we feel about it, or how we regard it, is another. We can accept it, assert it, believe it, command it, contest it, declare it, deny it, doubt it, enjoin it, exclaim it, expect it, imagine it, intend it, know it, observe it, prove it, question it, suggest it, or wish it were so. Different attitudes toward propositions are called ''propositional attitudes'', and they are also discussed under the headings of ''intentionality'' and ''linguistic modality''. The formal properties of verbs like ''assert'', ''believe'', ''command'', ''consider'', ''deny'', ''doubt'', ''hunt'', ''imagine'', ''judge'', ''know'', ''want'', ''wish'', and a host of others, are studied under these headings by linguists and logicians alike. Many problematic situations in real life arise from the circumstance that many different propositions in many different modalities are in the air at once. In order to compare propositions of different colors and flavors, as it were, we have no basis for comparison but to examine the underlying propositions themselves. Thus we are brought back to matters of language and logic. Despite the name, propositional attitudes are not regarded as psychological attitudes proper, since the formal disciplines of linguistics and logic are concerned with nothing more concrete than what can be said in general about their formal properties and their patterns of interaction. The variety of attitudes that a proposer can bear toward a single proposition is a critical factor in evaluating its truth. One topic of central concern is the relation between the modalities of assertion and belief, especially when viewed in the light of the proposer's intentions. For example, we frequently find ourselves faced with the question of whether a person's assertions conform to his or her beliefs. Discrepancies here can occur for many reasons, but when the departure of assertion from belief is intentional, we usually call that a ''lie''. Other comparisons of multiple modalities that frequently arise are the relationships between belief and knowledge and the discrepancies that occur among observations, expectations, and intentions. Deviations of observations from expectations are commonly perceived as ''surprises'', phenomena that call for ''explanations'' to reduce the shock of amazement. Deviations of observations from intentions are commonly experienced as ''problems'', situations that call for plans of action to reduce the drive of dissatisfaction. Either type of discrepancy forms an impulse to ''[[inquiry]]'' (Awbrey and Awbrey 1995). ===Reflection and quotation=== The study of propositional attitudes is no sooner begun than it leads to the all-important philosophical distinction between (1) using a meaning-bearer to bear its meaning in an active manner and (2) mentioning a meaning-bearer in a form that keeps its meaning in a more inert or inhibited state. The reasons for doing the latter are various, but involve the need to reflect on a potential meaning, to compare and contrast it with others, to criticize and evaluate both its logical implications and its practical consequences, all before deciding whether to put its meaning into action or not. The word &ldquo;quote&rdquo; derives from the Latin verb ''quotare'', which refers to the practice of numbering references and referring to pieces of text by marking their numbers. There is a certain aesthetic distance involved in this practice, and it leads, if only for moments at a time, to viewing each piece of text as a string of characters that bears its own litter of meanings, but meanings to be reflected on and critically compared with others, both in and out of their litter. It is hardly an accident, then, that matters of Gödel numbers, quotation, and reflection are bound up with each other in mathematical logic and computation theory. ==Varieties of truth theory== ===Nominal truth theories=== A ''nominal truth'' theory is defined by the axiom that the concept ''truth'' is a mere name. In traditional systems of logic, a concept is always a symbol, specifically, a mental symbol, and so the word ''mere'' in the nominal axiom says that ''truth'' is nothing more than a symbol. One of the aims of nominal philosophies, generally speaking, is to clear away the conceptual clutter of excess metaphysical ideas through the searching examination of their verbal formulations. Thus the question arises whether ''truth'' is one of the essentials or one of the excesses of rational thought. One method of critical analysis that is commonly brought to bear at this juncture is based on the nominal corollary that if one can do without the word in every linguistic context, then one can do without the concept, which is after all nothing but the word. ===Real truth theories=== ===Formal truth theories=== There is a generally acknowledged distinction between merely contemplating or entertaining a proposition, and actually asserting or believing it. This does not mean that there is general agreement as to the precise nature of the distinction. Although there are many ways of talking about the distinction, words alone do not guarantee clarity, and they often lead to the problem of having to decide which descriptions say the same thing and which say something different. For example, formal logic provides symbolic operators for indicating the assertion of a sentence, or the assertion of the proposition that comes from interpreting the sentence relative to a particular context of discussion. Another way of saying something about a sentence or the corresponding proposition is by means of various semantic predicates, including truth predicates as a special case. This raises the question of how these operators and predicates are related to one another. As noted before, one of the first questions of this sort is whether asserting a proposition amounts to the same thing as predicating truth of that proposition. ===Semantic relations=== A ''denotation relation'', or a ''name relation'', is a [[relation (mathematics)|relation]] between symbols (formulas, words, phrases) and the things that they are interpreted as denoting or naming in a particular context of discussion (Church 1962). The things denoted, which may be quite literally anything that can be talked about or thought about, are called the ''objects'' of denotation. Different theories of meaning vary in their use of denotation relations and the properties that they require of them. The following are two criteria that serve to distinguish particular theories of denotation: # How many things can a symbol denote? For instance, can a symbol denote more than one thing, or must a symbol always denote at most one thing? # Is denoting the same sort of relation as ''being true of'', and thus a state of affairs that can be described by a particular type of truth predicate, or is denoting a very different sort of relation than that? ==Truth and the conduct of life== <blockquote> Again, in a ship, if a man were at liberty to do what he chose, but were devoid of mind and excellence in navigation ([[arete (excellence)|&#945;&#961;&#949;&#964;&#951;s]] &#954;&#965;&#946;&#949;&#961;&#957;&#951;&#964;&#953;&#954;&#951;s), do you perceive what must happen to him and his fellow sailors? (Plato, ''Alcibiades'', 135A). </blockquote> ==References== * Aristotle, &ldquo;On Interpretation&rdquo;, Harold P. Cooke (trans.), pp. 111&ndash;179 in ''Aristotle, Volume&nbsp;1'', Loeb Classical Library, William Heinemann, London, UK, 1938. * Awbrey, J.L., and Awbrey, S.M. (Autumn 1995), &ldquo;Interpretation as Action : The Risk of Inquiry&rdquo;, ''Inquiry : Critical Thinking Across the Disciplines'' 15(1), pp. 40&ndash;52. [https://web.archive.org/web/19970626071826/http://chss.montclair.edu/inquiry/fall95/awbrey.html Archive]. [https://www.pdcnet.org/inquiryct/content/inquiryct_1995_0015_0001_0040_0052 Journal]. [https://www.academia.edu/1266493/Interpretation_as_Action_The_Risk_of_Inquiry Online]. * Blackburn, S. (1996), ''The Oxford Dictionary of Philosophy'', Oxford University Press, Oxford, UK, 1994. Paperback edition with new Chronology, 1996. * Blackburn, S., and Simmons, K. (eds., 1999), ''Truth'', Oxford University Press, Oxford, UK. * Church, A. (1962a), &ldquo;Name Relation, or Meaning Relation&rdquo;, p. 204 in Dagobert D. Runes (ed.), ''Dictionary of Philosophy'', Littlefield, Adams, and Company, Totowa, NJ. * Church, A. (1962b), &ldquo;Truth, Semantical&rdquo;, p. 322 in Dagobert D. Runes (ed.), ''Dictionary of Philosophy'', Littlefield, Adams, and Company, Totowa, NJ. * Kneale, W., and Kneale, M. (1962), ''The Development of Logic'', Oxford University Press, London, UK, 1962. Reprinted with corrections, 1975. * Plato, &ldquo;Alcibiades&nbsp;1&rdquo;, W.R.M. Lamb (trans.), pp. 93&ndash;223 in ''Plato, Volume 12'', Loeb Classical Library, William Heinemann, London, UK, 1927. * Russell, B. (1918), &ldquo;The Philosophy of Logical Atomism&rdquo;, ''The Monist'', 1918. Reprinted, pp. 177&ndash;281 in ''Logic and Knowledge: Essays 1901&ndash;1950'', Robert Charles Marsh (ed.), Unwin Hyman, London, UK, 1956. Reprinted, pp. 35&ndash;155 in ''The Philosophy of Logical Atomism'', David Pears (ed.), Open Court, La&nbsp;Salle, IL, 1985. ==Further reading== * Beaney, M. (ed., 1997), ''The Frege Reader'', Blackwell Publishers, Oxford, UK. * Dewey, J. (1900&ndash;1901), ''Lectures on Ethics 1900?1901'', Donald F. Koch (ed.), Southern Illinois University Press, Carbondale and Edwardsville, IL, 1991. * Dewey, John (1932), ''Theory of the Moral Life'', Part 2 of John Dewey and James H. Tufts, ''Ethics'', Henry Holt and Company, New York, NY, 1908. 2nd edition, Holt, Rinehart, and Winston, 1932. Reprinted, Arnold Isenberg (ed.), Victor Kestenbaum (Preface), Irvington Publishers, New&nbsp;York, NY, 1980. * Dummett, M. (1991), ''Frege and Other Philosophers'', Oxford University Press, Oxford, UK. * Dummett, M. (1993), ''Origins of Analytical Philosophy'', Harvard University Press, Cambridge, MA. * Foucault, M. (1997), ''Essential Works of Foucault, 1954&ndash;1984, Volume&nbsp;1, Ethics : Subjectivity and Truth'', Paul Rabinow (ed.), Robert Hurley et al. (trans.), The New Press, New&nbsp;York, NY. * Gadamer, H.-G. (1986), ''The Idea of the Good in Platonic&ndash;Aristotelian Philosophy'', P. Christopher Smith (trans.), Yale University Press, New Haven, CT. 1st published, ''Die Idee des Guten zwischen Plato und Aristoteles'', J.C.B. Mohr, Heidelberg, Germany, 1978. * Grover, Dorothy (1992), ''A Prosentential Theory of Truth'', Princeton University Press, Princeton, NJ. * Habermas, J. (1979), ''Communication and the Evolution of Society'', Thomas McCarthy (trans.), Beacon Press, Boston, MA. * Habermas, J. (1990), ''Moral Consciousness and Communicative Action'', Christian Lenhardt and Shierry Weber Nicholsen (trans.), Thomas McCarthy (intro.), MIT Press, Cambridge, MA. * Habermas, J. (2003), ''Truth and Justification'', Barbara Fultner (trans.), MIT Press, Cambridge, MA. * Kirkham, R.L. (1992), ''Theories of Truth'', MIT Press, Cambridge, MA. * Kripke, S.A. (1975), &ldquo;An Outline of a Theory of Truth&rdquo;, ''Journal of Philosophy'' 72 (1975), 690?716. * Kripke, S.A. (1980), ''Naming and Necessity'', Harvard University Press, Cambridge, MA. * Lewis, C.I. (1946), ''An Analysis of Knowledge and Valuation'', The Paul Carus Lectures, Series&nbsp;8, Open Court, La Salle, IL. * Linsky, L. (ed., 1971), ''Reference and Modality'', Oxford University Press, London, UK. * Martin, R.L. (ed., 1984), ''Recent Essays on Truth and the Liar Paradox'', Oxford University Press, Oxford, UK. * Moody, E.A. (1953), ''Truth and Consequence in Mediaeval Logic'', North-Holland, Amsterdam, Netherlands, 1953. Reprinted, Greenwood Press, Westport, CT, 1976. * Nietzsche, Friedrich (1873/1968). &ldquo;Uber Wahrheit und Lüge im aussermoralischen Sinn&rdquo;, (&ldquo;On Truth and Lying in an Extra-moral Sense&rdquo;), in Jürgen Habermas (ed.), ''Erkenntnistheoretische Schriften'', Suhrkamp, Frankfurt, Germany. * Putnam, Hilary (1981), ''Reason, Truth, and History'', Cambridge University Press, Cambridge, UK. * Quine, W.V. (1982), ''Methods of Logic'', (1st ed. 1950), (2nd ed. 1959), (3rd ed. 1972), 4th edition, Harvard University Press, Cambridge, MA. * Quine, W.V. (1992), ''Pursuit of Truth'', Harvard University Press, Cambridge, MA, 1990. Revised edition, Harvard University Press, Cambridge, MA, 1992. * Quine, W.V., and Ullian, J.S. (1978), ''The Web of Belief'', Random House, New York, NY, 1970. 2nd edition, Random House, New York, NY, 1978. * Rawls, J. (2000), ''Lectures on the History of Moral Philosophy'', Barbara Herman (ed.), Harvard University Press, Cambridge, MA. * Rescher, N. (1973), ''The Coherence Theory of Truth'', Oxford University Press, Oxford, UK. * Rorty, R. (1991), ''Objectivity, Relativism, and Truth : Philosophical Papers, Volume 1'', Cambridge University Press, Cambridge, UK. * Russell, B. (1913), ''Theory of Knowledge (The 1913 Manuscript)'', Elizabeth Ramsden Eames (ed.), Kenneth Blackwell (collab.), George Allen & Unwin, 1984. Reprinted, Routledge, London, UK, 1992. * Russell, B. (1940), ''An Inquiry into Meaning and Truth'', 'The William James Lectures for 1940 Delivered at Harvard University', George Allen & Unwin, 1950. Reprinted, Thomas Baldwin (intro.), Routledge, London, UK, 1992. * Salmon, N., and [[Scott Soames|Soames, Scott]] (eds., 1988), ''Propositions and Attitudes'', Oxford University Press, Oxford, UK. * Smart, N. (1969), ''The Religious Experience of Mankind'', Charles Scribner's Sons, New York, NY. * Tarski, A. (1944), &ldquo;The Semantic Conception of Truth and the Foundations of Semantics&rdquo;, ''Philosophy and Phenomenological Research'' 4 (3), 341&ndash;376. * Wallace, A.F.C.]] (1966), ''Religion, An Anthropological View'', Random House, New York, NY. * Williams, B. (2002), ''Truth and Truthfulness: An Essay in Genealogy'', Princeton University Press, Princeton, NJ. ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Truth_theory Truth Theory @ InterSciWiki] * [http://mywikibiz.com/Truth_theory Truth Theory @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Truth_theory Truth Theory @ Subject Wikis] * [http://en.wikiversity.org/wiki/Truth_theory Truth Theory @ Wikiversity] * [http://beta.wikiversity.org/wiki/Truth_theory Truth Theory @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Semiotic_Information Semiotic Information] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Truth_theory Truth Theory], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Truth_theory Truth Theory], [http://mywikibiz.com/ MyWikiBiz] * [http://semanticweb.org/wiki/Truth_theory Truth Theory], [http://semanticweb.org/ Semantic Web] * [http://ref.subwiki.org/wiki/Truth_theory Truth Theory], [http://ref.subwiki.org/ Subject Wikis] * [http://wikinfo.org/w/index.php/Truth_theory Truth Theory], [http://wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Truth_theory Truth Theory], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Truth_theory Truth Theory], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://web.archive.org/web/20060913000000/http://en.wikipedia.org/wiki/Truth_Theory Truth Theory], [http://en.wikipedia.org/ Wikipedia] This article contains material from an earlier version of the former Wikipedia article, [http://web.archive.org/web/20060913000000/http://en.wikipedia.org/wiki/Truth_Theory Truth Theory], no longer extant. The Wikipedia article was deleted and its edit history destroyed by Wikipedia administrators, in violation of the GNU Free Documentation License. A record of the Wikipedia AFD (Article For Deletion) proceedings can be found at the following locations: {{col-begin}} {{col-break}} * [http://en.wikipedia.org/w/index.php?title=Wikipedia:Articles_for_deletion/Truth_theory&oldid=54630517 1st AFD proceeding] {{col-break}} * [http://en.wikipedia.org/w/index.php?title=Wikipedia:Articles_for_deletion/Truth_theory(2)&oldid=71295142 2nd AFD proceeding] {{col-break}} * [http://en.wikipedia.org/w/index.php?title=Wikipedia:Articles_for_deletion/Truth_theory_(3rd_nomination)&oldid=81774680 3rd AFD proceeding] {{col-end}} [[Category:Artificial Intelligence]] [[Category:Charles Sanders Peirce]] [[Category:Critical Thinking]] [[Category:Cybernetics]] [[Category:Education]] [[Category:Hermeneutics]] [[Category:Information Systems]] [[Category:Inquiry]] [[Category:Intelligence Amplification]] [[Category:Learning Organizations]] [[Category:Knowledge Representation]] [[Category:Logic]] [[Category:Philosophy]] [[Category:Pragmatics]] [[Category:Pragmatism]] [[Category:Semantics]] [[Category:Semiotics]] [[Category:Systems Science]] 9ee870734deed46b0c7cbc9bec05efee768b1b15 0 180 769 703 2015-11-17T15:28:52Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page serves as a '''focal node''' for a collection of related resources. ==Participants== Interested parties may add their names on [[Inquiry Live/Participants|this page]]. ==Rudiments of organization== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] '''Focal nodes''' serve as hubs for collections of related resources, in particular, activity sites and article contents. ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Live Inquiry Live @ InterSciWiki] * [http://mywikibiz.com/Inquiry_Live Inquiry Live @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Inquiry_Live Inquiry Live @ Subject Wikis] * [http://en.wikiversity.org/wiki/Inquiry_Live Inquiry Live @ Wikiversity] * [http://beta.wikiversity.org/wiki/Inquiry_Live Inquiry Live @ Wikiversity Beta] '''Peer nodes''' are roughly parallel pages on different sites that are not necessarily identical in content &mdash; especially as they develop in time across different environments through interaction with diverse populations &mdash; but they should preserve enough information to reproduce each other, more or less, in case of damage or loss. [[Category:Inquiry]] [[Category:Logic]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] fb18b6a41f107c7e9b147cdb5a878f94038f48e2 783 769 2020-09-17T13:15:57Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page serves as a '''focal node''' for a collection of related resources. ==Participants== Interested parties may add their names on [[Inquiry Live/Participants|this page]]. ==Rudiments of organization== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] '''Focal nodes''' serve as hubs for collections of related resources, in particular, activity sites and article contents. ===Peer nodes=== * [http://mywikibiz.com/Inquiry_Live Inquiry Live @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Inquiry_Live Inquiry Live @ Subject Wikis] * [http://en.wikiversity.org/wiki/Inquiry_Live Inquiry Live @ Wikiversity] * [http://beta.wikiversity.org/wiki/Inquiry_Live Inquiry Live @ Wikiversity Beta] '''Peer nodes''' are roughly parallel pages on different sites that are not necessarily identical in content &mdash; especially as they develop in time across different environments through interaction with diverse populations &mdash; but they should preserve enough information to reproduce each other, more or less, in case of damage or loss. ===Archive=== * [http://web.archive.org/web/20191024043046/http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Live Inquiry Live @ InterSciWiki] [[Category:Inquiry]] [[Category:Logic]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] f259791faddc3358f6f48977098a4c274f1210e9 0 181 770 701 2015-11-17T15:35:19Z Jon Awbrey 3 wikitext text/x-wiki <font size="3">&#9758;</font> This page serves as a '''focal node''' for a collection of related resources. ==Participants== * Interested parties may add their names on [[Logic Live/Participants|this page]]. ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Logic_Live Logic Live @ InterSciWiki] * [http://mywikibiz.com/Logic_Live Logic Live @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Logic_Live Logic Live @ Subject Wikis] * [http://en.wikiversity.org/wiki/Logic_Live Logic Live @ Wikiversity] * [http://beta.wikiversity.org/wiki/Logic_Live Logic Live @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} [[Category:Inquiry]] [[Category:Logic]] [[Category:Open Educational Resource]] [[Category:Peer Educational Resource]] cff2a0e84e38521f42861f3853abf204cda5cd9f 0 301 771 728 2015-11-18T16:14:14Z Jon Awbrey 3 update wikitext text/x-wiki <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. '''Peirce's law''' is a formula in [[propositional calculus]] that is commonly expressed in the following form: {| align="center" cellpadding="10" | <math>((p \Rightarrow q) \Rightarrow p) \Rightarrow p</math> |} Peirce's law holds in classical propositional calculus, but not in intuitionistic propositional calculus. The precise axiom system that one chooses for classical propositional calculus determines whether Peirce's law is taken as an axiom or proven as a theorem. ==History== Here is Peirce's own statement and proof of the law: {| align="center" cellpadding="8" width="90%" | <p>A ''fifth icon'' is required for the principle of excluded middle and other propositions connected with it. One of the simplest formulae of this kind is:</p> <center> <p><math>\{ (x \,-\!\!\!< y) \,-\!\!\!< x \} \,-\!\!\!< x.</math></p> </center> <p>This is hardly axiomatical. That it is true appears as follows. It can only be false by the final consequent <math>x\!</math> being false while its antecedent <math>(x \,-\!\!\!< y) \,-\!\!\!< x</math> is true. If this is true, either its consequent, <math>x,\!</math> is true, when the whole formula would be true, or its antecedent <math>x \,-\!\!\!< y</math> is false. But in the last case the antecedent of <math>x \,-\!\!\!< y,</math> that is <math>x,\!</math> must be true. (Peirce, CP&nbsp;3.384).</p> |} Peirce goes on to point out an immediate application of the law: {| align="center" cellpadding="8" width="90%" | <p>From the formula just given, we at once get:</p> <center> <p><math>\{ (x \,-\!\!\!< y) \,-\!\!\!< a \} \,-\!\!\!< x,</math></p> </center> <p>where the <math>a\!</math> is used in such a sense that <math>(x \,-\!\!\!< y) \,-\!\!\!< a</math> means that from <math>(x \,-\!\!\!< y)</math> every proposition follows. With that understanding, the formula states the principle of excluded middle, that from the falsity of the denial of <math>x\!</math> follows the truth of <math>x.\!</math> (Peirce, CP&nbsp;3.384).</p> |} '''Note.''' Peirce uses the ''sign of illation'' “<math>-\!\!\!<</math>” for implication. In one place he explains “<math>-\!\!\!<</math>” as a variant of the sign “<math>\le</math>” for ''less than or equal to''; in another place he suggests that <math>A \,-\!\!\!< B</math> is an iconic way of representing a state of affairs where <math>A,\!</math> in every way that it can be, is <math>B.\!</math> ==Graphical proof== Under the existential interpretation of Peirce's [[logical graphs]], Peirce's law is represented by means of the following formal equivalence or logical equation. {| align="center" border="0" cellpadding="10" cellspacing="0" | [[Image:Peirce's Law 1.0 Splash Page.png|500px]] || (1) |} '''Proof.''' Using the axiom set given in the entry for [[logical graphs]], Peirce's law may be proved in the following manner. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Peirce's Law 1.0 Marquee Title.png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 1.png|500px]] |- | [[Image:Equational Inference Band Collect p.png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 2.png|500px]] |- | [[Image:Equational Inference Band Quit ((q)).png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 3.png|500px]] |- | [[Image:Equational Inference Band Cancel (( )).png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 4.png|500px]] |- | [[Image:Equational Inference Band Delete p.png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 5.png|500px]] |- | [[Image:Equational Inference Band Cancel (( )).png|500px]] |- | [[Image:Peirce's Law 1.0 Storyboard 6.png|500px]] |- | [[Image:Equational Inference Marquee QED.png|500px]] |} | (2) |} The following animation replays the steps of the proof. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Peirce's Law 2.0 Animation.gif]] |} | (3) |} ==Equational form== A stronger form of Peirce's law also holds, in which the final implication is observed to be reversible: {| align="center" cellpadding="10" | <math>((p \Rightarrow q) \Rightarrow p) \Leftrightarrow p</math> |} ===Proof 1=== Given what precedes, it remains to show that: {| align="center" cellpadding="10" | <math>p \Rightarrow ((p \Rightarrow q) \Rightarrow p)</math> |} But this is immediate, since <math>p \Rightarrow (r \Rightarrow p)</math> for any proposition <math>r.\!</math> ===Proof 2=== Representing propositions as logical graphs under the existential interpretation, the strong form of Peirce's law is expressed by the following equation: {| align="center" border="0" cellpadding="10" cellspacing="0" | [[Image:Peirce's Law Strong Form 1.0 Splash Page.png|500px]] || (4) |} Using the axioms and theorems listed in the article on [[logical graphs]], the equational form of Peirce's law may be proved in the following manner: {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Peirce's Law Strong Form 1.0 Marquee Title.png|500px]] |- | [[Image:Peirce's Law Strong Form 1.0 Storyboard 1.png|500px]] |- | [[Image:Equational Inference Rule Collect p.png|500px]] |- | [[Image:Peirce's Law Strong Form 1.0 Storyboard 2.png|500px]] |- | [[Image:Equational Inference Rule Quit ((q)).png|500px]] |- | [[Image:Peirce's Law Strong Form 1.0 Storyboard 3.png|500px]] |- | [[Image:Equational Inference Rule Cancel (( )).png|500px]] |- | [[Image:Peirce's Law Strong Form 1.0 Storyboard 4.png|500px]] |- | [[Image:Equational Inference Marquee QED.png|500px]] |} | (5) |} The following animation replays the steps of the proof. {| align="center" cellpadding="8" | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" |- | [[Image:Peirce's Law Strong Form 2.0 Animation.gif]] |} | (6) |} ==Bibliography== * [[Charles Sanders Peirce|Peirce, Charles Sanders]] (1885), "On the Algebra of Logic : A Contribution to the Philosophy of Notation", ''American Journal of Mathematics'' 7 (1885), 180&ndash;202. Reprinted (CP&nbsp;3.359&ndash;403), (CE&nbsp;5, 162&ndash;190). * Peirce, Charles Sanders (1931&ndash;1935, 1958), ''Collected Papers of Charles Sanders Peirce'', vols. 1&ndash;6, Charles Hartshorne and Paul Weiss (eds.), vols. 7&ndash;8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA. Cited as (CP&nbsp;volume.paragraph). * Peirce, Charles Sanders (1981&ndash;), ''Writings of Charles S. Peirce : A Chronological Edition'', Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN. Cited as (CE&nbsp;volume,&nbsp;page). ==Syllabus== ===Focal nodes=== * [[Inquiry Live]] * [[Logic Live]] ===Peer nodes=== * [http://intersci.ss.uci.edu/wiki/index.php/Peirce's_law Peirce's Law @ InterSciWiki] * [http://mywikibiz.com/Peirce's_law Peirce's Law @ MyWikiBiz] * [http://ref.subwiki.org/wiki/Peirce's_law Peirce's Law @ Subject Wikis] * [http://en.wikiversity.org/wiki/Peirce's_law Peirce's Law @ Wikiversity] * [http://beta.wikiversity.org/wiki/Peirce's_law Peirce's Law @ Wikiversity Beta] ===Logical operators=== {{col-begin}} {{col-break}} * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] {{col-break}} * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Logical negation|Negation]] {{col-end}} ===Related topics=== {{col-begin}} {{col-break}} * [[Ampheck]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean-valued function]] * [[Differential logic]] {{col-break}} * [[Logical graph]] * [[Minimal negation operator]] * [[Multigrade operator]] * [[Parametric operator]] * [[Peirce's law]] {{col-break}} * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Truth table]] * [[Universe of discourse]] * [[Zeroth order logic]] {{col-end}} ===Relational concepts=== {{col-begin}} {{col-break}} * [[Continuous predicate]] * [[Hypostatic abstraction]] * [[Logic of relatives]] * [[Logical matrix]] {{col-break}} * [[Relation (mathematics)|Relation]] * [[Relation composition]] * [[Relation construction]] * [[Relation reduction]] {{col-break}} * [[Relation theory]] * [[Relative term]] * [[Sign relation]] * [[Triadic relation]] {{col-end}} ===Information, Inquiry=== {{col-begin}} {{col-break}} * [[Inquiry]] * [[Dynamics of inquiry]] {{col-break}} * [[Semeiotic]] * [[Logic of information]] {{col-break}} * [[Descriptive science]] * [[Normative science]] {{col-break}} * [[Pragmatic maxim]] * [[Truth theory]] {{col-end}} ===Related articles=== {{col-begin}} {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] {{col-break}} * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] {{col-end}} ==Document history== Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. * [http://intersci.ss.uci.edu/wiki/index.php/Peirce's_law Peirce's Law], [http://intersci.ss.uci.edu/ InterSciWiki] * [http://mywikibiz.com/Peirce's_law Peirce's Law], [http://mywikibiz.com/ MyWikiBiz] * [http://planetmath.org/PeircesLaw Peirce's Law], [http://planetmath.org/ PlanetMath] * [http://wikinfo.org/w/index.php/Peirce's_law Peirce's Law], [http://www.wikinfo.org/w/ Wikinfo] * [http://en.wikiversity.org/wiki/Peirce's_law Peirce's Law], [http://en.wikiversity.org/ Wikiversity] * [http://beta.wikiversity.org/wiki/Peirce's_law Peirce's Law], [http://beta.wikiversity.org/ Wikiversity Beta] * [http://en.wikipedia.org/w/index.php?title=Peirce's_law&oldid=60606482 Peirce's Law], [http://en.wikipedia.org/ Wikipedia] [[Category:Artificial Intelligence]] [[Category:Charles Sanders Peirce]] [[Category:Combinatorics]] [[Category:Computer Science]] [[Category:Cybernetics]] [[Category:Equational Reasoning]] [[Category:Formal Languages]] [[Category:Formal Systems]] [[Category:Graph Theory]] [[Category:History of Logic]] [[Category:History of Mathematics]] [[Category:Inquiry]] [[Category:Knowledge Representation]] [[Category:Logic]] [[Category:Logical Graphs]] [[Category:Mathematics]] [[Category:Philosophy]] [[Category:Semiotics]] [[Category:Visualization]] afbedca4fee7259a46c3c1c61177bea307402f9c 106 22 772 696 2016-03-13T17:01:38Z Vipul 2 wikitext text/x-wiki {| class="sortable" border="1" ! Wiki !! Main topic !! Page count (lower bound) !! Creation month !! State of development !! Content being added? !! Error log !! 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In cases where the wiki was ported from elsewhere, the creation month may be earlier; in other cases, it may be later if the first edits happened long after installation. '''Pageview count note''': Estimates are based on Google Analytics data, ''not'' based on MediaWiki internal logs, which record about 2-3X the pageviews recorded by Google Analytics. e1e9b639c392dec221b83b8fc7ca7387989ccc57 773 772 2016-09-06T17:58:03Z Vipul 2 wikitext text/x-wiki {| class="sortable" border="1" ! Wiki !! Main topic !! Page count (lower bound) !! Creation month !! State of development !! Content being added? !! Error log !! Pageview count (2015) (Google Analytics) |- | [[Groupprops:Main Page|Groupprops]] || Group theory || [[Groupprops:Special:Statistics|7200]] || December 2006 || beta || Occasionally || [[Groupprops:Groupprops:Error log|here]] || 881,735 |- | [[Topospaces:Main Page|Topospaces]] || Topology (point set, algebraic) || [[Topospaces:Special:Statistics|600]] || May 2007 || alpha || No || [[Topospaces:Topospaces:Error log|here]] || 96,944 |- | [[Commalg:Main Page|Commalg]] || Commutative algebra || [[Commalg:Special:Statistics|400]] || January 2007 || alpha || No || Not yet || 12,309 |- | [[Diffgeom:Main Page|Diffgeom]] || Differential geometry || [[Diffgeom:Special:Statistics|400]] || February 2007 || alpha || No || Not yet || 4,317 |- | [[Number:Main Page|Number]] || Number theory || [[Number:Special:Statistics|200]] || March 2009 || alpha || No || Not yet || 5,271 |- | [[Market:Main Page|Market]] || Economic theory of markets, choices, and prices || [[Market:Special:Statistics|100]] || January 2009 || alpha || Occasionally || [[Market:Market:Error log|here]] || 209,792 |- | [[Mech:Main Page|Mech]] || Classical mechanics || [[Mech:Special:Statistics|50]] || January 2009 || alpha || No || [[Mech:Mech:Error log|here]] || 53,526 |- | [[Calculus:Main Page|Calculus]] || Calculus || [[Calculus:Special:Statistics|200]] || August 2011 || alpha || Occasionally || [[Calculus:Calculus:Error log|here]] || 210,826 |- | [[Cellbio:Main Page|Cellbio]] || Cellbio || [[Cellbio:Special:Statistics|30]] || September 2011 || alpha || No || Not yet || 843 |- | [[Learning:Main Page|Learning]] || Learning, teaching, and mastery || [[Learning:Special:Statistics|25]] || September 2013 || alpha || Occasionally || Not yet || 873 |} '''NOTE''': The creation month need not coincide with the month of MediaWiki installation. In cases where the wiki was ported from elsewhere, the creation month may be earlier; in other cases, it may be later if the first edits happened long after installation. '''Pageview count note''': Estimates are based on Google Analytics data, ''not'' based on MediaWiki internal logs, which record about 2-3X the pageviews recorded by Google Analytics. 26bdfe7f7a5da8f094d09c669aec72c52e67d7f6 774 773 2017-02-18T22:51:37Z Vipul 2 wikitext text/x-wiki {| class="sortable" border="1" ! Wiki !! Main topic !! Page count (lower bound) !! Creation month !! State of development !! Content being added? !! Error log !! Pageview count (2016) (Google Analytics) |- | [[Groupprops:Main Page|Groupprops]] || Group theory || [[Groupprops:Special:Statistics|7200]] || December 2006 || beta || Occasionally || [[Groupprops:Groupprops:Error log|here]] || 931,796 |- | [[Topospaces:Main Page|Topospaces]] || Topology (point set, algebraic) || [[Topospaces:Special:Statistics|600]] || May 2007 || alpha || No || [[Topospaces:Topospaces:Error log|here]] || 107,635 |- | [[Commalg:Main Page|Commalg]] || Commutative algebra || [[Commalg:Special:Statistics|400]] || January 2007 || alpha || No || Not yet || 13,287 |- | [[Diffgeom:Main Page|Diffgeom]] || Differential geometry || [[Diffgeom:Special:Statistics|400]] || February 2007 || alpha || No || Not yet || 5,527 |- | [[Number:Main Page|Number]] || Number theory || [[Number:Special:Statistics|200]] || March 2009 || alpha || No || Not yet || 7,950 |- | [[Market:Main Page|Market]] || Economic theory of markets, choices, and prices || [[Market:Special:Statistics|100]] || January 2009 || alpha || Occasionally || [[Market:Market:Error log|here]] || 146,678 |- | [[Mech:Main Page|Mech]] || Classical mechanics || [[Mech:Special:Statistics|50]] || January 2009 || alpha || No || [[Mech:Mech:Error log|here]] || 52,409 |- | [[Calculus:Main Page|Calculus]] || Calculus || [[Calculus:Special:Statistics|200]] || August 2011 || alpha || Occasionally || [[Calculus:Calculus:Error log|here]] || 203,083 |- | [[Cellbio:Main Page|Cellbio]] || Cellbio || [[Cellbio:Special:Statistics|30]] || September 2011 || alpha || No || Not yet || 1,327 |- | [[Learning:Main Page|Learning]] || Learning, teaching, and mastery || [[Learning:Special:Statistics|25]] || September 2013 || alpha || Occasionally || Not yet || 3,774 |} '''NOTE''': The creation month need not coincide with the month of MediaWiki installation. In cases where the wiki was ported from elsewhere, the creation month may be earlier; in other cases, it may be later if the first edits happened long after installation. '''Pageview count note''': Estimates are based on Google Analytics data, ''not'' based on MediaWiki internal logs, which record about 2-3X the pageviews recorded by Google Analytics. c4e90f9c2efa72cdb55fe626c31a4c7c117cdd1b 778 774 2017-04-09T01:35:16Z Vipul 2 wikitext text/x-wiki {| class="sortable" border="1" ! Wiki !! Main topic !! Page count (lower bound) !! Creation month !! State of development !! Content being added? !! Error log !! Pageview count (2016) (Google Analytics) !! Quantcast? |- | [[Groupprops:Main Page|Groupprops]] || Group theory || [[Groupprops:Special:Statistics|7200]] || December 2006 || beta || Occasionally || [[Groupprops:Groupprops:Error log|here]] || 931,796 || [https://www.quantcast.com/groupprops.subwiki.org Starting April 2, 2017] |- | [[Topospaces:Main Page|Topospaces]] || Topology (point set, algebraic) || [[Topospaces:Special:Statistics|600]] || May 2007 || alpha || No || [[Topospaces:Topospaces:Error log|here]] || 107,635 || [https://www.quantcast.com/topospaces.subwiki.org Starting April 2, 2017] |- | [[Commalg:Main Page|Commalg]] || Commutative algebra || [[Commalg:Special:Statistics|400]] || January 2007 || alpha || No || Not yet || 13,287 || No |- | [[Diffgeom:Main Page|Diffgeom]] || Differential geometry || [[Diffgeom:Special:Statistics|400]] || February 2007 || alpha || No || Not yet || 5,527 || No |- | [[Number:Main Page|Number]] || Number theory || [[Number:Special:Statistics|200]] || March 2009 || alpha || No || Not yet || 7,950 || No |- | [[Market:Main Page|Market]] || Economic theory of markets, choices, and prices || [[Market:Special:Statistics|100]] || January 2009 || alpha || Occasionally || [[Market:Market:Error log|here]] || 146,678 || [https://www.quantcast.com/market.subwiki.org Starting November 27, 2015] |- | [[Mech:Main Page|Mech]] || Classical mechanics || [[Mech:Special:Statistics|50]] || January 2009 || alpha || No || [[Mech:Mech:Error log|here]] || 52,409 || [https://www.quantcast.com/mech.subwiki.org Starting April 3, 2017] |- | [[Calculus:Main Page|Calculus]] || Calculus || [[Calculus:Special:Statistics|200]] || August 2011 || alpha || Occasionally || [[Calculus:Calculus:Error log|here]] || 203,083 || [https://www.quantcast.com/calculus.subwiki.org Starting November 27, 2015] |- | [[Cellbio:Main Page|Cellbio]] || Cellbio || [[Cellbio:Special:Statistics|30]] || September 2011 || alpha || No || Not yet || 1,327 || No |- | [[Learning:Main Page|Learning]] || Learning, teaching, and mastery || [[Learning:Special:Statistics|25]] || September 2013 || alpha || Occasionally || Not yet || 3,774 || No |} '''NOTE''': The creation month need not coincide with the month of MediaWiki installation. In cases where the wiki was ported from elsewhere, the creation month may be earlier; in other cases, it may be later if the first edits happened long after installation. '''Pageview count note''': Estimates are based on Google Analytics data, ''not'' based on MediaWiki internal logs, which record about 2-3X the pageviews recorded by Google Analytics. bde06bdf1fa82eb2c0923500fd3c68e0ef4c3bd2 779 778 2017-04-09T01:39:53Z Vipul 2 wikitext text/x-wiki {| class="sortable" border="1" ! Wiki !! Main topic !! Page count (lower bound) !! Creation month !! State of development !! Content being added? !! Error log !! Pageview count (2016) (Google Analytics) !! Quantcast? |- | [[Groupprops:Main Page|Groupprops]] || Group theory || [[Groupprops:Special:Statistics|7200]] || December 2006 || beta || Occasionally || [[Groupprops:Groupprops:Error log|here]] || 931,796 || [https://www.quantcast.com/groupprops.subwiki.org Starting April 2, 2017] |- | [[Topospaces:Main Page|Topospaces]] || Topology (point set, algebraic) || [[Topospaces:Special:Statistics|600]] || May 2007 || alpha || No || [[Topospaces:Topospaces:Error log|here]] || 107,635 || [https://www.quantcast.com/topospaces.subwiki.org Starting April 2, 2017] |- | [[Commalg:Main Page|Commalg]] || Commutative algebra || [[Commalg:Special:Statistics|400]] || January 2007 || alpha || No || Not yet || 13,287 || No (pending data checking; might be too little data) |- | [[Diffgeom:Main Page|Diffgeom]] || Differential geometry || [[Diffgeom:Special:Statistics|400]] || February 2007 || alpha || No || Not yet || 5,527 || No (pending data checking; might be too little data) |- | [[Number:Main Page|Number]] || Number theory || [[Number:Special:Statistics|200]] || March 2009 || alpha || No || Not yet || 7,950 || No (too little data) |- | [[Market:Main Page|Market]] || Economic theory of markets, choices, and prices || [[Market:Special:Statistics|100]] || January 2009 || alpha || Occasionally || [[Market:Market:Error log|here]] || 146,678 || [https://www.quantcast.com/market.subwiki.org Starting November 27, 2015] |- | [[Mech:Main Page|Mech]] || Classical mechanics || [[Mech:Special:Statistics|50]] || January 2009 || alpha || No || [[Mech:Mech:Error log|here]] || 52,409 || [https://www.quantcast.com/mech.subwiki.org Starting April 3, 2017] |- | [[Calculus:Main Page|Calculus]] || Calculus || [[Calculus:Special:Statistics|200]] || August 2011 || alpha || Occasionally || [[Calculus:Calculus:Error log|here]] || 203,083 || [https://www.quantcast.com/calculus.subwiki.org Starting November 27, 2015] |- | [[Cellbio:Main Page|Cellbio]] || Cellbio || [[Cellbio:Special:Statistics|30]] || September 2011 || alpha || No || Not yet || 1,327 || No |- | [[Learning:Main Page|Learning]] || Learning, teaching, and mastery || [[Learning:Special:Statistics|25]] || September 2013 || alpha || Occasionally || Not yet || 3,774 || No |} '''NOTE''': The creation month need not coincide with the month of MediaWiki installation. In cases where the wiki was ported from elsewhere, the creation month may be earlier; in other cases, it may be later if the first edits happened long after installation. '''Pageview count note''': Estimates are based on Google Analytics data, ''not'' based on MediaWiki internal logs, which record about 2-3X the pageviews recorded by Google Analytics. 445654613cebaa0d3f2953f421e061a5305b3414 780 779 2017-04-09T01:41:19Z Vipul 2 wikitext text/x-wiki {| class="sortable" border="1" ! Wiki !! Main topic !! Page count (lower bound) !! Creation month !! State of development !! Content being added? !! Error log !! Pageview count (2016) (Google Analytics) !! Quantcast? |- | [[Groupprops:Main Page|Groupprops]] || Group theory || [[Groupprops:Special:Statistics|7200]] || December 2006 || beta || Occasionally || [[Groupprops:Groupprops:Error log|here]] || 931,796 || [https://www.quantcast.com/groupprops.subwiki.org Starting April 2, 2017] |- | [[Topospaces:Main Page|Topospaces]] || Topology (point set, algebraic) || [[Topospaces:Special:Statistics|600]] || May 2007 || alpha || No || [[Topospaces:Topospaces:Error log|here]] || 107,635 || [https://www.quantcast.com/topospaces.subwiki.org Starting April 2, 2017] |- | [[Commalg:Main Page|Commalg]] || Commutative algebra || [[Commalg:Special:Statistics|400]] || January 2007 || alpha || No || Not yet || 13,287 || No (pending data checking; might be too little data) |- | [[Diffgeom:Main Page|Diffgeom]] || Differential geometry || [[Diffgeom:Special:Statistics|400]] || February 2007 || alpha || No || Not yet || 5,527 || No (too little data) |- | [[Number:Main Page|Number]] || Number theory || [[Number:Special:Statistics|200]] || March 2009 || alpha || No || Not yet || 7,950 || No (too little data) |- | [[Market:Main Page|Market]] || Economic theory of markets, choices, and prices || [[Market:Special:Statistics|100]] || January 2009 || alpha || Occasionally || [[Market:Market:Error log|here]] || 146,678 || [https://www.quantcast.com/market.subwiki.org Starting November 27, 2015] |- | [[Mech:Main Page|Mech]] || Classical mechanics || [[Mech:Special:Statistics|50]] || January 2009 || alpha || No || [[Mech:Mech:Error log|here]] || 52,409 || [https://www.quantcast.com/mech.subwiki.org Starting April 3, 2017] |- | [[Calculus:Main Page|Calculus]] || Calculus || [[Calculus:Special:Statistics|200]] || August 2011 || alpha || Occasionally || [[Calculus:Calculus:Error log|here]] || 203,083 || [https://www.quantcast.com/calculus.subwiki.org Starting November 27, 2015] |- | [[Cellbio:Main Page|Cellbio]] || Cellbio || [[Cellbio:Special:Statistics|30]] || September 2011 || alpha || No || Not yet || 1,327 || No (too little data) |- | [[Learning:Main Page|Learning]] || Learning, teaching, and mastery || [[Learning:Special:Statistics|25]] || September 2013 || alpha || Occasionally || Not yet || 3,774 || No (too little data) |} '''NOTE''': The creation month need not coincide with the month of MediaWiki installation. In cases where the wiki was ported from elsewhere, the creation month may be earlier; in other cases, it may be later if the first edits happened long after installation. '''Pageview count note''': Estimates are based on Google Analytics data, ''not'' based on MediaWiki internal logs, which record about 2-3X the pageviews recorded by Google Analytics. 574b49ed784abfd6382f0302df7ec2509f111a6b 781 780 2017-10-09T20:47:25Z Vipul 2 wikitext text/x-wiki {| class="sortable" border="1" ! Wiki !! Main topic !! Page count (lower bound) !! Creation month !! State of development !! Content being added? !! Error log !! Pageview count (2016) (Google Analytics) !! Quantcast? |- | [[Groupprops:Main Page|Groupprops]] || Group theory || [[Groupprops:Special:Statistics|7200]] || December 2006 || beta || Occasionally || [[Groupprops:Groupprops:Error log|here]] || 931,796 || [https://www.quantcast.com/groupprops.subwiki.org Starting April 2, 2017] |- | [[Topospaces:Main Page|Topospaces]] || Topology (point set, algebraic) || [[Topospaces:Special:Statistics|600]] || May 2007 || alpha || No || [[Topospaces:Topospaces:Error log|here]] || 107,635 || [https://www.quantcast.com/topospaces.subwiki.org Starting April 2, 2017] |- | [[Commalg:Main Page|Commalg]] || Commutative algebra || [[Commalg:Special:Statistics|400]] || January 2007 || alpha || No || Not yet || 13,287 || No (pending data checking; might be too little data) |- | [[Diffgeom:Main Page|Diffgeom]] || Differential geometry || [[Diffgeom:Special:Statistics|400]] || February 2007 || alpha || No || Not yet || 5,527 || No (too little data) |- | [[Number:Main Page|Number]] || Number theory || [[Number:Special:Statistics|200]] || March 2009 || alpha || No || Not yet || 7,950 || No (too little data) |- | [[Market:Main Page|Market]] || Economic theory of markets, choices, and prices || [[Market:Special:Statistics|100]] || January 2009 || alpha || Occasionally || [[Market:Market:Error log|here]] || 146,678 || [https://www.quantcast.com/market.subwiki.org Starting November 27, 2015] |- | [[Mech:Main Page|Mech]] || Classical mechanics || [[Mech:Special:Statistics|50]] || January 2009 || alpha || No || [[Mech:Mech:Error log|here]] || 52,409 || [https://www.quantcast.com/mech.subwiki.org Starting April 3, 2017] |- | [[Calculus:Main Page|Calculus]] || Calculus || [[Calculus:Special:Statistics|200]] || August 2011 || alpha || Occasionally || [[Calculus:Calculus:Error log|here]] || 203,083 || [https://www.quantcast.com/calculus.subwiki.org Starting November 27, 2015] |- | [[Cellbio:Main Page|Cellbio]] || Cellbio || [[Cellbio:Special:Statistics|30]] || September 2011 || alpha || No || Not yet || 1,327 || No (too little data) |- | [[Learning:Main Page|Learning]] || Learning, teaching, and mastery || [[Learning:Special:Statistics|25]] || September 2013 || alpha || Occasionally || Not yet || 3,774 || No (too little data) |- | [[Devec:Main Page|Devec]] || Development economics || [[Devec:Special:Statistics|21]] || February 2014 || alpha || Occasionally || Not yet || Not recorded || No (too little data) |- | [[Demography:Main Page|Demography]] || Demography || [[Demography:Special:Statistics|191]] || September 2013 || alpha || No || Not yet || 3,594 || No (too little data) |} '''NOTE''': The creation month need not coincide with the month of MediaWiki installation. In cases where the wiki was ported from elsewhere, the creation month may be earlier; in other cases, it may be later if the first edits happened long after installation. '''Pageview count note''': Estimates are based on Google Analytics data, ''not'' based on MediaWiki internal logs, which record about 2-3X the pageviews recorded by Google Analytics. 5622d8a4ccc4645809bb3dd6b74f07b881553ef3 4 133 775 628 2017-03-18T06:09:12Z Vipul 2 /* Examples of personal use that necessitate use of the license terms */ wikitext text/x-wiki This is a common copyright notice to all subject wikis. ==General license information== All content is put up under the [http://creativecommons.org/licenses/by-sa/3.0/ Creative Commons Attribute Share-Alike 3.0 license (unported)] (shorthand: CC-by-SA). 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Attribution formats are discussed below. * Giving handouts of pages or collections of pages to a group of people, such as students in a course: In this case, the handouts should be accompanied by an attribution that makes clear both the source and the original license terms, as well as a link to the Uniform Resource Identifier (URI) of the subject wiki. Attribution formats: * For websites: ''This content is from [{{fullurl:Main Page}} {{SITENAME}}]'' or ''This content is from [{{fullurl:Main Page}} {{fullsitetitle}}]''. Preferably, also a link to the specific pages (if any) used. The language of the attribution may be modified somewhat to indicate whether the content has been used as such, or whether changes have been made. * For print materials: ''This content is from {{SITENAME}}. The site URL is https://subwiki.org'' Attribution of individuals who have contributed content to the subject wiki entries is ''not'' necessary. (However, this list of individuals can be tracked down, as discussed in the next section). ==Uses not within the ambit of the license or fair use rights== Those who have contributed all the content to a particular subject wiki entry can republish the same content anywhere else ''without restriction'' and can also release the content elsewhere under a different license, since they own the full copyright to their content. To use the content on the wiki in a manner that is outside the terms of the license or beyond fair use rights, the individuals who have contributed that particular content need to be contacted. For any particular page, this information can be accessed using the history tab of the page. The users who have made edits to the page are the ones who need to be contacted. In case of difficulty doing this, please contact vipul.wikis@gmail.com or vipul@math.uchicago.edu for more information. ==Copyright violations== All users using subject wikis are requested not to violate copyright of other authors or publishers. This includes providing links that may aid and abet copyright violation. Examples of things that may be considered copyright violation: * Putting up links to personal copies of restricted-access journal articles acquired through a personal or library subscription: This usually goes against the Terms of Service of the subscription. The exception is when the content of the articles is out of copyright. * Giving links to personal copies, or copies on filesharing systems, of books that are under copyright and where either owning or sharing a copy in that manner is against the terms of copyright. In case copyright violations are detected, please email vipul.wikis@gmail.com and vipul@math.uchicago.edu to have the matter looked into immediately. cb4ff0112563fe213d4fdf72c31ebaa29e03c1de 776 775 2017-03-18T06:10:07Z Vipul 2 /* Uses not within the ambit of the license or fair use rights */ wikitext text/x-wiki This is a common copyright notice to all subject wikis. ==General license information== All content is put up under the [http://creativecommons.org/licenses/by-sa/3.0/ Creative Commons Attribute Share-Alike 3.0 license (unported)] (shorthand: CC-by-SA). Read the [http://creativehttp://ref.subwiki.org/w/skins/common/images/button_media.pngcommons.org/licenses/by-sa/3.0/legalcode full legal code here]. ===Exceptions=== The software engine that runs the wiki is MediaWiki, and this software is licensed under the GNU General Public License, which is distinct from and currently incompatible with the Creative Commons license. Similarly, some of the extensions to the software engine are licensed by their developers using the GPL, which is incompatible with Creative Commons licenses. More information on the version of MediaWiki and extensions: [[Special:Version]]. In case of material (such as videos) embedded or copied from other websites, and subject to different license restrictions, the information on the differences in license are clearly indicated on the page. ===Short description of the license=== The CC-by-SA license has the following features: # Content can be used and reused for any purposes, commercial or noncommercial. # Any public use or reuse of the material requires attribution of the original source (for attribution formats, see later in the document). # In case of adaptation or creation of derivative works, the newly created works must also be licensed under the same license (CC-by-SA) unless the creator of the derivative work takes ''explicit'' permission from the creator of the original work. ===Fair use and other rights=== The license cannot and does not limit fair use rights. Fair use may include quoting or copying select passages for the purposes of criticism and review. The license also cannot limit the use of data or information contained in the content on subject wikis. This means that anybody can use or reuse the facts or knowledge they gain from the subject wiki in any way they want. ==Applicability of license to the subject wiki== ===Examples of personal use that do not necessitate use of the license terms=== * Saving, making copies, or taking printouts of pages on the subject wiki, as long as these copies are meant for personal use and are not to be distributed to others. * Learning facts from the subject wikis and using those facts as part of instruction or research. * Providing links to pages on the subject wikis. * Quoting short passages from the subject wiki in a criticism or review (it is preferable, though not legally enforceable, that a link to the original content being reviewed be provided). ===Examples of personal use that necessitate use of the license terms=== * Creating a mirror website by exporting pages or content in bulk: In this case, the mirror website must release the mirrored content under the same license terms and ''also'' attribute the original source with a link. Attribution formats are discussed below. * Giving handouts of pages or collections of pages to a group of people, such as students in a course: In this case, the handouts should be accompanied by an attribution that makes clear both the source and the original license terms, as well as a link to the Uniform Resource Identifier (URI) of the subject wiki. Attribution formats: * For websites: ''This content is from [{{fullurl:Main Page}} {{SITENAME}}]'' or ''This content is from [{{fullurl:Main Page}} {{fullsitetitle}}]''. Preferably, also a link to the specific pages (if any) used. The language of the attribution may be modified somewhat to indicate whether the content has been used as such, or whether changes have been made. * For print materials: ''This content is from {{SITENAME}}. The site URL is https://subwiki.org'' Attribution of individuals who have contributed content to the subject wiki entries is ''not'' necessary. (However, this list of individuals can be tracked down, as discussed in the next section). ==Uses not within the ambit of the license or fair use rights== Those who have contributed all the content to a particular subject wiki entry can republish the same content anywhere else ''without restriction'' and can also release the content elsewhere under a different license, since they own the full copyright to their content. To use the content on the wiki in a manner that is outside the terms of the license or beyond fair use rights, the individuals who have contributed that particular content need to be contacted. For any particular page, this information can be accessed using the history tab of the page. The users who have made edits to the page are the ones who need to be contacted. In case of difficulty doing this, please contact vipulnaik1@gmail.com for more information. ==Copyright violations== All users using subject wikis are requested not to violate copyright of other authors or publishers. This includes providing links that may aid and abet copyright violation. Examples of things that may be considered copyright violation: * Putting up links to personal copies of restricted-access journal articles acquired through a personal or library subscription: This usually goes against the Terms of Service of the subscription. The exception is when the content of the articles is out of copyright. * Giving links to personal copies, or copies on filesharing systems, of books that are under copyright and where either owning or sharing a copy in that manner is against the terms of copyright. In case copyright violations are detected, please email vipul.wikis@gmail.com and vipul@math.uchicago.edu to have the matter looked into immediately. cebca2d93648a06524f6ee571803553934f2dd43 777 776 2017-03-18T06:10:17Z Vipul 2 /* Copyright violations */ wikitext text/x-wiki This is a common copyright notice to all subject wikis. ==General license information== All content is put up under the [http://creativecommons.org/licenses/by-sa/3.0/ Creative Commons Attribute Share-Alike 3.0 license (unported)] (shorthand: CC-by-SA). Read the [http://creativehttp://ref.subwiki.org/w/skins/common/images/button_media.pngcommons.org/licenses/by-sa/3.0/legalcode full legal code here]. ===Exceptions=== The software engine that runs the wiki is MediaWiki, and this software is licensed under the GNU General Public License, which is distinct from and currently incompatible with the Creative Commons license. Similarly, some of the extensions to the software engine are licensed by their developers using the GPL, which is incompatible with Creative Commons licenses. More information on the version of MediaWiki and extensions: [[Special:Version]]. In case of material (such as videos) embedded or copied from other websites, and subject to different license restrictions, the information on the differences in license are clearly indicated on the page. ===Short description of the license=== The CC-by-SA license has the following features: # Content can be used and reused for any purposes, commercial or noncommercial. # Any public use or reuse of the material requires attribution of the original source (for attribution formats, see later in the document). # In case of adaptation or creation of derivative works, the newly created works must also be licensed under the same license (CC-by-SA) unless the creator of the derivative work takes ''explicit'' permission from the creator of the original work. ===Fair use and other rights=== The license cannot and does not limit fair use rights. Fair use may include quoting or copying select passages for the purposes of criticism and review. The license also cannot limit the use of data or information contained in the content on subject wikis. This means that anybody can use or reuse the facts or knowledge they gain from the subject wiki in any way they want. ==Applicability of license to the subject wiki== ===Examples of personal use that do not necessitate use of the license terms=== * Saving, making copies, or taking printouts of pages on the subject wiki, as long as these copies are meant for personal use and are not to be distributed to others. * Learning facts from the subject wikis and using those facts as part of instruction or research. * Providing links to pages on the subject wikis. * Quoting short passages from the subject wiki in a criticism or review (it is preferable, though not legally enforceable, that a link to the original content being reviewed be provided). ===Examples of personal use that necessitate use of the license terms=== * Creating a mirror website by exporting pages or content in bulk: In this case, the mirror website must release the mirrored content under the same license terms and ''also'' attribute the original source with a link. Attribution formats are discussed below. * Giving handouts of pages or collections of pages to a group of people, such as students in a course: In this case, the handouts should be accompanied by an attribution that makes clear both the source and the original license terms, as well as a link to the Uniform Resource Identifier (URI) of the subject wiki. Attribution formats: * For websites: ''This content is from [{{fullurl:Main Page}} {{SITENAME}}]'' or ''This content is from [{{fullurl:Main Page}} {{fullsitetitle}}]''. Preferably, also a link to the specific pages (if any) used. The language of the attribution may be modified somewhat to indicate whether the content has been used as such, or whether changes have been made. * For print materials: ''This content is from {{SITENAME}}. The site URL is https://subwiki.org'' Attribution of individuals who have contributed content to the subject wiki entries is ''not'' necessary. (However, this list of individuals can be tracked down, as discussed in the next section). ==Uses not within the ambit of the license or fair use rights== Those who have contributed all the content to a particular subject wiki entry can republish the same content anywhere else ''without restriction'' and can also release the content elsewhere under a different license, since they own the full copyright to their content. To use the content on the wiki in a manner that is outside the terms of the license or beyond fair use rights, the individuals who have contributed that particular content need to be contacted. For any particular page, this information can be accessed using the history tab of the page. The users who have made edits to the page are the ones who need to be contacted. In case of difficulty doing this, please contact vipulnaik1@gmail.com for more information. ==Copyright violations== All users using subject wikis are requested not to violate copyright of other authors or publishers. This includes providing links that may aid and abet copyright violation. Examples of things that may be considered copyright violation: * Putting up links to personal copies of restricted-access journal articles acquired through a personal or library subscription: This usually goes against the Terms of Service of the subscription. The exception is when the content of the articles is out of copyright. * Giving links to personal copies, or copies on filesharing systems, of books that are under copyright and where either owning or sharing a copy in that manner is against the terms of copyright. In case copyright violations are detected, please email vipulnaik1@gmail.com to have the matter looked into immediately. 4824766e59c52aab421e232015c592fee19af4ed 799 777 2024-08-06T01:41:40Z Vipul 2 /* Applicability of license to the subject wiki */ wikitext text/x-wiki This is a common copyright notice to all subject wikis. ==General license information== All content is put up under the [http://creativecommons.org/licenses/by-sa/3.0/ Creative Commons Attribute Share-Alike 3.0 license (unported)] (shorthand: CC-by-SA). Read the [http://creativehttp://ref.subwiki.org/w/skins/common/images/button_media.pngcommons.org/licenses/by-sa/3.0/legalcode full legal code here]. ===Exceptions=== The software engine that runs the wiki is MediaWiki, and this software is licensed under the GNU General Public License, which is distinct from and currently incompatible with the Creative Commons license. Similarly, some of the extensions to the software engine are licensed by their developers using the GPL, which is incompatible with Creative Commons licenses. More information on the version of MediaWiki and extensions: [[Special:Version]]. In case of material (such as videos) embedded or copied from other websites, and subject to different license restrictions, the information on the differences in license are clearly indicated on the page. ===Short description of the license=== The CC-by-SA license has the following features: # Content can be used and reused for any purposes, commercial or noncommercial. # Any public use or reuse of the material requires attribution of the original source (for attribution formats, see later in the document). # In case of adaptation or creation of derivative works, the newly created works must also be licensed under the same license (CC-by-SA) unless the creator of the derivative work takes ''explicit'' permission from the creator of the original work. ===Fair use and other rights=== The license cannot and does not limit fair use rights. Fair use may include quoting or copying select passages for the purposes of criticism and review. The license also cannot limit the use of data or information contained in the content on subject wikis. This means that anybody can use or reuse the facts or knowledge they gain from the subject wiki in any way they want. ==Applicability of license to the subject wikis== ===Examples of personal use that do not necessitate use of the license terms=== * Saving, making copies, or taking printouts of pages on the subject wiki, as long as these copies are meant for personal use and are not to be distributed to others. * Learning facts from the subject wikis and using those facts as part of instruction or research. * Providing links to pages on the subject wikis. * Quoting short passages from the subject wiki in a criticism or review (it is preferable, though not legally enforceable, that a link to the original content being reviewed be provided). ===Examples of personal use that necessitate use of the license terms=== * Creating a mirror website by exporting pages or content in bulk: In this case, the mirror website must release the mirrored content under the same license terms and ''also'' attribute the original source with a link. Attribution formats are discussed below. * Giving handouts of pages or collections of pages to a group of people, such as students in a course: In this case, the handouts should be accompanied by an attribution that makes clear both the source and the original license terms, as well as a link to the Uniform Resource Identifier (URI) of the subject wiki. Attribution formats: * For websites: ''This content is from [{{fullurl:Main Page}} {{SITENAME}}]'' or ''This content is from [{{fullurl:Main Page}} {{fullsitetitle}}]''. Preferably, also a link to the specific pages (if any) used. The language of the attribution may be modified somewhat to indicate whether the content has been used as such, or whether changes have been made. * For print materials: ''This content is from {{SITENAME}}. The site URL is https://subwiki.org'' Attribution of individuals who have contributed content to the subject wiki entries is ''not'' necessary. (However, this list of individuals can be tracked down, as discussed in the next section). ==Uses not within the ambit of the license or fair use rights== Those who have contributed all the content to a particular subject wiki entry can republish the same content anywhere else ''without restriction'' and can also release the content elsewhere under a different license, since they own the full copyright to their content. To use the content on the wiki in a manner that is outside the terms of the license or beyond fair use rights, the individuals who have contributed that particular content need to be contacted. For any particular page, this information can be accessed using the history tab of the page. The users who have made edits to the page are the ones who need to be contacted. In case of difficulty doing this, please contact vipulnaik1@gmail.com for more information. ==Copyright violations== All users using subject wikis are requested not to violate copyright of other authors or publishers. This includes providing links that may aid and abet copyright violation. Examples of things that may be considered copyright violation: * Putting up links to personal copies of restricted-access journal articles acquired through a personal or library subscription: This usually goes against the Terms of Service of the subscription. The exception is when the content of the articles is out of copyright. * Giving links to personal copies, or copies on filesharing systems, of books that are under copyright and where either owning or sharing a copy in that manner is against the terms of copyright. In case copyright violations are detected, please email vipulnaik1@gmail.com to have the matter looked into immediately. 061cb89208631878dce27aeff9b18d471fa65d5c 800 799 2024-08-06T01:43:23Z Vipul 2 /* Applicability of license to the subject wikis */ wikitext text/x-wiki This is a common copyright notice to all subject wikis. ==General license information== All content is put up under the [http://creativecommons.org/licenses/by-sa/3.0/ Creative Commons Attribute Share-Alike 3.0 license (unported)] (shorthand: CC-by-SA). Read the [http://creativehttp://ref.subwiki.org/w/skins/common/images/button_media.pngcommons.org/licenses/by-sa/3.0/legalcode full legal code here]. ===Exceptions=== The software engine that runs the wiki is MediaWiki, and this software is licensed under the GNU General Public License, which is distinct from and currently incompatible with the Creative Commons license. Similarly, some of the extensions to the software engine are licensed by their developers using the GPL, which is incompatible with Creative Commons licenses. More information on the version of MediaWiki and extensions: [[Special:Version]]. In case of material (such as videos) embedded or copied from other websites, and subject to different license restrictions, the information on the differences in license are clearly indicated on the page. ===Short description of the license=== The CC-by-SA license has the following features: # Content can be used and reused for any purposes, commercial or noncommercial. # Any public use or reuse of the material requires attribution of the original source (for attribution formats, see later in the document). # In case of adaptation or creation of derivative works, the newly created works must also be licensed under the same license (CC-by-SA) unless the creator of the derivative work takes ''explicit'' permission from the creator of the original work. ===Fair use and other rights=== The license cannot and does not limit fair use rights. Fair use may include quoting or copying select passages for the purposes of criticism and review. The license also cannot limit the use of data or information contained in the content on subject wikis. This means that anybody can use or reuse the facts or knowledge they gain from the subject wiki in any way they want. ==Applicability of license to the subject wiki== ===Examples of personal use that do not necessitate use of the license terms=== * Saving, making copies, or taking printouts of pages on the subject wiki, as long as these copies are meant for personal use and are not to be distributed to others. * Learning facts from the subject wikis and using those facts as part of instruction or research. * Providing links to pages on the subject wikis. * Quoting short passages from the subject wiki in a criticism or review (it is preferable, though not legally enforceable, that a link to the original content being reviewed be provided). ===Examples of personal use that necessitate use of the license terms=== * Creating a mirror website by exporting pages or content in bulk: In this case, the mirror website must release the mirrored content under the same license terms and ''also'' attribute the original source with a link. Attribution formats are discussed below. * Giving handouts of pages or collections of pages to a group of people, such as students in a course: In this case, the handouts should be accompanied by an attribution that makes clear both the source and the original license terms, as well as a link to the Uniform Resource Identifier (URI) of the subject wiki. Attribution formats: * For websites: ''This content is from [{{fullurl:Main Page}} {{SITENAME}}]'' or ''This content is from [{{fullurl:Main Page}} {{fullsitetitle}}]''. Preferably, also a link to the specific pages (if any) used. The language of the attribution may be modified somewhat to indicate whether the content has been used as such, or whether changes have been made. * For print materials: ''This content is from {{SITENAME}}. The site URL is https://subwiki.org'' Attribution of individuals who have contributed content to the subject wiki entries is ''not'' necessary. (However, this list of individuals can be tracked down, as discussed in the next section). ==Uses not within the ambit of the license or fair use rights== Those who have contributed all the content to a particular subject wiki entry can republish the same content anywhere else ''without restriction'' and can also release the content elsewhere under a different license, since they own the full copyright to their content. To use the content on the wiki in a manner that is outside the terms of the license or beyond fair use rights, the individuals who have contributed that particular content need to be contacted. For any particular page, this information can be accessed using the history tab of the page. The users who have made edits to the page are the ones who need to be contacted. In case of difficulty doing this, please contact vipulnaik1@gmail.com for more information. ==Copyright violations== All users using subject wikis are requested not to violate copyright of other authors or publishers. This includes providing links that may aid and abet copyright violation. Examples of things that may be considered copyright violation: * Putting up links to personal copies of restricted-access journal articles acquired through a personal or library subscription: This usually goes against the Terms of Service of the subscription. The exception is when the content of the articles is out of copyright. * Giving links to personal copies, or copies on filesharing systems, of books that are under copyright and where either owning or sharing a copy in that manner is against the terms of copyright. In case copyright violations are detected, please email vipulnaik1@gmail.com to have the matter looked into immediately. 4824766e59c52aab421e232015c592fee19af4ed 0 1 782 694 2019-01-02T13:48:13Z Vipul 2 wikitext text/x-wiki {{DISPLAYTITLE:The Subject Wikis Reference Guide}} To see more up-to-date and regularly updated pageview counts for the subwiki sites, see [https://analytics.vipulnaik.com analytics.vipulnaik.com]. Here is our list of subject wikis: {{:Subwiki:List of subject wikis}} You can access the history of subject wikis [[Subwiki:History of subject wikis|here]]. 651ac5e64c95190398426e2a5573220b73d709dd 4 135 784 262 2022-09-25T15:28:48Z Vipul 2 wikitext text/x-wiki ==Privacy for readers== If you are surfing this website, your actions are logged in our usage logs. These usage logs are accessible to: * The site's administrators and technical support group. For a full list of administrators, contact [[User:Vipul|Vipul Naik]] by email: vipulnaik1@gmail.com. * The service that hosts the data and servers, which is currently [http://www.linode.com Linode]. * Google Analytics, which has been integrated to collect site statistics. View Google's privacy policy here: http://www.google.com/intl/en_ALL/privacypolicy.html Your usage logs are not made available to other parties. Aggregated data from logs, such as general usage patterns, may be used by the MediaWiki software as well as by site administrators in decision making. For instance, MediaWiki keeps track of the number of times each page is viewed. ==Privacy for editors== Editing on subject wikis is generally permitted only for registered users. Registered users must, at the time of registration, provide their real name, and enter basic information about their reason for interest. ''No'' private information such as date of birth, social security or taxation number, or home address is sought. Regarding personal information: * The email IDs of registered users are visible to site administrators only. For information about site administrators, contact vipulnaik1@gmail.com with the particular subject wiki and the reason for request. * All editing activity by registered users is recorded on the site and is visible to all site users. However, this information is not indexed by search engines that follow robots.txt. * For edits made by registered users when logged in, the originating IP addresses for the edits can be accessed only by the site administrators. * Passwords chosen by registered users are not humanly accessible, even to site administrators. ca856f5dfe44f1d4d34b883fa932f778c2bb19be 785 784 2022-09-25T15:30:19Z Vipul 2 /* Privacy for readers */ wikitext text/x-wiki ==Privacy for readers== If you are surfing this website, your actions are logged in our usage logs. These usage logs are accessible to: * The site's administrators and technical support group. For a full list of administrators, contact [[User:Vipul|Vipul Naik]] by email: vipulnaik1@gmail.com. * The service that hosts the data and servers, which is currently [http://www.linode.com Linode]. * Google Analytics, which has been integrated to collect site statistics. View Google's privacy policy here: http://www.google.com/intl/en_ALL/privacypolicy.html * Other third-party JS scripts that collect user activity; none of these should collect any personally identifiable information (PII). For a list of all scripts running at the current time, contact [[User:Vipul|Vipul Naik]] by email: vipulnaik1@gmail.com. ==Privacy for editors== Editing on subject wikis is generally permitted only for registered users. Registered users must, at the time of registration, provide their real name, and enter basic information about their reason for interest. ''No'' private information such as date of birth, social security or taxation number, or home address is sought. Regarding personal information: * The email IDs of registered users are visible to site administrators only. For information about site administrators, contact vipulnaik1@gmail.com with the particular subject wiki and the reason for request. * All editing activity by registered users is recorded on the site and is visible to all site users. However, this information is not indexed by search engines that follow robots.txt. * For edits made by registered users when logged in, the originating IP addresses for the edits can be accessed only by the site administrators. * Passwords chosen by registered users are not humanly accessible, even to site administrators. 1bcf54ebaeedbfcb9c248ca92c3ac59a1161339a 786 785 2022-09-25T15:36:33Z Vipul 2 wikitext text/x-wiki ==Privacy for readers== If you are surfing this website, your actions are logged in our usage logs. These usage logs are accessible to: * The site's administrators and technical support group. For a full list of administrators, contact [[User:Vipul|Vipul Naik]] by email: vipulnaik1@gmail.com. * The service that hosts the data and servers, which is currently [http://www.linode.com Linode]. * Google Analytics, which has been integrated to collect site statistics. View Google's privacy policy here: https://policies.google.com/privacy?hl=en * Other third-party JS scripts that collect user activity; none of these should collect any personally identifiable information (PII). For a list of all scripts running at the current time, contact [[User:Vipul|Vipul Naik]] by email: vipulnaik1@gmail.com. ==Privacy for editors== Editing on subject wikis is generally permitted only for registered users. Registered users must, at the time of registration, provide their real name, and enter basic information about their reason for interest. ''No'' private information such as date of birth, social security or taxation number, or home address is sought. Regarding personal information: * The email IDs of registered users are visible to site administrators only. For information about site administrators, contact vipulnaik1@gmail.com with the particular subject wiki and the reason for request. * All editing activity by registered users is recorded on the site and is visible to all site users. However, this information is not indexed by search engines that follow robots.txt. * For edits made by registered users when logged in, the originating IP addresses for the edits can be accessed only by the site administrators. * Passwords chosen by registered users are not humanly accessible, even to site administrators. 99f819973f45b857981f3d257ac1e079e3ef4485 142 378 789 2024-07-21T23:15:25Z 127.0.0.1 0 Semantic MediaWiki group import smw/schema application/json { "type": "PROPERTY_GROUP_SCHEMA", "groups": { "schema_group": { "canonical_name": "Schema properties", "message_key": "smw-property-group-label-schema-group", "property_keys": [ "_SCHEMA_TYPE", "_SCHEMA_DEF", "_SCHEMA_DESC", "_SCHEMA_TAG", "_SCHEMA_LINK", "_FORMAT_SCHEMA", "_CONSTRAINT_SCHEMA", "_PROFILE_SCHEMA" ] } }, "tags": [ "group", "property group" ] } fdba38d9db40d0248af036f81b4b7fdb74f6170d 142 379 790 2024-07-21T23:15:26Z 127.0.0.1 0 Semantic MediaWiki group import smw/schema application/json { "type": "PROPERTY_GROUP_SCHEMA", "groups": { "administrative_group": { "canonical_name": "Adminstrative properties", "message_key": "smw-property-group-label-administrative-properties", "property_keys": [ "_MDAT", "_CDAT", "_NEWP", "_LEDT", "_DTITLE", "_CHGPRO", "_EDIP", "_ERRC" ] }, "classification_group": { "canonical_name": "Classification properties", "message_key": "smw-property-group-label-classification-properties", "property_keys": [ "_INST", "_PPGR", "_SUBP", "_SUBC" ] }, "content_group": { "canonical_name": "Content properties", "message_key": "smw-property-group-label-content-properties", "property_keys": [ "_SOBJ", "_ASK", "_MEDIA", "_MIME", "_ATTCH_LINK", "_FILE_ATTCH", "_CONT_TYPE", "_CONT_AUTHOR", "_CONT_LEN", "_CONT_LANG", "_CONT_TITLE", "_CONT_DATE", "_CONT_KEYW", "_TRANS", "_TRANS_SOURCE", "_TRANS_GROUP" ] }, "declarative_group": { "canonical_name": "Declarative properties", "message_key": "smw-property-group-label-declarative-properties", "property_keys": [ "_TYPE", "_UNIT", "_IMPO", "_CONV", "_SERV", "_PVAL", "_LIST", "_PREC", "_PDESC", "_PPLB", "_PVAP", "_PVALI", "_PVUC", "_PEID", "_PEFU" ] } }, "tags": [ "group", "property group" ] } 8536eb8767d6dc5a87a0b4f7791b9cd791acbd75 8 380 791 2024-07-21T23:15:26Z 127.0.0.1 0 Semantic MediaWiki default vocabulary import wikitext text/x-wiki http://www.w3.org/2004/02/skos/core#|[http://www.w3.org/TR/skos-reference/skos.rdf Simple Knowledge Organization System (SKOS)] altLabel|Type:Monolingual text broader|Type:Annotation URI broaderTransitive|Type:Annotation URI broadMatch|Type:Annotation URI changeNote|Type:Text closeMatch|Type:Annotation URI Collection|Class Concept|Class ConceptScheme|Class definition|Type:Text editorialNote|Type:Text exactMatch|Type:Annotation URI example|Type:Text hasTopConcept|Type:Page hiddenLabel|Type:String historyNote|Type:Text inScheme|Type:Page mappingRelation|Type:Page member|Type:Page memberList|Type:Page narrower|Type:Annotation URI narrowerTransitive|Type:Annotation URI narrowMatch|Type:Annotation URI notation|Type:Text note|Type:Text OrderedCollection|Class prefLabel|Type:String related|Type:Annotation URI relatedMatch|Type:Annotation URI scopeNote|Type:Text semanticRelation|Type:Page topConceptOf|Type:Page [[Category:Imported vocabulary]] 4327e3118f75f756b955108e04693a361d19c2cb 8 381 792 2024-07-21T23:15:26Z 127.0.0.1 0 Semantic MediaWiki default vocabulary import wikitext text/x-wiki http://xmlns.com/foaf/0.1/|[http://www.foaf-project.org/ Friend Of A Friend] name|Type:Text homepage|Type:URL mbox|Type:Email mbox_sha1sum|Type:Text depiction|Type:URL phone|Type:Text Person|Category Organization|Category knows|Type:Page member|Type:Page [[Category:Imported vocabulary]] 2be18fc91e334e0c7f23bea734cdc2a301fd86e8 8 382 793 2024-07-21T23:15:26Z 127.0.0.1 0 Semantic MediaWiki default vocabulary import wikitext text/x-wiki http://www.w3.org/2002/07/owl#|[http://www.w3.org/2002/07/owl Web Ontology Language (OWL)] AllDifferent|Category allValuesFrom|Type:Page AnnotationProperty|Category backwardCompatibleWith|Type:Page cardinality|Type:Number Class|Category comment|Type:Page complementOf|Type:Page DataRange|Category DatatypeProperty|Category DeprecatedClass|Category DeprecatedProperty|Category differentFrom|Type:Page disjointWith|Type:Page distinctMembers|Type:Page equivalentClass|Type:Page equivalentProperty|Type:Page FunctionalProperty|Category hasValue|Type:Page imports|Type:Page incompatibleWith|Type:Page intersectionOf|Type:Page InverseFunctionalProperty|Category inverseOf|Type:Page isDefinedBy|Type:Page label|Type:Page maxCardinality|Type:Number minCardinality|Type:Number Nothing|Category ObjectProperty|Category oneOf|Type:Page onProperty|Type:Page Ontology|Category OntologyProperty|Category owl|Type:Page priorVersion|Type:Page Restriction|Category sameAs|Type:Page seeAlso|Type:Page someValuesFrom|Type:Page SymmetricProperty|Category Thing|Category TransitiveProperty|Category unionOf|Type:Page versionInfo|Type:Page [[Category:Imported vocabulary]] c109cc4c667590611dc35b3d06655129c572809a 132 383 794 2024-07-21T23:15:26Z 127.0.0.1 0 Semantic MediaWiki default vocabulary import wikitext text/x-wiki * [[Imported from::foaf:knows]] * [[Property description::A person known by this person (indicating some level of reciprocated interaction between the parties).@en]] [[Category:Imported vocabulary]] {{DISPLAYTITLE:foaf:knows}} e9134ab265b9bc923266ffa2bbcde2b59557202a 132 384 795 2024-07-21T23:15:26Z 127.0.0.1 0 Semantic MediaWiki default vocabulary import wikitext text/x-wiki * [[Imported from::foaf:name]] * [[Property description::A name for some thing or agent.@en]] [[Category:Imported vocabulary]] {{DISPLAYTITLE:foaf:name}} 2a8b3537cd6d95e741d56a2fe5b824216824c2e3 132 385 796 2024-07-21T23:15:26Z 127.0.0.1 0 Semantic MediaWiki default vocabulary import wikitext text/x-wiki * [[Imported from::foaf:homepage]] * [[Property description::URL of the homepage of something, which is a general web resource.@en]] [[Category:Imported vocabulary]] {{DISPLAYTITLE:foaf:homepage}} 083058f1760bcc251820336dbe29dee9a38516e6 132 386 797 2024-07-21T23:15:26Z 127.0.0.1 0 Semantic MediaWiki default vocabulary import wikitext text/x-wiki * [[Imported from::owl:differentFrom]] * [[Property description::The property that determines that two given individuals are different.@en]] [[Category:Imported vocabulary]] {{DISPLAYTITLE:owl:differentFrom}} d706757d4fb8eff4bb5622ea05ea7cab2f3b1a0c 2 387 798 2024-07-21T23:16:56Z Vipul 2 Created page with "* <math>\sqrt{7 + 2}!! + 3 = 723</math>" wikitext text/x-wiki * <math>\sqrt{7 + 2}!! + 3 = 723</math> b4af7c9678a8b85a33a63383963a1533af612d6b 801 798 2024-08-06T01:58:32Z Vipul 2 wikitext text/x-wiki * <math>\sqrt{7 + 2}!! + 3 = 723</math> * <math>(7 + 2)^{\sqrt{9}} = 729</math> 894e93713a7a28c9aa54d3b826bf53cdfb0b9fb0 142 388 802 2024-08-06T02:00:56Z Maintenance script 117 Semantic MediaWiki search import smw/schema application/json { "type": "FACETEDSEARCH_PROFILE_SCHEMA", "profiles": { "default_profile": { "message_key": "smw-facetedsearch-profile-label-default", "debug_output": false, "theme": "default-theme", "result": { "default_limit": 50, "paging_limit": [ 10, 20, 50, 250, 500 ] }, "filters": { "property_filter": { "hierarchy_tree": false, "filter_input": { "min_item": 10 } }, "category_filter": { "hierarchy_tree": false, "filter_input": { "min_item": 10 } }, "value_filter": { "default_filter": "list_filter", "condition_field": false, "filter_input": { "min_item": 10 } } } } }, "tags": [ "faceted search" ] } d0ca4220843b002d381de0ddd29fdd00758f4ff8 8 389 803 2024-08-06T02:00:56Z Maintenance script 117 Semantic MediaWiki default vocabulary import wikitext text/x-wiki https://schema.org/ | [https://schema.org/version/latest Schema.org], V 14.0 about|Type:Text abridged|Type:Text abstract|Type:Text accelerationTime|Type:Text acceptedAnswer|Type:Text acceptedOffer|Type:Text acceptedPaymentMethod|Type:Text acceptsReservations|Type:Text accessCode|Type:Text accessMode|Type:Text accessModeSufficient|Type:Text accessibilityAPI|Type:Text accessibilityControl|Type:Text accessibilityFeature|Type:Text accessibilityHazard|Type:Text accessibilitySummary|Type:Text accommodationCategory|Type:Text accommodationFloorPlan|Type:Text accountId|Type:Text accountMinimumInflow|Type:Text accountOverdraftLimit|Type:Text accountablePerson|Type:Text acquireLicensePage|Type:Text acquiredFrom|Type:Text acrissCode|Type:Text actionAccessibilityRequirement|Type:Text actionApplication|Type:Text actionOption|Type:Text actionPlatform|Type:Text actionStatus|Type:Text actionableFeedbackPolicy|Type:Text activeIngredient|Type:Text activityDuration|Type:Text activityFrequency|Type:Text actor|Type:Text actors|Type:Text addOn|Type:Text additionalName|Type:Text additionalNumberOfGuests|Type:Text additionalProperty|Type:Text additionalType|Type:Text additionalVariable|Type:Text address|Type:Text addressCountry|Type:Text addressLocality|Type:Text addressRegion|Type:Text administrationRoute|Type:Text advanceBookingRequirement|Type:Text adverseOutcome|Type:Text affectedBy|Type:Text affiliation|Type:Text afterMedia|Type:Text agent|Type:Text aggregateRating|Type:Text aircraft|Type:Text album|Type:Text albumProductionType|Type:Text albumRelease|Type:Text albumReleaseType|Type:Text albums|Type:Text alcoholWarning|Type:Text algorithm|Type:Text alignmentType|Type:Text alternateName|Type:Text alternativeHeadline|Type:Text alternativeOf|Type:Text alumni|Type:Text alumniOf|Type:Text amenityFeature|Type:Text amount|Type:Text amountOfThisGood|Type:Text announcementLocation|Type:Text annualPercentageRate|Type:Text answerCount|Type:Text answerExplanation|Type:Text antagonist|Type:Text appearance|Type:Text applicableCountry|Type:Text applicableLocation|Type:Text applicantLocationRequirements|Type:Text application|Type:Text applicationCategory|Type:Text applicationContact|Type:Text applicationDeadline|Type:Text applicationStartDate|Type:Date applicationSubCategory|Type:Text applicationSuite|Type:Text appliesToDeliveryMethod|Type:Text appliesToPaymentMethod|Type:Text archiveHeld|Type:Text archivedAt|Type:Text area|Type:Text areaServed|Type:Text arrivalAirport|Type:Text arrivalBoatTerminal|Type:Text arrivalBusStop|Type:Text arrivalGate|Type:Text arrivalPlatform|Type:Text arrivalStation|Type:Text arrivalTerminal|Type:Text arrivalTime|Type:Text artEdition|Type:Text artMedium|Type:Text arterialBranch|Type:Text artform|Type:Text articleBody|Type:Text articleSection|Type:Text artist|Type:Text artworkSurface|Type:Text aspect|Type:Text assembly|Type:Text assemblyVersion|Type:Text assesses|Type:Text associatedAnatomy|Type:Text associatedArticle|Type:Text associatedClaimReview|Type:Text associatedDisease|Type:Text associatedMedia|Type:Text associatedMediaReview|Type:Text associatedPathophysiology|Type:Text associatedReview|Type:Text athlete|Type:Text attendee|Type:Text attendees|Type:Text audience|Type:Text audienceType|Type:Text audio|Type:Text authenticator|Type:Text author|Type:Text availability|Type:Text availabilityEnds|Type:Text availabilityStarts|Type:Text availableAtOrFrom|Type:Text availableChannel|Type:Text availableDeliveryMethod|Type:Text availableFrom|Type:Text availableIn|Type:Text availableLanguage|Type:Text availableOnDevice|Type:Text availableService|Type:Text availableStrength|Type:Text availableTest|Type:Text availableThrough|Type:Text award|Type:Text awards|Type:Text awayTeam|Type:Text backstory|Type:Text bankAccountType|Type:Text baseSalary|Type:Text bccRecipient|Type:Text bed|Type:Text beforeMedia|Type:Text beneficiaryBank|Type:Text benefits|Type:Text benefitsSummaryUrl|Type:URL bestRating|Type:Text billingAddress|Type:Text billingDuration|Type:Text billingIncrement|Type:Text billingPeriod|Type:Text billingStart|Type:Text bioChemInteraction|Type:Text bioChemSimilarity|Type:Text biologicalRole|Type:Text biomechnicalClass|Type:Text birthDate|Type:Date birthPlace|Type:Text bitrate|Type:Text blogPost|Type:Text blogPosts|Type:Text bloodSupply|Type:Text boardingGroup|Type:Text boardingPolicy|Type:Text bodyLocation|Type:Text bodyType|Type:Text bookEdition|Type:Text bookFormat|Type:Text bookingAgent|Type:Text bookingTime|Type:Text borrower|Type:Text box|Type:Text branch|Type:Text branchCode|Type:Text branchOf|Type:Text brand|Type:Text breadcrumb|Type:Text breastfeedingWarning|Type:Text broadcastAffiliateOf|Type:Text broadcastChannelId|Type:Text broadcastDisplayName|Type:Text broadcastFrequency|Type:Text broadcastFrequencyValue|Type:Text broadcastOfEvent|Type:Text broadcastServiceTier|Type:Text broadcastSignalModulation|Type:Text broadcastSubChannel|Type:Text broadcastTimezone|Type:Text broadcaster|Type:Text broker|Type:Text browserRequirements|Type:Text busName|Type:Text busNumber|Type:Text businessDays|Type:Text businessFunction|Type:Text buyer|Type:Text byArtist|Type:Text byDay|Type:Text byMonth|Type:Text byMonthDay|Type:Text byMonthWeek|Type:Text callSign|Type:Text calories|Type:Text candidate|Type:Text caption|Type:Text carbohydrateContent|Type:Text cargoVolume|Type:Text carrier|Type:Text carrierRequirements|Type:Text cashBack|Type:Text catalog|Type:Text catalogNumber|Type:Text category|Type:Text causeOf|Type:Text ccRecipient|Type:Text character|Type:Text characterAttribute|Type:Text characterName|Type:Text cheatCode|Type:Text checkinTime|Type:Text checkoutTime|Type:Text chemicalComposition|Type:Text chemicalRole|Type:Text childMaxAge|Type:Text childMinAge|Type:Text childTaxon|Type:Text children|Type:Text cholesterolContent|Type:Text circle|Type:Text citation|Type:Text claimInterpreter|Type:Text claimReviewed|Type:Text clincalPharmacology|Type:Text clinicalPharmacology|Type:Text clipNumber|Type:Text closes|Type:Text coach|Type:Text code|Type:Text codeRepository|Type:Text codeSampleType|Type:Text codeValue|Type:Text codingSystem|Type:Text colleague|Type:Text colleagues|Type:Text collection|Type:Text collectionSize|Type:Text color|Type:Text colorist|Type:Text comment|Type:Text commentCount|Type:Text commentText|Type:Text commentTime|Type:Text competencyRequired|Type:Text competitor|Type:Text composer|Type:Text comprisedOf|Type:Text conditionsOfAccess|Type:Text confirmationNumber|Type:Text connectedTo|Type:Text constrainingProperty|Type:Text contactOption|Type:Text contactPoint|Type:Text contactPoints|Type:Text contactType|Type:Text contactlessPayment|Type:Text containedIn|Type:Text containedInPlace|Type:Text containsPlace|Type:Text containsSeason|Type:Text contentLocation|Type:Text contentRating|Type:Text contentReferenceTime|Type:Text contentSize|Type:Text contentType|Type:Text contentUrl|Type:URL contraindication|Type:Text contributor|Type:Text cookTime|Type:Text cookingMethod|Type:Text copyrightHolder|Type:Text copyrightNotice|Type:Text copyrightYear|Type:Text correction|Type:Text correctionsPolicy|Type:Text costCategory|Type:Text costCurrency|Type:Text costOrigin|Type:Text costPerUnit|Type:Text countriesNotSupported|Type:Text countriesSupported|Type:Text countryOfAssembly|Type:Text countryOfLastProcessing|Type:Text countryOfOrigin|Type:Text course|Type:Text courseCode|Type:Text courseMode|Type:Text coursePrerequisites|Type:Text courseWorkload|Type:Text coverageEndTime|Type:Text coverageStartTime|Type:Text creativeWorkStatus|Type:Text creator|Type:Text credentialCategory|Type:Text creditText|Type:Text creditedTo|Type:Text cssSelector|Type:Text currenciesAccepted|Type:Text currency|Type:Text currentExchangeRate|Type:Text customer|Type:Text customerRemorseReturnFees|Type:Text customerRemorseReturnLabelSource|Type:Text customerRemorseReturnShippingFeesAmount|Type:Text cutoffTime|Type:Text cvdCollectionDate|Type:Date cvdFacilityCounty|Type:Text cvdFacilityId|Type:Text cvdNumBeds|Type:Text cvdNumBedsOcc|Type:Text cvdNumC19Died|Type:Text cvdNumC19HOPats|Type:Text cvdNumC19HospPats|Type:Text cvdNumC19MechVentPats|Type:Text cvdNumC19OFMechVentPats|Type:Text cvdNumC19OverflowPats|Type:Text cvdNumICUBeds|Type:Text cvdNumICUBedsOcc|Type:Text cvdNumTotBeds|Type:Text cvdNumVent|Type:Text cvdNumVentUse|Type:Text dataFeedElement|Type:Text dataset|Type:Text datasetTimeInterval|Type:Text dateCreated|Type:Date dateDeleted|Type:Date dateIssued|Type:Date dateModified|Type:Date datePosted|Type:Date datePublished|Type:Date dateRead|Type:Date dateReceived|Type:Date dateSent|Type:Date dateVehicleFirstRegistered|Type:Date dateline|Type:Text dayOfWeek|Type:Text deathDate|Type:Date deathPlace|Type:Text defaultValue|Type:Text deliveryAddress|Type:Text deliveryLeadTime|Type:Text deliveryMethod|Type:Text deliveryStatus|Type:Text deliveryTime|Type:Text department|Type:Text departureAirport|Type:Text departureBoatTerminal|Type:Text departureBusStop|Type:Text departureGate|Type:Text departurePlatform|Type:Text departureStation|Type:Text departureTerminal|Type:Text departureTime|Type:Text dependencies|Type:Text depth|Type:Text description|Type:Text device|Type:Text diagnosis|Type:Text diagram|Type:Text diet|Type:Text dietFeatures|Type:Text differentialDiagnosis|Type:Text directApply|Type:Text director|Type:Text directors|Type:Text disambiguatingDescription|Type:Text discount|Type:Text discountCode|Type:Text discountCurrency|Type:Text discusses|Type:Text discussionUrl|Type:URL diseasePreventionInfo|Type:Text diseaseSpreadStatistics|Type:Text dissolutionDate|Type:Date distance|Type:Text distinguishingSign|Type:Text distribution|Type:Text diversityPolicy|Type:Text diversityStaffingReport|Type:Text documentation|Type:Text doesNotShip|Type:Text domainIncludes|Type:Text domiciledMortgage|Type:Text doorTime|Type:Text dosageForm|Type:Text doseSchedule|Type:Text doseUnit|Type:Text doseValue|Type:Text downPayment|Type:Text downloadUrl|Type:URL downvoteCount|Type:Text drainsTo|Type:Text driveWheelConfiguration|Type:Text dropoffLocation|Type:Text dropoffTime|Type:Text drug|Type:Text drugClass|Type:Text drugUnit|Type:Text duns|Type:Text duplicateTherapy|Type:Text duration|Type:Text durationOfWarranty|Type:Text duringMedia|Type:Text earlyPrepaymentPenalty|Type:Text editEIDR|Type:Text editor|Type:Text eduQuestionType|Type:Text educationRequirements|Type:Text educationalAlignment|Type:Text educationalCredentialAwarded|Type:Text educationalFramework|Type:Text educationalLevel|Type:Text educationalProgramMode|Type:Text educationalRole|Type:Text educationalUse|Type:Text elevation|Type:Text eligibilityToWorkRequirement|Type:Text eligibleCustomerType|Type:Text eligibleDuration|Type:Text eligibleQuantity|Type:Text eligibleRegion|Type:Text eligibleTransactionVolume|Type:Text email|Type:Email embedUrl|Type:URL embeddedTextCaption|Type:Text emissionsCO2|Type:Text employee|Type:Text employees|Type:Text employerOverview|Type:Text employmentType|Type:Text employmentUnit|Type:Text encodesBioChemEntity|Type:Text encodesCreativeWork|Type:Text encoding|Type:Text encodingFormat|Type:Text encodingType|Type:Text encodings|Type:Text endDate|Type:Date endOffset|Type:Text endTime|Type:Text endorsee|Type:Text endorsers|Type:Text energyEfficiencyScaleMax|Type:Text energyEfficiencyScaleMin|Type:Text engineDisplacement|Type:Text enginePower|Type:Text engineType|Type:Text entertainmentBusiness|Type:Text epidemiology|Type:Text episode|Type:Text episodeNumber|Type:Text episodes|Type:Text equal|Type:Text error|Type:Text estimatedCost|Type:Text estimatedFlightDuration|Type:Text estimatedSalary|Type:Text estimatesRiskOf|Type:Text ethicsPolicy|Type:Text event|Type:Text eventAttendanceMode|Type:Text eventSchedule|Type:Text eventStatus|Type:Text events|Type:Text evidenceLevel|Type:Text evidenceOrigin|Type:Text exampleOfWork|Type:Text exceptDate|Type:Date exchangeRateSpread|Type:Text executableLibraryName|Type:Text exerciseCourse|Type:Text exercisePlan|Type:Text exerciseRelatedDiet|Type:Text exerciseType|Type:Text exifData|Type:Text expectedArrivalFrom|Type:Text expectedArrivalUntil|Type:Text expectedPrognosis|Type:Text expectsAcceptanceOf|Type:Text experienceInPlaceOfEducation|Type:Text experienceRequirements|Type:Text expertConsiderations|Type:Text expires|Type:Text expressedIn|Type:Text familyName|Type:Text fatContent|Type:Text faxNumber|Type:Text featureList|Type:Text feesAndCommissionsSpecification|Type:Text fiberContent|Type:Text fileFormat|Type:Text fileSize|Type:Text financialAidEligible|Type:Text firstAppearance|Type:Text firstPerformance|Type:Text flightDistance|Type:Text flightNumber|Type:Text floorLevel|Type:Text floorLimit|Type:Text floorSize|Type:Text followee|Type:Text follows|Type:Text followup|Type:Text foodEstablishment|Type:Text foodEvent|Type:Text foodWarning|Type:Text founder|Type:Text founders|Type:Text foundingDate|Type:Date foundingLocation|Type:Text free|Type:Text freeShippingThreshold|Type:Text frequency|Type:Text fromLocation|Type:Text fuelCapacity|Type:Text fuelConsumption|Type:Text fuelEfficiency|Type:Text fuelType|Type:Text functionalClass|Type:Text fundedItem|Type:Text funder|Type:Text funding|Type:Text game|Type:Text gameAvailabilityType|Type:Text gameEdition|Type:Text gameItem|Type:Text gameLocation|Type:Text gamePlatform|Type:Text gameServer|Type:Text gameTip|Type:Text gender|Type:Text genre|Type:Text geo|Type:Text geoContains|Type:Text geoCoveredBy|Type:Text geoCovers|Type:Text geoCrosses|Type:Text geoDisjoint|Type:Text geoEquals|Type:Text geoIntersects|Type:Text geoMidpoint|Type:Text geoOverlaps|Type:Text geoRadius|Type:Text geoTouches|Type:Text geoWithin|Type:Text geographicArea|Type:Text gettingTestedInfo|Type:Text givenName|Type:Text globalLocationNumber|Type:Text governmentBenefitsInfo|Type:Text gracePeriod|Type:Text grantee|Type:Text greater|Type:Text greaterOrEqual|Type:Text gtin|Type:Text gtin12|Type:Text gtin13|Type:Text gtin14|Type:Text gtin8|Type:Text guideline|Type:Text guidelineDate|Type:Date guidelineSubject|Type:Text handlingTime|Type:Text hasAdultConsideration|Type:Text hasBioChemEntityPart|Type:Text hasBioPolymerSequence|Type:Text hasBroadcastChannel|Type:Text hasCategoryCode|Type:Text hasCourse|Type:Text hasCourseInstance|Type:Text hasCredential|Type:Text hasDefinedTerm|Type:Text hasDeliveryMethod|Type:Text hasDigitalDocumentPermission|Type:Text hasDriveThroughService|Type:Text hasEnergyConsumptionDetails|Type:Text hasEnergyEfficiencyCategory|Type:Text hasHealthAspect|Type:Text hasMap|Type:Text hasMeasurement|Type:Text hasMenu|Type:Text hasMenuItem|Type:Text hasMenuSection|Type:Text hasMerchantReturnPolicy|Type:Text hasMolecularFunction|Type:Text hasOccupation|Type:Text hasOfferCatalog|Type:Text hasPOS|Type:Text hasPart|Type:Text hasRepresentation|Type:Text hasVariant|Type:Text headline|Type:Text healthCondition|Type:Text healthPlanCoinsuranceOption|Type:Text healthPlanCoinsuranceRate|Type:Text healthPlanCopay|Type:Text healthPlanCopayOption|Type:Text healthPlanCostSharing|Type:Text healthPlanDrugOption|Type:Text healthPlanDrugTier|Type:Text healthPlanId|Type:Text healthPlanMarketingUrl|Type:URL healthPlanNetworkId|Type:Text healthPlanNetworkTier|Type:Text healthPlanPharmacyCategory|Type:Text healthcareReportingData|Type:Text height|Type:Text highPrice|Type:Text hiringOrganization|Type:Text holdingArchive|Type:Text homeLocation|Type:Text homeTeam|Type:Text honorificPrefix|Type:Text honorificSuffix|Type:Text hospitalAffiliation|Type:Text hostingOrganization|Type:Text hoursAvailable|Type:Text howPerformed|Type:Text httpMethod|Type:Text iataCode|Type:Text icaoCode|Type:Text identifier|Type:Text identifyingExam|Type:Text identifyingTest|Type:Text illustrator|Type:Text image|Type:Text imagingTechnique|Type:Text inAlbum|Type:Text inBroadcastLineup|Type:Text inChI|Type:Text inChIKey|Type:Text inCodeSet|Type:Text inDefinedTermSet|Type:Text inLanguage|Type:Text inPlaylist|Type:Text inProductGroupWithID|Type:Text inStoreReturnsOffered|Type:Text inSupportOf|Type:Text incentiveCompensation|Type:Text incentives|Type:Text includedComposition|Type:Text includedDataCatalog|Type:Text includedInDataCatalog|Type:Text includedInHealthInsurancePlan|Type:Text includedRiskFactor|Type:Text includesAttraction|Type:Text includesHealthPlanFormulary|Type:Text includesHealthPlanNetwork|Type:Text includesObject|Type:Text increasesRiskOf|Type:Text industry|Type:Text ineligibleRegion|Type:Text infectiousAgent|Type:Text infectiousAgentClass|Type:Text ingredients|Type:Text inker|Type:Text insertion|Type:Text installUrl|Type:URL instructor|Type:Text instrument|Type:Text intensity|Type:Text interactingDrug|Type:Text interactionCount|Type:Text interactionService|Type:Text interactionStatistic|Type:Text interactionType|Type:Text interactivityType|Type:Text interestRate|Type:Text interpretedAsClaim|Type:Text inventoryLevel|Type:Text inverseOf|Type:Text isAcceptingNewPatients|Type:Text isAccessibleForFree|Type:Text isAccessoryOrSparePartFor|Type:Text isAvailableGenerically|Type:Text isBasedOn|Type:Text isBasedOnUrl|Type:URL isConsumableFor|Type:Text isEncodedByBioChemEntity|Type:Text isFamilyFriendly|Type:Text isGift|Type:Text isInvolvedInBiologicalProcess|Type:Text isLiveBroadcast|Type:Text isLocatedInSubcellularLocation|Type:Text isPartOf|Type:Text isPartOfBioChemEntity|Type:Text isPlanForApartment|Type:Text isProprietary|Type:Text isRelatedTo|Type:Text isResizable|Type:Text isSimilarTo|Type:Text isUnlabelledFallback|Type:Text isVariantOf|Type:Text isbn|Type:Text isicV4|Type:Text iso6523Code|Type:Text isrcCode|Type:Text issn|Type:Text issueNumber|Type:Text issuedBy|Type:Text issuedThrough|Type:Text iswcCode|Type:Text item|Type:Text itemCondition|Type:Text itemDefectReturnFees|Type:Text itemDefectReturnLabelSource|Type:Text itemDefectReturnShippingFeesAmount|Type:Text itemListElement|Type:Text itemListOrder|Type:Text itemLocation|Type:Text itemOffered|Type:Text itemReviewed|Type:Text itemShipped|Type:Text itinerary|Type:Text iupacName|Type:Text jobBenefits|Type:Text jobImmediateStart|Type:Text jobLocation|Type:Text jobLocationType|Type:Text jobStartDate|Type:Date jobTitle|Type:Text jurisdiction|Type:Text keywords|Type:Text knownVehicleDamages|Type:Text knows|Type:Text knowsAbout|Type:Text knowsLanguage|Type:Text labelDetails|Type:Text landlord|Type:Text language|Type:Text lastReviewed|Type:Text latitude|Type:Text layoutImage|Type:Text learningResourceType|Type:Text leaseLength|Type:Text legalName|Type:Text legalStatus|Type:Text legislationApplies|Type:Text legislationChanges|Type:Text legislationConsolidates|Type:Text legislationDate|Type:Date legislationDateVersion|Type:Text legislationIdentifier|Type:Text legislationJurisdiction|Type:Text legislationLegalForce|Type:Text legislationLegalValue|Type:Text legislationPassedBy|Type:Text legislationResponsible|Type:Text legislationTransposes|Type:Text legislationType|Type:Text leiCode|Type:Text lender|Type:Text lesser|Type:Text lesserOrEqual|Type:Text letterer|Type:Text license|Type:Text line|Type:Text 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Nonprofit501d|Category Nonprofit501e|Category Nonprofit501f|Category Nonprofit501k|Category Nonprofit501n|Category Nonprofit501q|Category Nonprofit527|Category NonprofitANBI|Category NonprofitSBBI|Category NonprofitType|Category Nose|Category NotInForce|Category NotYetRecruiting|Category Notary|Category NoteDigitalDocument|Category Number|Category Nursing|Category NutritionInformation|Category OTC|Category Observation|Category Observational|Category Obstetric|Category Occupation|Category OccupationalActivity|Category OccupationalExperienceRequirements|Category OccupationalTherapy|Category OceanBodyOfWater|Category Offer|Category OfferCatalog|Category OfferForLease|Category OfferForPurchase|Category OfferItemCondition|Category OfferShippingDetails|Category OfficeEquipmentStore|Category OfficialLegalValue|Category OfflineEventAttendanceMode|Category OfflinePermanently|Category OfflineTemporarily|Category OnDemandEvent|Category OnSitePickup|Category Oncologic|Category OneTimePayments|Category Online|Category OnlineBusiness|Category OnlineEventAttendanceMode|Category OnlineFull|Category OnlineOnly|Category OnlineStore|Category OpenTrial|Category OpeningHoursSpecification|Category OpinionNewsArticle|Category Optician|Category Optometric|Category Order|Category OrderAction|Category OrderCancelled|Category OrderDelivered|Category OrderInTransit|Category OrderItem|Category OrderPaymentDue|Category OrderPickupAvailable|Category OrderProblem|Category OrderProcessing|Category OrderReturned|Category OrderStatus|Category Organization|Category OrganizationRole|Category OrganizeAction|Category OriginalMediaContent|Category OriginalShippingFees|Category Osteopathic|Category Otolaryngologic|Category OutOfStock|Category OutletStore|Category OverviewHealthAspect|Category OwnershipInfo|Category PET|Category PaidLeave|Category PaintAction|Category Painting|Category PalliativeProcedure|Category Paperback|Category ParcelDelivery|Category ParcelService|Category ParentAudience|Category ParentalSupport|Category Park|Category ParkingFacility|Category ParkingMap|Category PartiallyInForce|Category Pathology|Category PathologyTest|Category Patient|Category PatientExperienceHealthAspect|Category PawnShop|Category PayAction|Category PaymentAutomaticallyApplied|Category PaymentCard|Category PaymentChargeSpecification|Category PaymentComplete|Category PaymentDeclined|Category PaymentDue|Category PaymentMethod|Category PaymentPastDue|Category PaymentService|Category PaymentStatusType|Category Pediatric|Category PeopleAudience|Category PercutaneousProcedure|Category PerformAction|Category PerformanceRole|Category PerformingArtsTheater|Category PerformingGroup|Category Periodical|Category Permit|Category Person|Category PetStore|Category Pharmacy|Category PharmacySpecialty|Category Photograph|Category PhotographAction|Category PhysicalActivity|Category PhysicalActivityCategory|Category PhysicalExam|Category PhysicalTherapy|Category Physician|Category Physiotherapy|Category Place|Category PlaceOfWorship|Category PlaceboControlledTrial|Category PlanAction|Category PlasticSurgery|Category Play|Category PlayAction|Category PlayGameAction|Category Playground|Category Plumber|Category PodcastEpisode|Category PodcastSeason|Category PodcastSeries|Category Podiatric|Category PoliceStation|Category Pond|Category PostOffice|Category PostalAddress|Category PostalCodeRangeSpecification|Category Poster|Category PotentialActionStatus|Category PreOrder|Category PreOrderAction|Category PreSale|Category PregnancyHealthAspect|Category PrependAction|Category Preschool|Category PrescriptionOnly|Category PresentationDigitalDocument|Category PreventionHealthAspect|Category PreventionIndication|Category PriceComponentTypeEnumeration|Category PriceSpecification|Category PriceTypeEnumeration|Category PrimaryCare|Category Prion|Category Product|Category ProductCollection|Category ProductGroup|Category ProductModel|Category ProfessionalService|Category ProfilePage|Category PrognosisHealthAspect|Category ProgramMembership|Category Project|Category PronounceableText|Category Property|Category PropertyValue|Category PropertyValueSpecification|Category Protein|Category Protozoa|Category Psychiatric|Category PsychologicalTreatment|Category PublicHealth|Category PublicHolidays|Category PublicSwimmingPool|Category PublicToilet|Category PublicationEvent|Category PublicationIssue|Category PublicationVolume|Category Pulmonary|Category QAPage|Category QualitativeValue|Category QuantitativeValue|Category QuantitativeValueDistribution|Category Quantity|Category Question|Category Quiz|Category Quotation|Category QuoteAction|Category RVPark|Category RadiationTherapy|Category RadioBroadcastService|Category RadioChannel|Category RadioClip|Category RadioEpisode|Category RadioSeason|Category RadioSeries|Category RadioStation|Category Radiography|Category RandomizedTrial|Category Rating|Category ReactAction|Category ReadAction|Category ReadPermission|Category RealEstateAgent|Category RealEstateListing|Category RearWheelDriveConfiguration|Category ReceiveAction|Category Recipe|Category Recommendation|Category RecommendedDoseSchedule|Category Recruiting|Category RecyclingCenter|Category ReducedRelevanceForChildrenConsideration|Category RefundTypeEnumeration|Category RefurbishedCondition|Category RegisterAction|Category Registry|Category ReimbursementCap|Category RejectAction|Category RelatedTopicsHealthAspect|Category RemixAlbum|Category Renal|Category RentAction|Category RentalCarReservation|Category RentalVehicleUsage|Category RepaymentSpecification|Category ReplaceAction|Category ReplyAction|Category Report|Category ReportageNewsArticle|Category ReportedDoseSchedule|Category ResearchOrganization|Category ResearchProject|Category Researcher|Category Reservation|Category ReservationCancelled|Category ReservationConfirmed|Category ReservationHold|Category ReservationPackage|Category ReservationPending|Category ReservationStatusType|Category ReserveAction|Category Reservoir|Category Residence|Category Resort|Category RespiratoryTherapy|Category Restaurant|Category RestockingFees|Category RestrictedDiet|Category ResultsAvailable|Category ResultsNotAvailable|Category ResumeAction|Category Retail|Category ReturnAction|Category ReturnAtKiosk|Category ReturnByMail|Category ReturnFeesCustomerResponsibility|Category ReturnFeesEnumeration|Category ReturnInStore|Category ReturnLabelCustomerResponsibility|Category ReturnLabelDownloadAndPrint|Category ReturnLabelInBox|Category ReturnLabelSourceEnumeration|Category ReturnMethodEnumeration|Category ReturnShippingFees|Category Review|Category ReviewAction|Category ReviewNewsArticle|Category Rheumatologic|Category RightHandDriving|Category RisksOrComplicationsHealthAspect|Category RiverBodyOfWater|Category Role|Category RoofingContractor|Category Room|Category RsvpAction|Category RsvpResponseMaybe|Category RsvpResponseNo|Category RsvpResponseType|Category RsvpResponseYes|Category SRP|Category SafetyHealthAspect|Category SaleEvent|Category SalePrice|Category SatireOrParodyContent|Category SatiricalArticle|Category Saturday|Category Schedule|Category ScheduleAction|Category ScholarlyArticle|Category School|Category SchoolDistrict|Category ScreeningEvent|Category ScreeningHealthAspect|Category Sculpture|Category SeaBodyOfWater|Category SearchAction|Category SearchRescueOrganization|Category SearchResultsPage|Category Season|Category Seat|Category SeatingMap|Category SeeDoctorHealthAspect|Category SeekToAction|Category SelfCareHealthAspect|Category SelfStorage|Category SellAction|Category SendAction|Category Series|Category Service|Category ServiceChannel|Category SexualContentConsideration|Category ShareAction|Category SheetMusic|Category ShippingDeliveryTime|Category ShippingRateSettings|Category ShoeStore|Category ShoppingCenter|Category ShortStory|Category SideEffectsHealthAspect|Category SingleBlindedTrial|Category SingleCenterTrial|Category SingleFamilyResidence|Category SinglePlayer|Category SingleRelease|Category SiteNavigationElement|Category SizeGroupEnumeration|Category SizeSpecification|Category SizeSystemEnumeration|Category SizeSystemImperial|Category SizeSystemMetric|Category SkiResort|Category Skin|Category SocialEvent|Category SocialMediaPosting|Category SoftwareApplication|Category SoftwareSourceCode|Category SoldOut|Category SolveMathAction|Category SomeProducts|Category SoundtrackAlbum|Category SpeakableSpecification|Category SpecialAnnouncement|Category Specialty|Category SpeechPathology|Category SpokenWordAlbum|Category SportingGoodsStore|Category SportsActivityLocation|Category SportsClub|Category SportsEvent|Category SportsOrganization|Category SportsTeam|Category SpreadsheetDigitalDocument|Category StadiumOrArena|Category StagedContent|Category StagesHealthAspect|Category State|Category Statement|Category StatisticalPopulation|Category StatusEnumeration|Category SteeringPositionValue|Category Store|Category StoreCreditRefund|Category StrengthTraining|Category StructuredValue|Category StudioAlbum|Category SubscribeAction|Category Subscription|Category Substance|Category SubwayStation|Category Suite|Category Sunday|Category SuperficialAnatomy|Category Surgical|Category SurgicalProcedure|Category SuspendAction|Category Suspended|Category SymptomsHealthAspect|Category Synagogue|Category TVClip|Category TVEpisode|Category TVSeason|Category TVSeries|Category Table|Category TakeAction|Category TattooParlor|Category Taxi|Category TaxiReservation|Category TaxiService|Category TaxiStand|Category TaxiVehicleUsage|Category Taxon|Category TechArticle|Category TelevisionChannel|Category TelevisionStation|Category TennisComplex|Category Terminated|Category Text|Category TextDigitalDocument|Category TheaterEvent|Category TheaterGroup|Category Therapeutic|Category TherapeuticProcedure|Category Thesis|Category Thing|Category Throat|Category Thursday|Category Ticket|Category TieAction|Category Time|Category TipAction|Category TireShop|Category TobaccoNicotineConsideration|Category TollFree|Category TouristAttraction|Category TouristDestination|Category TouristInformationCenter|Category TouristTrip|Category Toxicologic|Category ToyStore|Category TrackAction|Category TradeAction|Category TraditionalChinese|Category TrainReservation|Category TrainStation|Category TrainTrip|Category TransferAction|Category TransformedContent|Category TransitMap|Category TravelAction|Category TravelAgency|Category TreatmentIndication|Category TreatmentsHealthAspect|Category Trip|Category TripleBlindedTrial|Category True|Category Tuesday|Category TypeAndQuantityNode|Category TypesHealthAspect|Category UKNonprofitType|Category UKTrust|Category URL|Category USNonprofitType|Category Ultrasound|Category UnRegisterAction|Category UnclassifiedAdultConsideration|Category UnemploymentSupport|Category UnincorporatedAssociationCharity|Category UnitPriceSpecification|Category UnofficialLegalValue|Category UpdateAction|Category Urologic|Category UsageOrScheduleHealthAspect|Category UseAction|Category UsedCondition|Category UserBlocks|Category UserCheckins|Category UserComments|Category UserDownloads|Category UserInteraction|Category UserLikes|Category UserPageVisits|Category UserPlays|Category UserPlusOnes|Category UserReview|Category UserTweets|Category VeganDiet|Category VegetarianDiet|Category Vehicle|Category Vein|Category VenueMap|Category Vessel|Category VeterinaryCare|Category VideoGallery|Category VideoGame|Category VideoGameClip|Category VideoGameSeries|Category VideoObject|Category VideoObjectSnapshot|Category ViewAction|Category VinylFormat|Category ViolenceConsideration|Category VirtualLocation|Category Virus|Category VisualArtsEvent|Category VisualArtwork|Category VitalSign|Category Volcano|Category VoteAction|Category WPAdBlock|Category WPFooter|Category WPHeader|Category WPSideBar|Category WantAction|Category WarrantyPromise|Category WarrantyScope|Category WatchAction|Category Waterfall|Category WeaponConsideration|Category WearAction|Category WearableMeasurementBack|Category WearableMeasurementChestOrBust|Category WearableMeasurementCollar|Category WearableMeasurementCup|Category WearableMeasurementHeight|Category WearableMeasurementHips|Category WearableMeasurementInseam|Category WearableMeasurementLength|Category WearableMeasurementOutsideLeg|Category WearableMeasurementSleeve|Category WearableMeasurementTypeEnumeration|Category WearableMeasurementWaist|Category WearableMeasurementWidth|Category WearableSizeGroupBig|Category WearableSizeGroupBoys|Category WearableSizeGroupEnumeration|Category WearableSizeGroupExtraShort|Category WearableSizeGroupExtraTall|Category WearableSizeGroupGirls|Category WearableSizeGroupHusky|Category WearableSizeGroupInfants|Category WearableSizeGroupJuniors|Category WearableSizeGroupMaternity|Category WearableSizeGroupMens|Category WearableSizeGroupMisses|Category WearableSizeGroupPetite|Category WearableSizeGroupPlus|Category WearableSizeGroupRegular|Category WearableSizeGroupShort|Category WearableSizeGroupTall|Category WearableSizeGroupWomens|Category WearableSizeSystemAU|Category WearableSizeSystemBR|Category WearableSizeSystemCN|Category WearableSizeSystemContinental|Category WearableSizeSystemDE|Category WearableSizeSystemEN13402|Category WearableSizeSystemEnumeration|Category WearableSizeSystemEurope|Category WearableSizeSystemFR|Category WearableSizeSystemGS1|Category WearableSizeSystemIT|Category WearableSizeSystemJP|Category WearableSizeSystemMX|Category WearableSizeSystemUK|Category WearableSizeSystemUS|Category WebAPI|Category WebApplication|Category WebContent|Category WebPage|Category WebPageElement|Category WebSite|Category Wednesday|Category WesternConventional|Category Wholesale|Category WholesaleStore|Category WinAction|Category Winery|Category Withdrawn|Category WorkBasedProgram|Category WorkersUnion|Category WriteAction|Category WritePermission|Category XPathType|Category XRay|Category ZoneBoardingPolicy|Category Zoo|Category [[Category:Imported vocabulary]] 06698762b8bbc41a468ff3ad688308233b1f2543 2 387 804 801 2024-08-06T02:05:36Z Vipul 2 wikitext text/x-wiki * <math>\sqrt{7 + 2}!! + 3 = 723</math> * <math>(7 + 2)^{\sqrt{9}} = 729</math> * <math>2^{8 - 1} = 128</math> 02a5df6b5d9e8e7368dc9c61c5a6873811339b8b 821 804 2024-09-08T05:11:03Z Vipul 2 wikitext text/x-wiki * <math>\sqrt{7 + 2}!! + 3 = 723</math> * <math>(7 + 2)^{\sqrt{9}} = 729</math> * <math>2^{8 - 1} = 128</math> * <math>2^7 - 1 = 127</math> 7f066cad57b0f2863ab5d6db436e76c137178b28 4 390 805 2024-08-06T02:30:26Z Vipul 2 Created page with "Site search autocompletion is currently broken by default on this site. This page includes details on how to get it to work, and what's going on. ==What's wrong with site search autcompletion== When you start typing something in the site search bar, you'll see it stuck at "Loading search suggestions":" wikitext text/x-wiki Site search autocompletion is currently broken by default on this site. This page includes details on how to get it to work, and what's going on. ==What's wrong with site search autcompletion== When you start typing something in the site search bar, you'll see it stuck at "Loading search suggestions": 1071004012c1dee0890c4bcaf91c9a312973a32b 806 805 2024-08-06T02:30:48Z Vipul 2 wikitext text/x-wiki Site search autocompletion is currently broken by default on this site. This page includes details on how to get it to work, and what's going on. ==What's wrong with site search autocompletion== When you start typing something in the site search bar, you'll see it stuck at "Loading search suggestions": 5fcd0b724d062100856a8b323529665ee3966077 809 806 2024-08-06T02:55:05Z Vipul 2 /* What's wrong with site search autocompletion */ wikitext text/x-wiki Site search autocompletion is currently broken by default on this site. This page includes details on how to get it to work, and what's going on. ==What's wrong with site search autocompletion and how to fix it== ===What's wrong=== When you start typing something in the site search bar, you'll see it stuck at "Loading search suggestions" as shown in the screenshot below: [[File:Site search autocompletion broken.png]] Note that the actual search is still working -- you just have to hit Enter after typing the search query and it'll go to the search results page. It's the autocompletion before you hit Enter that is broken. ===How to fix it=== To fix it, you need to follow these steps: * Write to vipulnaik1@gmail.com asking for a login to the site. Please include the following with your request: preferred username, preferred initial password (you can change it after logging in), real name (if you want it entered), email address to use (if you want an actual email address by which you can be contacted), and whether you want edit access as well. You don't need edit access for enabling site search autocompletion. * Log in to the site, and go to [[Special:Preferences]]. Go to the Appearance section and switch the Skin from "Vector (2022)" to "Vector legacy (2010)". * Make sure to hit "Save" at the bottom. * Now you can load or reload. Site search autocompletion should now work. Here's an example: [[File:Site search autocompletion working.png]] ==More background== We've recently upgraded the MediaWiki version of this wiki from 1.35.13 to 1.41.2 (see [[Special:Version]]). With the current setup for MediaWiki 1.41.2, we're in this situation: * The "Vector legacy (2010)" skin has site search autocompletion working, but it doesn't render well on small screens. Specifically, even on small mobile screens, it still shows the left menu, and doesn't properly use the MobileFrontend extension settings. * The "Vector (2022)" skin doesn't have site search autocompletion working (see screenshots in preceding section) but it does render fine on mobile devices. It is possible to set only one default skin (that is applicable to all non-logged-in users and is the default for logged-in users who have not configured a skin for themselves). So, the selection of default skin comes down to whether it's more important for casual users to have the mobile experience working or to have site search autocompletion working. Based on a general understanding of user behavior, we believe that having a usable mobile experience is more important for casual users than having site search autocompletion. However, for power users who are using the site extensively, site search autocompletion may be important. That's why we've written this page giving guidance on how to set up site search autocompletion. fb1931f35ce8b2794719de3f36d06664b98e80a2 813 809 2024-08-06T05:54:30Z Vipul 2 /* More background */ wikitext text/x-wiki Site search autocompletion is currently broken by default on this site. This page includes details on how to get it to work, and what's going on. ==What's wrong with site search autocompletion and how to fix it== ===What's wrong=== When you start typing something in the site search bar, you'll see it stuck at "Loading search suggestions" as shown in the screenshot below: [[File:Site search autocompletion broken.png]] Note that the actual search is still working -- you just have to hit Enter after typing the search query and it'll go to the search results page. It's the autocompletion before you hit Enter that is broken. ===How to fix it=== To fix it, you need to follow these steps: * Write to vipulnaik1@gmail.com asking for a login to the site. Please include the following with your request: preferred username, preferred initial password (you can change it after logging in), real name (if you want it entered), email address to use (if you want an actual email address by which you can be contacted), and whether you want edit access as well. You don't need edit access for enabling site search autocompletion. * Log in to the site, and go to [[Special:Preferences]]. Go to the Appearance section and switch the Skin from "Vector (2022)" to "Vector legacy (2010)". * Make sure to hit "Save" at the bottom. * Now you can load or reload. Site search autocompletion should now work. Here's an example: [[File:Site search autocompletion working.png]] ==More background== We've recently upgraded the MediaWiki version of this wiki from 1.35.13 to 1.41.2 (see [[Special:Version]]). The upgrade allows us to migrate the wiki to a more modern operating system version running PHP 8. With the current setup for MediaWiki 1.41.2, we're in this situation: * The "Vector legacy (2010)" skin has site search autocompletion working, but it doesn't render well on small screens. Specifically, even on small mobile screens, it still shows the left menu, and doesn't properly use the MobileFrontend extension settings. * The "Vector (2022)" skin doesn't have site search autocompletion working (see screenshots in preceding section) but it does render fine on mobile devices. It is possible to set only one default skin (that is applicable to all non-logged-in users and is the default for logged-in users who have not configured a skin for themselves). So, the selection of default skin comes down to whether it's more important for casual users to have the mobile experience working or to have site search autocompletion working. Based on a general understanding of user behavior, we believe that having a usable mobile experience is more important for casual users than having site search autocompletion. However, for power users who are using the site extensively, site search autocompletion may be important. That's why we've written this page giving guidance on how to set up site search autocompletion. 9db4c816b1fcd4f285f0d0b7afe4b058033fbaf8 815 813 2024-08-06T16:20:41Z Vipul 2 /* How to fix it */ wikitext text/x-wiki Site search autocompletion is currently broken by default on this site. This page includes details on how to get it to work, and what's going on. ==What's wrong with site search autocompletion and how to fix it== ===What's wrong=== When you start typing something in the site search bar, you'll see it stuck at "Loading search suggestions" as shown in the screenshot below: [[File:Site search autocompletion broken.png]] Note that the actual search is still working -- you just have to hit Enter after typing the search query and it'll go to the search results page. It's the autocompletion before you hit Enter that is broken. ===How to fix it=== To fix it, you need to follow these steps: * Write to vipulnaik1@gmail.com asking for a login to the site. Please include the following with your request: preferred username, preferred initial password (you can change it after logging in), real name (if you want it entered), email address to use (if you want an actual email address by which you can be contacted), and whether you want edit access as well. You don't need edit access for enabling site search autocompletion. * Log in to the site, and go to [[Special:Preferences]]. Go to the Appearance section and switch the Skin from "Vector (2022)" to "Vector legacy (2010)". * Make sure to hit "Save" at the bottom. * Now you can reload the page or load a new page. Site search autocompletion should now work. Here's an example: [[File:Site search autocompletion working.png]] ==More background== We've recently upgraded the MediaWiki version of this wiki from 1.35.13 to 1.41.2 (see [[Special:Version]]). The upgrade allows us to migrate the wiki to a more modern operating system version running PHP 8. With the current setup for MediaWiki 1.41.2, we're in this situation: * The "Vector legacy (2010)" skin has site search autocompletion working, but it doesn't render well on small screens. Specifically, even on small mobile screens, it still shows the left menu, and doesn't properly use the MobileFrontend extension settings. * The "Vector (2022)" skin doesn't have site search autocompletion working (see screenshots in preceding section) but it does render fine on mobile devices. It is possible to set only one default skin (that is applicable to all non-logged-in users and is the default for logged-in users who have not configured a skin for themselves). So, the selection of default skin comes down to whether it's more important for casual users to have the mobile experience working or to have site search autocompletion working. Based on a general understanding of user behavior, we believe that having a usable mobile experience is more important for casual users than having site search autocompletion. However, for power users who are using the site extensively, site search autocompletion may be important. That's why we've written this page giving guidance on how to set up site search autocompletion. d565ecd7a606a31b88c7b58ed6d3af9de5c7e547 819 815 2024-08-07T21:30:29Z Vipul 2 /* How to fix it */ wikitext text/x-wiki Site search autocompletion is currently broken by default on this site. This page includes details on how to get it to work, and what's going on. ==What's wrong with site search autocompletion and how to fix it== ===What's wrong=== When you start typing something in the site search bar, you'll see it stuck at "Loading search suggestions" as shown in the screenshot below: [[File:Site search autocompletion broken.png]] Note that the actual search is still working -- you just have to hit Enter after typing the search query and it'll go to the search results page. It's the autocompletion before you hit Enter that is broken. ===How to fix it=== To fix it, you need to follow these steps: * Write to vipulnaik1@gmail.com asking for a login to the site. Please include the following with your request: preferred username, preferred initial password (you can change it after logging in), real name (if you want it entered), email address to use (if you want an actual email address by which you can be contacted), and whether you want edit access as well. You don't need edit access for enabling site search autocompletion. * Log in to the site. Then go to [[Special:Preferences]]. Go to the Appearance section and switch the Skin from "Vector (2022)" to "Vector legacy (2010)". * Make sure to hit "Save" at the bottom. * Now you can reload the page or load a new page. Site search autocompletion should now work. Here's an example: [[File:Site search autocompletion working.png]] ==More background== We've recently upgraded the MediaWiki version of this wiki from 1.35.13 to 1.41.2 (see [[Special:Version]]). The upgrade allows us to migrate the wiki to a more modern operating system version running PHP 8. With the current setup for MediaWiki 1.41.2, we're in this situation: * The "Vector legacy (2010)" skin has site search autocompletion working, but it doesn't render well on small screens. Specifically, even on small mobile screens, it still shows the left menu, and doesn't properly use the MobileFrontend extension settings. * The "Vector (2022)" skin doesn't have site search autocompletion working (see screenshots in preceding section) but it does render fine on mobile devices. It is possible to set only one default skin (that is applicable to all non-logged-in users and is the default for logged-in users who have not configured a skin for themselves). So, the selection of default skin comes down to whether it's more important for casual users to have the mobile experience working or to have site search autocompletion working. Based on a general understanding of user behavior, we believe that having a usable mobile experience is more important for casual users than having site search autocompletion. However, for power users who are using the site extensively, site search autocompletion may be important. That's why we've written this page giving guidance on how to set up site search autocompletion. b1f2dbaee25a75c4f6e6c4a9106e1ad7e1459afa 6 391 807 2024-08-06T02:36:53Z Vipul 2 wikitext text/x-wiki da39a3ee5e6b4b0d3255bfef95601890afd80709 6 392 808 2024-08-06T02:42:46Z Vipul 2 wikitext text/x-wiki da39a3ee5e6b4b0d3255bfef95601890afd80709 4 393 810 2024-08-06T04:57:28Z Vipul 2 Created page with "If you get a 429 Too Many Requests error when browsing this site: You're probably seeing this error because a large number of requests have been made from your IP address over a short period of time. That's probably a lot of requests from you or others who share your IP address (such as your home wi-fi network). Waiting a minute and then retrying should generally work. If you are an actual human being with a legitimate reason to be browsing the site heavily, first, tha..." wikitext text/x-wiki If you get a 429 Too Many Requests error when browsing this site: You're probably seeing this error because a large number of requests have been made from your IP address over a short period of time. That's probably a lot of requests from you or others who share your IP address (such as your home wi-fi network). Waiting a minute and then retrying should generally work. If you are an actual human being with a legitimate reason to be browsing the site heavily, first, thank you and sorry about this! We set rate limits to prevent bots, spiders, spammers, and malicious actors from consuming too much of our server's resources so that our server's resources can be devoted to real humans like you. Consider writing to vipulnaik1@gmail.com with your IP address to have the IP address whitelisted. You can get your IP address by [https://www.google.com/search?q=my+ip+address Googling "my IP address"] (scroll down a little bit to where Google includes the IP address in a box). 040a253b19d7e8957ab91006b063a259b315affc 814 810 2024-08-06T15:58:33Z Vipul 2 wikitext text/x-wiki If you get a 429 Too Many Requests error when browsing this site: You're probably seeing this error because a large number of requests have been made from your IP address over a short period of time. That's probably a lot of requests from you or others who share your IP address (such as your home wi-fi network). Waiting a minute and then retrying should generally work. If you are an actual human being with a legitimate reason to be browsing the site heavily, first, thank you and sorry about this! We set rate limits to prevent bots, spiders, spammers, and malicious actors from consuming too much of our server's resources so that our server's resources can be devoted to real humans like you. Consider writing to vipulnaik1@gmail.com with your IP address to have the IP address whitelisted. You can get your IP address by [https://www.google.com/search?q=my+ip+address Googling "my IP address"] (scroll down a little bit to where Google includes the IP address in a box). NOTE: If you have both an IPv4 address and an IPv6 address, you may need to send both; the server uses IPv6 if your client has both addresses. To check if you have an IPv6 address, try visiting [https://ipv6.google.com/ ipv6.google.com]. af9c4beff8d0e3ed014715174fb05de4310eb7ae 816 814 2024-08-06T18:36:09Z Vipul 2 wikitext text/x-wiki If you get a 429 Too Many Requests error when browsing this site: You're probably seeing this error because a large number of requests have been made from your IP address over a short period of time. That's probably a lot of requests from you or others who share your IP address (such as your home wi-fi network). Waiting a minute and then retrying should generally work. If you are an actual human being with a legitimate reason to be browsing the site heavily, first, thank you and sorry about this! We set rate limits to prevent bots, spiders, spammers, and malicious actors from consuming too much of our server's resources so that our server's resources can be devoted to real humans like you. Consider writing to vipulnaik1@gmail.com with your IP address to have the IP address whitelisted. You can get your IP address by [https://www.google.com/search?q=my+ip+address Googling "my IP address"] (scroll down a little bit to where Google includes the IP address in a box). NOTE: If you have both an IPv4 address and an IPv6 address, you may need to send both; the server uses IPv6 if your client has both addresses. To check if you have an IPv6 address, try visiting [https://ipv6.google.com/ ipv6.google.com]. If your IP address changes, or you are away from your home network, then you'll get rate-limited again. So if you find yourself getting rate-limited after already having been whitelisted, check if you are on a different IP address than the one for which you requested whitelisting. 6411fa016532757f256951c6fd355476fc0f3bad 817 816 2024-08-06T18:36:44Z Vipul 2 wikitext text/x-wiki If you get a 429 Too Many Requests error when browsing this site, read on. You're probably seeing this error because a large number of requests have been made from your IP address over a short period of time. That's probably a lot of requests from you or others who share your IP address (such as your home wi-fi network). Waiting a minute and then retrying should generally work. If you are an actual human being with a legitimate reason to be browsing the site heavily, first, thank you and sorry about this! We set rate limits to prevent bots, spiders, spammers, and malicious actors from consuming too much of our server's resources so that our server's resources can be devoted to real humans like you. Consider writing to vipulnaik1@gmail.com with your IP address to have the IP address whitelisted. You can get your IP address by [https://www.google.com/search?q=my+ip+address Googling "my IP address"] (scroll down a little bit to where Google includes the IP address in a box). NOTE: If you have both an IPv4 address and an IPv6 address, you may need to send both; the server uses IPv6 if your client has both addresses. To check if you have an IPv6 address, try visiting [https://ipv6.google.com/ ipv6.google.com]. If your IP address changes, or you are away from your home network, then you'll get rate-limited again. So if you find yourself getting rate-limited after already having been whitelisted, check if you are on a different IP address than the one for which you requested whitelisting. 921e98cd8f6c85fd6fbbf0522f7f12eeaed9f1b3 818 817 2024-08-07T20:34:12Z Vipul 2 wikitext text/x-wiki If you get a 429 Too Many Requests error when browsing this site, read on. You're probably seeing this error because a large number of requests have been made from your IP address over a short period of time. That's probably a lot of requests from you or others who share your IP address (such as your home wi-fi network). Waiting a minute and then retrying should generally work. If you are an actual human being with a legitimate reason to be browsing the site heavily, first, thank you and sorry about this! We set rate limits to prevent bots, spiders, spammers, and malicious actors from consuming too much of our server's resources so that our server's resources can be devoted to real humans like you. Consider writing to vipulnaik1@gmail.com with your IP address to have the IP address whitelisted. You can get your IP address by [https://www.google.com/search?q=my+ip+address Googling "my IP address"] (scroll down a little bit to where Google includes the IP address in a box). NOTE: If you have both an IPv4 address and an IPv6 address, you should send both; the server supports both IPv4 and IPv6, so either may end up getting used. To check if you have an IPv6 address, try visiting [https://ipv6.google.com/ ipv6.google.com]. If your IP address changes, or you are away from your home network, then you'll get rate-limited again. So if you find yourself getting rate-limited after already having been whitelisted, check if you are on a different IP address than the one for which you requested whitelisting. 17dc0d1bad9c530e496fefe0a3cb51ade4dfc149 8 21 811 616 2024-08-06T04:58:53Z Vipul 2 wikitext text/x-wiki Want site search autocompletion? See [[Project:Enabling site search autocompletion|Here]]<br/> Encountering 429 Too Many Requests errors when browsing the site? See [[Project:429 Too Many Requests error|here]] cefda7e9a33ce259cfeb6db0f3fe7bb65f12624b 812 811 2024-08-06T04:59:04Z Vipul 2 wikitext text/x-wiki Want site search autocompletion? See [[Project:Enabling site search autocompletion|here]]<br/> Encountering 429 Too Many Requests errors when browsing the site? See [[Project:429 Too Many Requests error|here]] 24cb3da16a82785b8778e57405b489378dc4168f 820 812 2024-09-06T17:52:36Z Vipul 2 wikitext text/x-wiki '''This site is in the process of being migrated to a new server. Edits made until this notice has been removed may be lost.'''<br/> Want site search autocompletion? See [[Project:Enabling site search autocompletion|here]]<br/> Encountering 429 Too Many Requests errors when browsing the site? See [[Project:429 Too Many Requests error|here]] f662711002606fba12e7f2de72f480bc8da64694 822 820 2024-09-08T05:12:01Z Vipul 2 wikitext text/x-wiki Want site search autocompletion? See [[Project:Enabling site search autocompletion|here]]<br/> Encountering 429 Too Many Requests errors when browsing the site? See [[Project:429 Too Many Requests error|here]] 24cb3da16a82785b8778e57405b489378dc4168f